static char help[] = "Landau Damping test using Vlasov-Poisson equations\n";

/*
  To run the code with particles sinusoidally perturbed in x space use the test "pp_poisson_bsi_1d_4" or "pp_poisson_bsi_2d_4"
  According to Lukas, good damping results come at ~16k particles per cell

  To visualize the efield use

    -monitor_efield

  To visualize the swarm distribution use

    -ts_monitor_hg_swarm

  To visualize the particles, we can use

    -ts_monitor_sp_swarm -ts_monitor_sp_swarm_retain 0 -ts_monitor_sp_swarm_phase 1 -draw_size 500,500

*/
#include <petscts.h>
#include <petscdmplex.h>
#include <petscdmswarm.h>
#include <petscfe.h>
#include <petscds.h>
#include <petscbag.h>
#include <petscdraw.h>
#include <petsc/private/dmpleximpl.h>  /* For norm and dot */
#include <petsc/private/petscfeimpl.h> /* For interpolation */
#include "petscdm.h"
#include "petscdmlabel.h"

const char *EMTypes[] = {"primal", "mixed", "coulomb", "none", "EMType", "EM_", NULL};
typedef enum {
  EM_PRIMAL,
  EM_MIXED,
  EM_COULOMB,
  EM_NONE
} EMType;

typedef enum {
  V0,
  X0,
  T0,
  M0,
  Q0,
  PHI0,
  POISSON,
  VLASOV,
  SIGMA,
  NUM_CONSTANTS
} ConstantType;
typedef struct {
  PetscScalar v0; /* Velocity scale, often the thermal velocity */
  PetscScalar t0; /* Time scale */
  PetscScalar x0; /* Space scale */
  PetscScalar m0; /* Mass scale */
  PetscScalar q0; /* Charge scale */
  PetscScalar kb;
  PetscScalar epsi0;
  PetscScalar phi0;          /* Potential scale */
  PetscScalar poissonNumber; /* Non-Dimensional Poisson Number */
  PetscScalar vlasovNumber;  /* Non-Dimensional Vlasov Number */
  PetscReal   sigma;         /* Nondimensional charge per length in x */
} Parameter;

typedef struct {
  PetscBag    bag;            /* Problem parameters */
  PetscBool   error;          /* Flag for printing the error */
  PetscBool   efield_monitor; /* Flag to show electric field monitor */
  PetscBool   initial_monitor;
  PetscBool   periodic;          /* Use periodic boundaries */
  PetscBool   fake_1D;           /* Run simulation in 2D but zeroing second dimension */
  PetscBool   perturbed_weights; /* Uniformly sample x,v space with gaussian weights */
  PetscBool   poisson_monitor;
  PetscInt    ostep; /* print the energy at each ostep time steps */
  PetscInt    numParticles;
  PetscReal   timeScale;              /* Nondimensionalizing time scale */
  PetscReal   charges[2];             /* The charges of each species */
  PetscReal   masses[2];              /* The masses of each species */
  PetscReal   thermal_energy[2];      /* Thermal Energy (used to get other constants)*/
  PetscReal   cosine_coefficients[2]; /*(alpha, k)*/
  PetscReal   totalWeight;
  PetscReal   stepSize;
  PetscInt    steps;
  PetscReal   initVel;
  EMType      em; /* Type of electrostatic model */
  SNES        snes;
  PetscDraw   drawef;
  PetscDrawLG drawlg_ef;
  PetscDraw   drawic_x;
  PetscDraw   drawic_v;
  PetscDraw   drawic_w;
  PetscDrawHG drawhgic_x;
  PetscDrawHG drawhgic_v;
  PetscDrawHG drawhgic_w;
  PetscDraw   EDraw;
  PetscDraw   RhoDraw;
  PetscDraw   PotDraw;
  PetscDrawSP EDrawSP;
  PetscDrawSP RhoDrawSP;
  PetscDrawSP PotDrawSP;
  PetscBool   monitor_positions; /* Flag to show particle positins at each time step */
  PetscDraw   positionDraw;
  PetscDrawSP positionDrawSP;

} AppCtx;

static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
{
  PetscFunctionBeginUser;
  PetscInt d                      = 2;
  options->error                  = PETSC_FALSE;
  options->efield_monitor         = PETSC_FALSE;
  options->initial_monitor        = PETSC_FALSE;
  options->periodic               = PETSC_FALSE;
  options->fake_1D                = PETSC_FALSE;
  options->perturbed_weights      = PETSC_FALSE;
  options->poisson_monitor        = PETSC_FALSE;
  options->ostep                  = 100;
  options->timeScale              = 2.0e-14;
  options->charges[0]             = -1.0;
  options->charges[1]             = 1.0;
  options->masses[0]              = 1.0;
  options->masses[1]              = 1000.0;
  options->thermal_energy[0]      = 1.0;
  options->thermal_energy[1]      = 1.0;
  options->cosine_coefficients[0] = 0.01;
  options->cosine_coefficients[1] = 0.5;
  options->initVel                = 1;
  options->totalWeight            = 1.0;
  options->drawef                 = NULL;
  options->drawlg_ef              = NULL;
  options->drawic_x               = NULL;
  options->drawic_v               = NULL;
  options->drawic_w               = NULL;
  options->drawhgic_x             = NULL;
  options->drawhgic_v             = NULL;
  options->drawhgic_w             = NULL;
  options->EDraw                  = NULL;
  options->RhoDraw                = NULL;
  options->PotDraw                = NULL;
  options->EDrawSP                = NULL;
  options->RhoDrawSP              = NULL;
  options->PotDrawSP              = NULL;
  options->em                     = EM_COULOMB;
  options->numParticles           = 32768;
  options->monitor_positions      = PETSC_FALSE;
  options->positionDraw           = NULL;
  options->positionDrawSP         = NULL;

  PetscOptionsBegin(comm, "", "Central Orbit Options", "DMSWARM");
  PetscCall(PetscOptionsBool("-error", "Flag to print the error", "ex9.c", options->error, &options->error, NULL));
  PetscCall(PetscOptionsBool("-monitor_efield", "Flag to show efield plot", "ex9.c", options->efield_monitor, &options->efield_monitor, NULL));
  PetscCall(PetscOptionsBool("-monitor_ics", "Flag to show initial condition histograms", "ex9.c", options->initial_monitor, &options->initial_monitor, NULL));
  PetscCall(PetscOptionsBool("-monitor_positions", "The flag to show particle positions", "ex9.c", options->monitor_positions, &options->monitor_positions, NULL));
  PetscCall(PetscOptionsBool("-monitor_poisson", "The flag to show charges, Efield and potential solve", "ex9.c", options->poisson_monitor, &options->poisson_monitor, NULL));
  PetscCall(PetscOptionsBool("-periodic", "Flag to use periodic particle boundaries", "ex9.c", options->periodic, &options->periodic, NULL));
  PetscCall(PetscOptionsBool("-fake_1D", "Flag to run a 1D simulation (but really in 2D)", "ex9.c", options->fake_1D, &options->fake_1D, NULL));
  PetscCall(PetscOptionsBool("-perturbed_weights", "Flag to run uniform sampling with perturbed weights", "ex9.c", options->perturbed_weights, &options->perturbed_weights, NULL));
  PetscCall(PetscOptionsInt("-output_step", "Number of time steps between output", "ex9.c", options->ostep, &options->ostep, NULL));
  PetscCall(PetscOptionsReal("-timeScale", "Nondimensionalizing time scale", "ex9.c", options->timeScale, &options->timeScale, NULL));
  PetscCall(PetscOptionsReal("-initial_velocity", "Initial velocity of perturbed particle", "ex9.c", options->initVel, &options->initVel, NULL));
  PetscCall(PetscOptionsReal("-total_weight", "Total weight of all particles", "ex9.c", options->totalWeight, &options->totalWeight, NULL));
  PetscCall(PetscOptionsRealArray("-cosine_coefficients", "Amplitude and frequency of cosine equation used in initialization", "ex9.c", options->cosine_coefficients, &d, NULL));
  PetscCall(PetscOptionsRealArray("-charges", "Species charges", "ex9.c", options->charges, &d, NULL));
  PetscCall(PetscOptionsEnum("-em_type", "Type of electrostatic solver", "ex9.c", EMTypes, (PetscEnum)options->em, (PetscEnum *)&options->em, NULL));
  PetscOptionsEnd();

  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SetupContext(DM dm, DM sw, AppCtx *user)
{
  PetscFunctionBeginUser;
  if (user->efield_monitor) {
    PetscDrawAxis axis_ef;
    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "monitor_efield", 0, 300, 400, 300, &user->drawef));
    PetscCall(PetscDrawSetSave(user->drawef, "ex9_Efield.png"));
    PetscCall(PetscDrawSetFromOptions(user->drawef));
    PetscCall(PetscDrawLGCreate(user->drawef, 1, &user->drawlg_ef));
    PetscCall(PetscDrawLGGetAxis(user->drawlg_ef, &axis_ef));
    PetscCall(PetscDrawAxisSetLabels(axis_ef, "Electron Electric Field", "time", "E_max"));
    PetscCall(PetscDrawLGSetLimits(user->drawlg_ef, 0., user->steps * user->stepSize, -10., 0.));
    PetscCall(PetscDrawAxisSetLimits(axis_ef, 0., user->steps * user->stepSize, -10., 0.));
  }
  if (user->initial_monitor) {
    PetscDrawAxis axis1, axis2, axis3;
    PetscReal     dmboxlower[2], dmboxupper[2];
    PetscInt      dim, cStart, cEnd;
    PetscCall(DMGetDimension(sw, &dim));
    PetscCall(DMGetBoundingBox(dm, dmboxlower, dmboxupper));
    PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "monitor_initial_conditions_x", 0, 300, 400, 300, &user->drawic_x));
    PetscCall(PetscDrawSetSave(user->drawic_x, "ex9_ic_x.png"));
    PetscCall(PetscDrawSetFromOptions(user->drawic_x));
    PetscCall(PetscDrawHGCreate(user->drawic_x, dim, &user->drawhgic_x));
    PetscCall(PetscDrawHGGetAxis(user->drawhgic_x, &axis1));
    PetscCall(PetscDrawHGSetNumberBins(user->drawhgic_x, cEnd - cStart));
    PetscCall(PetscDrawAxisSetLabels(axis1, "Initial X Distribution", "X", "counts"));
    PetscCall(PetscDrawAxisSetLimits(axis1, dmboxlower[0], dmboxupper[0], 0, 1500));

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "monitor_initial_conditions_v", 400, 300, 400, 300, &user->drawic_v));
    PetscCall(PetscDrawSetSave(user->drawic_v, "ex9_ic_v.png"));
    PetscCall(PetscDrawSetFromOptions(user->drawic_v));
    PetscCall(PetscDrawHGCreate(user->drawic_v, dim, &user->drawhgic_v));
    PetscCall(PetscDrawHGGetAxis(user->drawhgic_v, &axis2));
    PetscCall(PetscDrawHGSetNumberBins(user->drawhgic_v, 1000));
    PetscCall(PetscDrawAxisSetLabels(axis2, "Initial V_x Distribution", "V", "counts"));
    PetscCall(PetscDrawAxisSetLimits(axis2, -1, 1, 0, 1500));

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "monitor_initial_conditions_w", 800, 300, 400, 300, &user->drawic_w));
    PetscCall(PetscDrawSetSave(user->drawic_w, "ex9_ic_w.png"));
    PetscCall(PetscDrawSetFromOptions(user->drawic_w));
    PetscCall(PetscDrawHGCreate(user->drawic_w, dim, &user->drawhgic_w));
    PetscCall(PetscDrawHGGetAxis(user->drawhgic_w, &axis3));
    PetscCall(PetscDrawHGSetNumberBins(user->drawhgic_w, 10));
    PetscCall(PetscDrawAxisSetLabels(axis3, "Initial W Distribution", "weight", "counts"));
    PetscCall(PetscDrawAxisSetLimits(axis3, 0, 0.01, 0, 5000));
  }
  if (user->monitor_positions) {
    PetscDrawAxis axis;

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "position_monitor_species1", 0, 0, 400, 300, &user->positionDraw));
    PetscCall(PetscDrawSetFromOptions(user->positionDraw));
    PetscCall(PetscDrawSPCreate(user->positionDraw, 10, &user->positionDrawSP));
    PetscCall(PetscDrawSPSetDimension(user->positionDrawSP, 1));
    PetscCall(PetscDrawSPGetAxis(user->positionDrawSP, &axis));
    PetscCall(PetscDrawSPReset(user->positionDrawSP));
    PetscCall(PetscDrawAxisSetLabels(axis, "Particles", "x", "v"));
    PetscCall(PetscDrawSetSave(user->positionDraw, "ex9_pos.png"));
  }
  if (user->poisson_monitor) {
    PetscDrawAxis axis_E, axis_Rho, axis_Pot;

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "Efield_monitor", 0, 0, 400, 300, &user->EDraw));
    PetscCall(PetscDrawSetFromOptions(user->EDraw));
    PetscCall(PetscDrawSPCreate(user->EDraw, 10, &user->EDrawSP));
    PetscCall(PetscDrawSPSetDimension(user->EDrawSP, 1));
    PetscCall(PetscDrawSPGetAxis(user->EDrawSP, &axis_E));
    PetscCall(PetscDrawSPReset(user->EDrawSP));
    PetscCall(PetscDrawAxisSetLabels(axis_E, "Particles", "x", "E"));
    PetscCall(PetscDrawSetSave(user->EDraw, "ex9_E_spatial.png"));

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "rho_monitor", 0, 0, 400, 300, &user->RhoDraw));
    PetscCall(PetscDrawSetFromOptions(user->RhoDraw));
    PetscCall(PetscDrawSPCreate(user->RhoDraw, 10, &user->RhoDrawSP));
    PetscCall(PetscDrawSPSetDimension(user->RhoDrawSP, 1));
    PetscCall(PetscDrawSPGetAxis(user->RhoDrawSP, &axis_Rho));
    PetscCall(PetscDrawSPReset(user->RhoDrawSP));
    PetscCall(PetscDrawAxisSetLabels(axis_Rho, "Particles", "x", "rho"));
    PetscCall(PetscDrawSetSave(user->RhoDraw, "ex9_rho_spatial.png"));

    PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "potential_monitor", 0, 0, 400, 300, &user->PotDraw));
    PetscCall(PetscDrawSetFromOptions(user->PotDraw));
    PetscCall(PetscDrawSPCreate(user->PotDraw, 10, &user->PotDrawSP));
    PetscCall(PetscDrawSPSetDimension(user->PotDrawSP, 1));
    PetscCall(PetscDrawSPGetAxis(user->PotDrawSP, &axis_Pot));
    PetscCall(PetscDrawSPReset(user->PotDrawSP));
    PetscCall(PetscDrawAxisSetLabels(axis_Pot, "Particles", "x", "potential"));
    PetscCall(PetscDrawSetSave(user->PotDraw, "ex9_phi_spatial.png"));
  }

  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DestroyContext(AppCtx *user)
{
  PetscFunctionBeginUser;
  PetscCall(PetscDrawLGDestroy(&user->drawlg_ef));
  PetscCall(PetscDrawDestroy(&user->drawef));
  PetscCall(PetscDrawHGDestroy(&user->drawhgic_x));
  PetscCall(PetscDrawDestroy(&user->drawic_x));
  PetscCall(PetscDrawHGDestroy(&user->drawhgic_v));
  PetscCall(PetscDrawDestroy(&user->drawic_v));
  PetscCall(PetscDrawHGDestroy(&user->drawhgic_w));
  PetscCall(PetscDrawDestroy(&user->drawic_w));
  PetscCall(PetscDrawSPDestroy(&user->positionDrawSP));
  PetscCall(PetscDrawDestroy(&user->positionDraw));

  PetscCall(PetscDrawSPDestroy(&user->EDrawSP));
  PetscCall(PetscDrawDestroy(&user->EDraw));
  PetscCall(PetscDrawSPDestroy(&user->RhoDrawSP));
  PetscCall(PetscDrawDestroy(&user->RhoDraw));
  PetscCall(PetscDrawSPDestroy(&user->PotDrawSP));
  PetscCall(PetscDrawDestroy(&user->PotDraw));

  PetscCall(PetscBagDestroy(&user->bag));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode computeParticleMoments(DM sw, PetscReal moments[3], AppCtx *user)
{
  DM                 dm;
  const PetscReal   *coords;
  const PetscScalar *w;
  PetscReal          mom[3] = {0.0, 0.0, 0.0};
  PetscInt           cell, cStart, cEnd, dim;

  PetscFunctionBeginUser;
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
  PetscCall(DMSwarmSortGetAccess(sw));
  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&w));
  for (cell = cStart; cell < cEnd; ++cell) {
    PetscInt *pidx;
    PetscInt  Np, p, d;

    PetscCall(DMSwarmSortGetPointsPerCell(sw, cell, &Np, &pidx));
    for (p = 0; p < Np; ++p) {
      const PetscInt   idx = pidx[p];
      const PetscReal *c   = &coords[idx * dim];

      mom[0] += PetscRealPart(w[idx]);
      mom[1] += PetscRealPart(w[idx]) * c[0];
      for (d = 0; d < dim; ++d) mom[2] += PetscRealPart(w[idx]) * c[d] * c[d];
    }
    PetscCall(PetscFree(pidx));
  }
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&w));
  PetscCall(DMSwarmSortRestoreAccess(sw));
  PetscCall(MPIU_Allreduce(mom, moments, 3, MPIU_REAL, MPI_SUM, PetscObjectComm((PetscObject)sw)));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static void f0_1(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  f0[0] = u[0];
}

static void f0_x(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  f0[0] = x[0] * u[0];
}

static void f0_r2(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  PetscInt d;

  f0[0] = 0.0;
  for (d = 0; d < dim; ++d) f0[0] += PetscSqr(x[d]) * u[0];
}

static PetscErrorCode computeFEMMoments(DM dm, Vec u, PetscReal moments[3], AppCtx *user)
{
  PetscDS     prob;
  PetscScalar mom;
  PetscInt    field = 0;

  PetscFunctionBeginUser;
  PetscCall(DMGetDS(dm, &prob));
  PetscCall(PetscDSSetObjective(prob, field, &f0_1));
  PetscCall(DMPlexComputeIntegralFEM(dm, u, &mom, user));
  moments[0] = PetscRealPart(mom);
  PetscCall(PetscDSSetObjective(prob, field, &f0_x));
  PetscCall(DMPlexComputeIntegralFEM(dm, u, &mom, user));
  moments[1] = PetscRealPart(mom);
  PetscCall(PetscDSSetObjective(prob, field, &f0_r2));
  PetscCall(DMPlexComputeIntegralFEM(dm, u, &mom, user));
  moments[2] = PetscRealPart(mom);
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode MonitorEField(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx)
{
  AppCtx    *user = (AppCtx *)ctx;
  DM         dm, sw;
  PetscReal *E;
  PetscReal  Enorm = 0., lgEnorm, lgEmax, sum = 0., Emax = 0., temp = 0., *weight, chargesum = 0.;
  PetscReal *x, *v;
  PetscInt  *species, dim, p, d, Np, cStart, cEnd;
  PetscReal  pmoments[3]; /* \int f, \int x f, \int r^2 f */
  PetscReal  fmoments[3]; /* \int \hat f, \int x \hat f, \int r^2 \hat f */
  Vec        rho;

  PetscFunctionBeginUser;
  if (step < 0) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMSwarmSortGetAccess(sw));
  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
  PetscCall(DMSwarmGetField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMSwarmGetField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&species));
  PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));

  for (p = 0; p < Np; ++p) {
    for (d = 0; d < 1; ++d) {
      temp = PetscAbsReal(E[p * dim + d]);
      if (temp > Emax) Emax = temp;
    }
    Enorm += PetscSqrtReal(E[p * dim] * E[p * dim]);
    sum += E[p * dim];
    chargesum += user->charges[0] * weight[p];
  }
  lgEnorm = Enorm != 0 ? PetscLog10Real(Enorm) : -16.;
  lgEmax  = Emax != 0 ? PetscLog10Real(Emax) : -16.;

  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
  PetscCall(DMSwarmRestoreField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMSwarmRestoreField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&species));

  Parameter *param;
  PetscCall(PetscBagGetData(user->bag, (void **)&param));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "charges", &rho));
  if (user->em == EM_PRIMAL) {
    PetscCall(computeParticleMoments(sw, pmoments, user));
    PetscCall(computeFEMMoments(dm, rho, fmoments, user));
  } else if (user->em == EM_MIXED) {
    DM       potential_dm;
    IS       potential_IS;
    PetscInt fields = 1;
    PetscCall(DMCreateSubDM(dm, 1, &fields, &potential_IS, &potential_dm));

    PetscCall(computeParticleMoments(sw, pmoments, user));
    PetscCall(computeFEMMoments(potential_dm, rho, fmoments, user));
    PetscCall(DMDestroy(&potential_dm));
    PetscCall(ISDestroy(&potential_IS));
  }
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "charges", &rho));

  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%f\t%+e\t%e\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\n", (double)t, (double)sum, (double)Enorm, (double)lgEnorm, (double)Emax, (double)lgEmax, (double)chargesum, (double)pmoments[0], (double)pmoments[1], (double)pmoments[2], (double)fmoments[0], (double)fmoments[1], (double)fmoments[2]));
  PetscCall(PetscDrawLGAddPoint(user->drawlg_ef, &t, &lgEmax));
  PetscCall(PetscDrawLGDraw(user->drawlg_ef));
  PetscCall(PetscDrawSave(user->drawef));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MonitorInitialConditions(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx)
{
  AppCtx            *user = (AppCtx *)ctx;
  DM                 dm, sw;
  const PetscScalar *u;
  PetscReal         *weight, *pos, *vel;
  PetscInt           dim, p, Np, cStart, cEnd;

  PetscFunctionBegin;
  if (step < 0) PetscFunctionReturn(PETSC_SUCCESS); /* -1 indicates interpolated solution */
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMSwarmSortGetAccess(sw));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));

  if (step == 0) {
    PetscCall(PetscDrawHGReset(user->drawhgic_x));
    PetscCall(PetscDrawHGGetDraw(user->drawhgic_x, &user->drawic_x));
    PetscCall(PetscDrawClear(user->drawic_x));
    PetscCall(PetscDrawFlush(user->drawic_x));

    PetscCall(PetscDrawHGReset(user->drawhgic_v));
    PetscCall(PetscDrawHGGetDraw(user->drawhgic_v, &user->drawic_v));
    PetscCall(PetscDrawClear(user->drawic_v));
    PetscCall(PetscDrawFlush(user->drawic_v));

    PetscCall(PetscDrawHGReset(user->drawhgic_w));
    PetscCall(PetscDrawHGGetDraw(user->drawhgic_w, &user->drawic_w));
    PetscCall(PetscDrawClear(user->drawic_w));
    PetscCall(PetscDrawFlush(user->drawic_w));

    PetscCall(VecGetArrayRead(U, &u));
    PetscCall(DMSwarmGetField(sw, "velocity", NULL, NULL, (void **)&vel));
    PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));
    PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&pos));

    PetscCall(VecGetLocalSize(U, &Np));
    Np /= dim * 2;
    for (p = 0; p < Np; ++p) {
      PetscCall(PetscDrawHGAddValue(user->drawhgic_x, pos[p * dim]));
      PetscCall(PetscDrawHGAddValue(user->drawhgic_v, vel[p * dim]));
      PetscCall(PetscDrawHGAddValue(user->drawhgic_w, weight[p]));
    }

    PetscCall(VecRestoreArrayRead(U, &u));
    PetscCall(PetscDrawHGDraw(user->drawhgic_x));
    PetscCall(PetscDrawHGSave(user->drawhgic_x));

    PetscCall(PetscDrawHGDraw(user->drawhgic_v));
    PetscCall(PetscDrawHGSave(user->drawhgic_v));

    PetscCall(PetscDrawHGDraw(user->drawhgic_w));
    PetscCall(PetscDrawHGSave(user->drawhgic_w));

    PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&pos));
    PetscCall(DMSwarmRestoreField(sw, "velocity", NULL, NULL, (void **)&vel));
    PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode MonitorPositions_2D(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx)
{
  AppCtx         *user = (AppCtx *)ctx;
  DM              dm, sw;
  PetscScalar    *x, *v, *weight;
  PetscReal       lower[3], upper[3], speed;
  const PetscInt *s;
  PetscInt        dim, cStart, cEnd, c;

  PetscFunctionBeginUser;
  if (step > 0 && step % user->ostep == 0) {
    PetscCall(TSGetDM(ts, &sw));
    PetscCall(DMSwarmGetCellDM(sw, &dm));
    PetscCall(DMGetDimension(dm, &dim));
    PetscCall(DMGetBoundingBox(dm, lower, upper));
    PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
    PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
    PetscCall(DMSwarmGetField(sw, "velocity", NULL, NULL, (void **)&v));
    PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));
    PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&s));
    PetscCall(DMSwarmSortGetAccess(sw));
    PetscCall(PetscDrawSPReset(user->positionDrawSP));
    PetscCall(PetscDrawSPSetLimits(user->positionDrawSP, lower[0], upper[0], lower[1], upper[1]));
    PetscCall(PetscDrawSPSetLimits(user->positionDrawSP, lower[0], upper[0], -12, 12));
    for (c = 0; c < cEnd - cStart; ++c) {
      PetscInt *pidx, Npc, q;
      PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Npc, &pidx));
      for (q = 0; q < Npc; ++q) {
        const PetscInt p = pidx[q];
        if (s[p] == 0) {
          speed = PetscSqrtReal(PetscSqr(v[p * dim]) + PetscSqr(v[p * dim + 1]));
          if (dim == 1 || user->fake_1D) {
            PetscCall(PetscDrawSPAddPointColorized(user->positionDrawSP, &x[p * dim], &x[p * dim + 1], &speed));
          } else {
            PetscCall(PetscDrawSPAddPointColorized(user->positionDrawSP, &x[p * dim], &v[p * dim], &speed));
          }
        } else if (s[p] == 1) {
          PetscCall(PetscDrawSPAddPoint(user->positionDrawSP, &x[p * dim], &v[p * dim]));
        }
      }
      PetscCall(PetscFree(pidx));
    }
    PetscCall(PetscDrawSPDraw(user->positionDrawSP, PETSC_TRUE));
    PetscCall(PetscDrawSave(user->positionDraw));
    PetscCall(DMSwarmSortRestoreAccess(sw));
    PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
    PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
    PetscCall(DMSwarmRestoreField(sw, "velocity", NULL, NULL, (void **)&v));
    PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&s));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode MonitorPoisson(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx)
{
  AppCtx      *user = (AppCtx *)ctx;
  DM           dm, sw;
  PetscScalar *x, *E, *weight, *pot, *charges;
  PetscReal    lower[3], upper[3], xval;
  PetscInt     dim, cStart, cEnd, c;

  PetscFunctionBeginUser;
  if (step > 0 && step % user->ostep == 0) {
    PetscCall(TSGetDM(ts, &sw));
    PetscCall(DMSwarmGetCellDM(sw, &dm));
    PetscCall(DMGetDimension(dm, &dim));
    PetscCall(DMGetBoundingBox(dm, lower, upper));
    PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));

    PetscCall(PetscDrawSPReset(user->RhoDrawSP));
    PetscCall(PetscDrawSPReset(user->EDrawSP));
    PetscCall(PetscDrawSPReset(user->PotDrawSP));
    PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
    PetscCall(DMSwarmGetField(sw, "E_field", NULL, NULL, (void **)&E));
    PetscCall(DMSwarmGetField(sw, "potential", NULL, NULL, (void **)&pot));
    PetscCall(DMSwarmGetField(sw, "charges", NULL, NULL, (void **)&charges));
    PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));

    PetscCall(DMSwarmSortGetAccess(sw));
    for (c = 0; c < cEnd - cStart; ++c) {
      PetscReal Esum = 0.0;
      PetscInt *pidx, Npc, q;
      PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Npc, &pidx));
      for (q = 0; q < Npc; ++q) {
        const PetscInt p = pidx[q];
        Esum += E[p * dim];
      }
      xval = (c + 0.5) * ((upper - lower) / (cEnd - cStart));
      PetscCall(PetscDrawSPAddPoint(user->EDrawSP, &xval, &Esum));
      PetscCall(PetscFree(pidx));
    }
    for (c = 0; c < (cEnd - cStart); ++c) {
      xval = (c + 0.5) * ((upper - lower) / (cEnd - cStart));
      PetscCall(PetscDrawSPAddPoint(user->RhoDrawSP, &xval, &charges[c]));
      PetscCall(PetscDrawSPAddPoint(user->PotDrawSP, &xval, &pot[c]));
    }
    PetscCall(PetscDrawSPDraw(user->RhoDrawSP, PETSC_TRUE));
    PetscCall(PetscDrawSave(user->RhoDraw));
    PetscCall(PetscDrawSPDraw(user->EDrawSP, PETSC_TRUE));
    PetscCall(PetscDrawSave(user->EDraw));
    PetscCall(PetscDrawSPDraw(user->PotDrawSP, PETSC_TRUE));
    PetscCall(PetscDrawSave(user->PotDraw));
    PetscCall(DMSwarmSortRestoreAccess(sw));
    PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
    PetscCall(DMSwarmRestoreField(sw, "potential", NULL, NULL, (void **)&pot));
    PetscCall(DMSwarmRestoreField(sw, "charges", NULL, NULL, (void **)&charges));
    PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
    PetscCall(DMSwarmRestoreField(sw, "E_field", NULL, NULL, (void **)&E));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SetupParameters(MPI_Comm comm, AppCtx *ctx)
{
  PetscBag   bag;
  Parameter *p;

  PetscFunctionBeginUser;
  /* setup PETSc parameter bag */
  PetscCall(PetscBagGetData(ctx->bag, (void **)&p));
  PetscCall(PetscBagSetName(ctx->bag, "par", "Vlasov-Poisson Parameters"));
  bag = ctx->bag;
  PetscCall(PetscBagRegisterScalar(bag, &p->v0, 1.0, "v0", "Velocity scale, m/s"));
  PetscCall(PetscBagRegisterScalar(bag, &p->t0, 1.0, "t0", "Time scale, s"));
  PetscCall(PetscBagRegisterScalar(bag, &p->x0, 1.0, "x0", "Space scale, m"));
  PetscCall(PetscBagRegisterScalar(bag, &p->v0, 1.0, "phi0", "Potential scale, kg*m^2/A*s^3"));
  PetscCall(PetscBagRegisterScalar(bag, &p->q0, 1.0, "q0", "Charge Scale, A*s"));
  PetscCall(PetscBagRegisterScalar(bag, &p->m0, 1.0, "m0", "Mass Scale, kg"));
  PetscCall(PetscBagRegisterScalar(bag, &p->epsi0, 1.0, "epsi0", "Permittivity of Free Space, kg"));
  PetscCall(PetscBagRegisterScalar(bag, &p->kb, 1.0, "kb", "Boltzmann Constant, m^2 kg/s^2 K^1"));

  PetscCall(PetscBagRegisterScalar(bag, &p->sigma, 1.0, "sigma", "Charge per unit area, C/m^3"));
  PetscCall(PetscBagRegisterScalar(bag, &p->poissonNumber, 1.0, "poissonNumber", "Non-Dimensional Poisson Number"));
  PetscCall(PetscBagRegisterScalar(bag, &p->vlasovNumber, 1.0, "vlasovNumber", "Non-Dimensional Vlasov Number"));
  PetscCall(PetscBagSetFromOptions(bag));
  {
    PetscViewer       viewer;
    PetscViewerFormat format;
    PetscBool         flg;

    PetscCall(PetscOptionsGetViewer(comm, NULL, NULL, "-param_view", &viewer, &format, &flg));
    if (flg) {
      PetscCall(PetscViewerPushFormat(viewer, format));
      PetscCall(PetscBagView(bag, viewer));
      PetscCall(PetscViewerFlush(viewer));
      PetscCall(PetscViewerPopFormat(viewer));
      PetscCall(PetscViewerDestroy(&viewer));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
{
  PetscFunctionBeginUser;
  PetscCall(DMCreate(comm, dm));
  PetscCall(DMSetType(*dm, DMPLEX));
  PetscCall(DMSetFromOptions(*dm));
  PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static void ion_f0(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  f0[0] = -constants[SIGMA];
}

static void laplacian_f1(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) f1[d] = u_x[d];
}

static void laplacian_g3(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0;
}

static PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  *u = 0.0;
  return PETSC_SUCCESS;
}

/*
   /  I   -grad\ / q \ = /0\
   \-div    0  / \phi/   \f/
*/
static void f0_q(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  for (PetscInt d = 0; d < dim; ++d) f0[d] += u[uOff[0] + d];
}

static void f1_q(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
{
  for (PetscInt d = 0; d < dim; ++d) f1[d * dim + d] = u[uOff[1]];
}

static void f0_phi_backgroundCharge(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  f0[0] += constants[SIGMA];
  for (PetscInt d = 0; d < dim; ++d) f0[0] += u_x[uOff_x[0] + d * dim + d];
}

/* Boundary residual. Dirichlet boundary for u means u_bdy=p*n */
static void g0_qq(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
{
  for (PetscInt d = 0; d < dim; ++d) g0[d * dim + d] = 1.0;
}

static void g2_qphi(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
{
  for (PetscInt d = 0; d < dim; ++d) g2[d * dim + d] = 1.0;
}

static void g1_phiq(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g1[])
{
  for (PetscInt d = 0; d < dim; ++d) g1[d * dim + d] = 1.0;
}

static PetscErrorCode CreateFEM(DM dm, AppCtx *user)
{
  PetscFE   fephi, feq;
  PetscDS   ds;
  PetscBool simplex;
  PetscInt  dim;

  PetscFunctionBeginUser;
  PetscCall(DMGetDimension(dm, &dim));
  PetscCall(DMPlexIsSimplex(dm, &simplex));
  if (user->em == EM_MIXED) {
    DMLabel        label;
    const PetscInt id = 1;

    PetscCall(PetscFECreateDefault(PETSC_COMM_SELF, dim, dim, simplex, "field_", PETSC_DETERMINE, &feq));
    PetscCall(PetscObjectSetName((PetscObject)feq, "field"));
    PetscCall(PetscFECreateDefault(PETSC_COMM_SELF, dim, 1, simplex, "potential_", PETSC_DETERMINE, &fephi));
    PetscCall(PetscObjectSetName((PetscObject)fephi, "potential"));
    PetscCall(PetscFECopyQuadrature(feq, fephi));
    PetscCall(DMSetField(dm, 0, NULL, (PetscObject)feq));
    PetscCall(DMSetField(dm, 1, NULL, (PetscObject)fephi));
    PetscCall(DMCreateDS(dm));
    PetscCall(PetscFEDestroy(&fephi));
    PetscCall(PetscFEDestroy(&feq));

    PetscCall(DMGetLabel(dm, "marker", &label));
    PetscCall(DMGetDS(dm, &ds));

    PetscCall(PetscDSSetResidual(ds, 0, f0_q, f1_q));
    PetscCall(PetscDSSetResidual(ds, 1, f0_phi_backgroundCharge, NULL));
    PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_qq, NULL, NULL, NULL));
    PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_qphi, NULL));
    PetscCall(PetscDSSetJacobian(ds, 1, 0, NULL, g1_phiq, NULL, NULL));

    PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void))zero, NULL, NULL, NULL));

  } else if (user->em == EM_PRIMAL) {
    MatNullSpace nullsp;
    PetscCall(PetscFECreateDefault(PETSC_COMM_SELF, dim, 1, simplex, NULL, PETSC_DETERMINE, &fephi));
    PetscCall(PetscObjectSetName((PetscObject)fephi, "potential"));
    PetscCall(DMSetField(dm, 0, NULL, (PetscObject)fephi));
    PetscCall(DMCreateDS(dm));
    PetscCall(DMGetDS(dm, &ds));
    PetscCall(PetscDSSetResidual(ds, 0, ion_f0, laplacian_f1));
    PetscCall(PetscDSSetJacobian(ds, 0, 0, NULL, NULL, NULL, laplacian_g3));
    PetscCall(MatNullSpaceCreate(PetscObjectComm((PetscObject)dm), PETSC_TRUE, 0, NULL, &nullsp));
    PetscCall(PetscObjectCompose((PetscObject)fephi, "nullspace", (PetscObject)nullsp));
    PetscCall(MatNullSpaceDestroy(&nullsp));
    PetscCall(PetscFEDestroy(&fephi));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreatePoisson(DM dm, AppCtx *user)
{
  SNES         snes;
  Mat          J;
  MatNullSpace nullSpace;

  PetscFunctionBeginUser;
  PetscCall(CreateFEM(dm, user));
  PetscCall(SNESCreate(PetscObjectComm((PetscObject)dm), &snes));
  PetscCall(SNESSetOptionsPrefix(snes, "em_"));
  PetscCall(SNESSetDM(snes, dm));
  PetscCall(DMPlexSetSNESLocalFEM(dm, user, user, user));
  PetscCall(SNESSetFromOptions(snes));

  PetscCall(DMCreateMatrix(dm, &J));
  PetscCall(MatNullSpaceCreate(PetscObjectComm((PetscObject)dm), PETSC_TRUE, 0, NULL, &nullSpace));
  PetscCall(MatSetNullSpace(J, nullSpace));
  PetscCall(MatNullSpaceDestroy(&nullSpace));
  PetscCall(SNESSetJacobian(snes, J, J, NULL, NULL));
  PetscCall(MatDestroy(&J));
  user->snes = snes;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PetscPDFPertubedConstant2D(const PetscReal x[], const PetscReal dummy[], PetscReal p[])
{
  p[0] = (1 + 0.01 * PetscCosReal(0.5 * x[0])) / (2 * PETSC_PI);
  p[1] = (1 + 0.01 * PetscCosReal(0.5 * x[1])) / (2 * PETSC_PI);
  return PETSC_SUCCESS;
}
PetscErrorCode PetscPDFPertubedConstant1D(const PetscReal x[], const PetscReal dummy[], PetscReal p[])
{
  p[0] = (1. + 0.01 * PetscCosReal(0.5 * x[0])) / (2 * PETSC_PI);
  return PETSC_SUCCESS;
}

PetscErrorCode PetscPDFCosine1D(const PetscReal x[], const PetscReal scale[], PetscReal p[])
{
  const PetscReal alpha = scale ? scale[0] : 0.0;
  const PetscReal k     = scale ? scale[1] : 1.;
  p[0]                  = (1 + alpha * PetscCosReal(k * x[0]));
  return PETSC_SUCCESS;
}

PetscErrorCode PetscPDFCosine2D(const PetscReal x[], const PetscReal scale[], PetscReal p[])
{
  const PetscReal alpha = scale ? scale[0] : 0.;
  const PetscReal k     = scale ? scale[0] : 1.;
  p[0]                  = (1 + alpha * PetscCosReal(k * (x[0] + x[1])));
  return PETSC_SUCCESS;
}

static PetscErrorCode InitializeParticles_PerturbedWeights(DM sw, AppCtx *user)
{
  DM           vdm, dm;
  PetscScalar *weight;
  PetscReal   *x, *v, vmin[3], vmax[3], gmin[3], gmax[3], xi0[3];
  PetscInt    *N, Ns, dim, *cellid, *species, Np, cStart, cEnd, Npc, n;
  PetscInt     p, q, s, c, d, cv;
  PetscBool    flg;
  PetscMPIInt  size, rank;
  Parameter   *param;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)sw), &size));
  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)sw), &rank));
  PetscOptionsBegin(PetscObjectComm((PetscObject)sw), "", "DMSwarm Options", "DMSWARM");
  PetscCall(DMSwarmGetNumSpecies(sw, &Ns));
  PetscCall(PetscOptionsInt("-dm_swarm_num_species", "The number of species", "DMSwarmSetNumSpecies", Ns, &Ns, &flg));
  if (flg) PetscCall(DMSwarmSetNumSpecies(sw, Ns));
  PetscCall(PetscCalloc1(Ns, &N));
  n = Ns;
  PetscCall(PetscOptionsIntArray("-dm_swarm_num_particles", "The target number of particles", "", N, &n, NULL));
  PetscOptionsEnd();

  Np = N[0];
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Np = %" PetscInt_FMT "\n", Np));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));

  PetscCall(DMCreate(PETSC_COMM_WORLD, &vdm));
  PetscCall(DMSetType(vdm, DMPLEX));
  PetscCall(DMPlexSetOptionsPrefix(vdm, "v"));
  PetscCall(DMSetFromOptions(vdm));
  PetscCall(DMViewFromOptions(vdm, NULL, "-vdm_view"));

  PetscCall(DMGetBoundingBox(dm, gmin, gmax));
  PetscCall(PetscBagGetData(user->bag, (void **)&param));
  PetscCall(DMSwarmSetLocalSizes(sw, Np, 0));
  Npc = Np / (cEnd - cStart);
  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_cellid, NULL, NULL, (void **)&cellid));
  for (c = 0, p = 0; c < cEnd - cStart; ++c) {
    for (s = 0; s < Ns; ++s) {
      for (q = 0; q < Npc; ++q, ++p) cellid[p] = c;
    }
  }
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_cellid, NULL, NULL, (void **)&cellid));
  PetscCall(PetscFree(N));

  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
  PetscCall(DMSwarmGetField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&species));

  PetscCall(DMSwarmSortGetAccess(sw));
  PetscInt vStart, vEnd;
  PetscCall(DMPlexGetHeightStratum(vdm, 0, &vStart, &vEnd));
  PetscCall(DMGetBoundingBox(vdm, vmin, vmax));
  for (c = 0; c < cEnd - cStart; ++c) {
    const PetscInt cell = c + cStart;
    PetscInt      *pidx, Npc;
    PetscReal      centroid[3], volume;

    PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Npc, &pidx));
    PetscCall(DMPlexComputeCellGeometryFVM(dm, cell, &volume, centroid, NULL));
    for (q = 0; q < Npc; ++q) {
      const PetscInt p = pidx[q];

      for (d = 0; d < dim; ++d) {
        x[p * dim + d] = centroid[d];
        v[p * dim + d] = vmin[0] + (q + 0.5) * (vmax[0] - vmin[0]) / Npc;
        if (user->fake_1D && d > 0) v[p * dim + d] = 0;
      }
    }
    PetscCall(PetscFree(pidx));
  }
  PetscCall(DMGetCoordinatesLocalSetUp(vdm));

  /* Setup Quadrature for spatial and velocity weight calculations*/
  PetscQuadrature  quad_x;
  PetscInt         Nq_x;
  const PetscReal *wq_x, *xq_x;
  PetscReal       *xq_x_extended;
  PetscReal        weightsum = 0., totalcellweight = 0., *weight_x, *weight_v;
  PetscReal        scale[2] = {user->cosine_coefficients[0], user->cosine_coefficients[1]};

  PetscCall(PetscCalloc2(cEnd - cStart, &weight_x, Np, &weight_v));
  if (user->fake_1D) PetscCall(PetscDTGaussTensorQuadrature(1, 1, 5, -1.0, 1.0, &quad_x));
  else PetscCall(PetscDTGaussTensorQuadrature(dim, 1, 5, -1.0, 1.0, &quad_x));
  PetscCall(PetscQuadratureGetData(quad_x, NULL, NULL, &Nq_x, &xq_x, &wq_x));
  if (user->fake_1D) {
    PetscCall(PetscCalloc1(Nq_x * dim, &xq_x_extended));
    for (PetscInt i = 0; i < Nq_x; ++i) xq_x_extended[i * dim] = xq_x[i];
  }
  /* Integrate the density function to get the weights of particles in each cell */
  for (d = 0; d < dim; ++d) xi0[d] = -1.0;
  for (c = cStart; c < cEnd; ++c) {
    PetscReal          v0_x[3], J_x[9], invJ_x[9], detJ_x, xr_x[3], den_x;
    PetscInt          *pidx, Npc, q;
    PetscInt           Ncx;
    const PetscScalar *array_x;
    PetscScalar       *coords_x = NULL;
    PetscBool          isDGx;
    weight_x[c] = 0.;

    PetscCall(DMPlexGetCellCoordinates(dm, c, &isDGx, &Ncx, &array_x, &coords_x));
    PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Npc, &pidx));
    PetscCall(DMPlexComputeCellGeometryFEM(dm, c, NULL, v0_x, J_x, invJ_x, &detJ_x));
    for (q = 0; q < Nq_x; ++q) {
      /*Transform quadrature points from ref space to real space (0,12.5664)*/
      if (user->fake_1D) CoordinatesRefToReal(dim, dim, xi0, v0_x, J_x, &xq_x_extended[q * dim], xr_x);
      else CoordinatesRefToReal(dim, dim, xi0, v0_x, J_x, &xq_x[q * dim], xr_x);

      /*Transform quadrature points from real space to ideal real space (0, 2PI/k)*/
      if (user->fake_1D) {
        PetscCall(PetscPDFCosine1D(xr_x, scale, &den_x));
        detJ_x = J_x[0];
      } else PetscCall(PetscPDFCosine2D(xr_x, scale, &den_x));
      /*We have to transform the quadrature weights as well*/
      weight_x[c] += den_x * (wq_x[q] * detJ_x);
    }
    PetscCall(PetscPrintf(PETSC_COMM_WORLD, "c:%" PetscInt_FMT " [x_a,x_b] = %1.15f,%1.15f -> cell weight = %1.15f\n", c, (double)PetscRealPart(coords_x[0]), (double)PetscRealPart(coords_x[2]), (double)weight_x[c]));
    totalcellweight += weight_x[c];
    PetscCheck(Npc / size == vEnd - vStart, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Number of particles %" PetscInt_FMT " in cell (rank %d/%d) != %" PetscInt_FMT " number of velocity vertices", Npc, rank, size, vEnd - vStart);

    /* Set weights to be gaussian in velocity cells (using exact solution) */
    for (cv = 0; cv < vEnd - vStart; ++cv) {
      PetscInt           Nc;
      const PetscScalar *array_v;
      PetscScalar       *coords_v = NULL;
      PetscBool          isDG;
      PetscCall(DMPlexGetCellCoordinates(vdm, cv, &isDG, &Nc, &array_v, &coords_v));

      const PetscInt p = pidx[cv];

      weight_v[p] = 0.5 * (PetscErfReal(coords_v[1] / PetscSqrtReal(2.)) - PetscErfReal(coords_v[0] / PetscSqrtReal(2.)));

      weight[p] = user->totalWeight * weight_v[p] * weight_x[c];
      weightsum += weight[p];

      PetscCall(DMPlexRestoreCellCoordinates(vdm, cv, &isDG, &Nc, &array_v, &coords_v));
    }
    PetscCall(DMPlexRestoreCellCoordinates(dm, c, &isDGx, &Ncx, &array_x, &coords_x));
    PetscCall(PetscFree(pidx));
  }
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "particle weight sum = %1.10f cell weight sum = %1.10f\n", (double)totalcellweight, (double)weightsum));
  if (user->fake_1D) PetscCall(PetscFree(xq_x_extended));
  PetscCall(PetscFree2(weight_x, weight_v));
  PetscCall(PetscQuadratureDestroy(&quad_x));
  PetscCall(DMSwarmSortRestoreAccess(sw));
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&x));
  PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&species));
  PetscCall(DMSwarmRestoreField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMDestroy(&vdm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode InitializeConstants(DM sw, AppCtx *user)
{
  DM         dm;
  PetscInt  *species;
  PetscReal *weight, totalCharge = 0., totalWeight = 0., gmin[3], gmax[3];
  PetscInt   Np, p, dim;

  PetscFunctionBegin;
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMGetBoundingBox(dm, gmin, gmax));
  PetscCall(DMSwarmGetField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&species));
  for (p = 0; p < Np; ++p) {
    totalWeight += weight[p];
    totalCharge += user->charges[species[p]] * weight[p];
  }
  PetscCall(DMSwarmRestoreField(sw, "w_q", NULL, NULL, (void **)&weight));
  PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&species));
  {
    Parameter *param;
    PetscReal  Area;

    PetscCall(PetscBagGetData(user->bag, (void **)&param));
    switch (dim) {
    case 1:
      Area = (gmax[0] - gmin[0]);
      break;
    case 2:
      if (user->fake_1D) {
        Area = (gmax[0] - gmin[0]);
      } else {
        Area = (gmax[0] - gmin[0]) * (gmax[1] - gmin[1]);
      }
      break;
    case 3:
      Area = (gmax[0] - gmin[0]) * (gmax[1] - gmin[1]) * (gmax[2] - gmin[2]);
      break;
    default:
      SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Dimension %" PetscInt_FMT " not supported", dim);
    }
    PetscCall(PetscPrintf(PETSC_COMM_WORLD, "dim = %" PetscInt_FMT "\ttotalWeight = %f, user->charges[species[p]] = %f\ttotalCharge = %f, Total Area = %f\n", dim, (double)totalWeight, (double)user->charges[0], (double)totalCharge, (double)Area));
    param->sigma = PetscAbsReal(totalCharge / (Area));

    PetscCall(PetscPrintf(PETSC_COMM_SELF, "sigma: %g\n", (double)param->sigma));
    PetscCall(PetscPrintf(PETSC_COMM_SELF, "(x0,v0,t0,m0,q0,phi0): (%e, %e, %e, %e, %e, %e) - (P, V) = (%e, %e)\n", (double)param->x0, (double)param->v0, (double)param->t0, (double)param->m0, (double)param->q0, (double)param->phi0, (double)param->poissonNumber,
                          (double)param->vlasovNumber));
  }
  /* Setup Constants */
  {
    PetscDS    ds;
    Parameter *param;
    PetscCall(PetscBagGetData(user->bag, (void **)&param));
    PetscScalar constants[NUM_CONSTANTS];
    constants[SIGMA]   = param->sigma;
    constants[V0]      = param->v0;
    constants[T0]      = param->t0;
    constants[X0]      = param->x0;
    constants[M0]      = param->m0;
    constants[Q0]      = param->q0;
    constants[PHI0]    = param->phi0;
    constants[POISSON] = param->poissonNumber;
    constants[VLASOV]  = param->vlasovNumber;
    PetscCall(DMGetDS(dm, &ds));
    PetscCall(PetscDSSetConstants(ds, NUM_CONSTANTS, constants));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode InitializeVelocites_Fake1D(DM sw, AppCtx *user)
{
  DM         dm;
  PetscReal *v;
  PetscInt  *species, cStart, cEnd;
  PetscInt   dim, Np, p;

  PetscFunctionBegin;
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMSwarmGetField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&species));
  PetscCall(DMSwarmGetCellDM(sw, &dm));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
  PetscRandom rnd;
  PetscCall(PetscRandomCreate(PetscObjectComm((PetscObject)sw), &rnd));
  PetscCall(PetscRandomSetInterval(rnd, 0, 1.));
  PetscCall(PetscRandomSetFromOptions(rnd));

  for (p = 0; p < Np; ++p) {
    PetscReal a[3] = {0., 0., 0.}, vel[3] = {0., 0., 0.};

    PetscCall(PetscRandomGetValueReal(rnd, &a[0]));
    if (user->perturbed_weights) {
      PetscCall(PetscPDFSampleConstant1D(a, NULL, vel));
    } else {
      PetscCall(PetscPDFSampleGaussian1D(a, NULL, vel));
    }
    v[p * dim] = vel[0];
  }
  PetscCall(PetscRandomDestroy(&rnd));
  PetscCall(DMSwarmRestoreField(sw, "velocity", NULL, NULL, (void **)&v));
  PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&species));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreateSwarm(DM dm, AppCtx *user, DM *sw)
{
  PetscReal v0[2] = {1., 0.};
  PetscInt  dim;

  PetscFunctionBeginUser;
  PetscCall(DMGetDimension(dm, &dim));
  PetscCall(DMCreate(PetscObjectComm((PetscObject)dm), sw));
  PetscCall(DMSetType(*sw, DMSWARM));
  PetscCall(DMSetDimension(*sw, dim));
  PetscCall(DMSwarmSetType(*sw, DMSWARM_PIC));
  PetscCall(DMSwarmSetCellDM(*sw, dm));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "w_q", 1, PETSC_SCALAR));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "velocity", dim, PETSC_REAL));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "species", 1, PETSC_INT));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "initCoordinates", dim, PETSC_REAL));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "initVelocity", dim, PETSC_REAL));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "E_field", dim, PETSC_REAL));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "potential", dim, PETSC_REAL));
  PetscCall(DMSwarmRegisterPetscDatatypeField(*sw, "charges", dim, PETSC_REAL));
  PetscCall(DMSwarmFinalizeFieldRegister(*sw));

  if (user->perturbed_weights) {
    PetscCall(InitializeParticles_PerturbedWeights(*sw, user));
  } else {
    PetscCall(DMSwarmComputeLocalSizeFromOptions(*sw));
    PetscCall(DMSwarmInitializeCoordinates(*sw));
    if (user->fake_1D) {
      PetscCall(InitializeVelocites_Fake1D(*sw, user));
    } else {
      PetscCall(DMSwarmInitializeVelocitiesFromOptions(*sw, v0));
    }
  }
  PetscCall(DMSetFromOptions(*sw));
  PetscCall(DMSetApplicationContext(*sw, user));
  PetscCall(PetscObjectSetName((PetscObject)*sw, "Particles"));
  PetscCall(DMViewFromOptions(*sw, NULL, "-sw_view"));
  {
    Vec gc, gc0, gv, gv0;

    PetscCall(DMSwarmCreateGlobalVectorFromField(*sw, DMSwarmPICField_coor, &gc));
    PetscCall(DMSwarmCreateGlobalVectorFromField(*sw, "initCoordinates", &gc0));
    PetscCall(VecCopy(gc, gc0));
    PetscCall(VecViewFromOptions(gc, NULL, "-ic_x_view"));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(*sw, DMSwarmPICField_coor, &gc));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(*sw, "initCoordinates", &gc0));
    PetscCall(DMSwarmCreateGlobalVectorFromField(*sw, "velocity", &gv));
    PetscCall(DMSwarmCreateGlobalVectorFromField(*sw, "initVelocity", &gv0));
    PetscCall(VecCopy(gv, gv0));
    PetscCall(VecViewFromOptions(gv, NULL, "-ic_v_view"));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(*sw, "velocity", &gv));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(*sw, "initVelocity", &gv0));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ComputeFieldAtParticles_Coulomb(SNES snes, DM sw, PetscReal E[])
{
  AppCtx     *user;
  PetscReal  *coords;
  PetscInt   *species, dim, d, Np, p, q, Ns;
  PetscMPIInt size;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)snes), &size));
  PetscCheck(size == 1, PetscObjectComm((PetscObject)snes), PETSC_ERR_SUP, "Coulomb code only works in serial");
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMSwarmGetNumSpecies(sw, &Ns));
  PetscCall(DMGetApplicationContext(sw, (void *)&user));

  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "species", NULL, NULL, (void **)&species));
  for (p = 0; p < Np; ++p) {
    PetscReal *pcoord = &coords[p * dim];
    PetscReal  pE[3]  = {0., 0., 0.};

    /* Calculate field at particle p due to particle q */
    for (q = 0; q < Np; ++q) {
      PetscReal *qcoord = &coords[q * dim];
      PetscReal  rpq[3], r, r3, q_q;

      if (p == q) continue;
      q_q = user->charges[species[q]] * 1.;
      for (d = 0; d < dim; ++d) rpq[d] = pcoord[d] - qcoord[d];
      r = DMPlex_NormD_Internal(dim, rpq);
      if (r < PETSC_SQRT_MACHINE_EPSILON) continue;
      r3 = PetscPowRealInt(r, 3);
      for (d = 0; d < dim; ++d) pE[d] += q_q * rpq[d] / r3;
    }
    for (d = 0; d < dim; ++d) E[p * dim + d] = pE[d];
  }
  PetscCall(DMSwarmRestoreField(sw, "species", NULL, NULL, (void **)&species));
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ComputeFieldAtParticles_Primal(SNES snes, DM sw, PetscReal E[])
{
  DM              dm;
  AppCtx         *user;
  PetscDS         ds;
  PetscFE         fe;
  Mat             M_p, M;
  Vec             phi, locPhi, rho, f;
  PetscReal      *coords;
  PetscInt        dim, d, cStart, cEnd, c, Np;
  PetscQuadrature q;

  PetscFunctionBegin;
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMGetApplicationContext(sw, (void *)&user));

  KSP ksp;
  Vec rho0;
  /* Create the charges rho */
  PetscCall(SNESGetDM(snes, &dm));

  PetscCall(DMCreateMassMatrix(sw, dm, &M_p));
  PetscCall(DMCreateMassMatrix(dm, dm, &M));
  PetscCall(DMGetGlobalVector(dm, &rho0));
  PetscCall(PetscObjectSetName((PetscObject)rho0, "Charge density (rho0) from Primal Compute"));
  PetscCall(DMGetGlobalVector(dm, &rho));
  PetscCall(PetscObjectSetName((PetscObject)rho, "rho"));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "w_q", &f));

  PetscCall(PetscObjectSetName((PetscObject)f, "particle weight"));
  PetscCall(MatMultTranspose(M_p, f, rho));
  PetscCall(MatViewFromOptions(M_p, NULL, "-mp_view"));
  PetscCall(MatViewFromOptions(M, NULL, "-m_view"));
  PetscCall(VecViewFromOptions(f, NULL, "-weights_view"));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "w_q", &f));

  PetscCall(KSPCreate(PetscObjectComm((PetscObject)dm), &ksp));
  PetscCall(KSPSetOptionsPrefix(ksp, "em_proj_"));
  PetscCall(KSPSetOperators(ksp, M, M));
  PetscCall(KSPSetFromOptions(ksp));
  PetscCall(KSPSolve(ksp, rho, rho0));
  PetscCall(VecViewFromOptions(rho0, NULL, "-rho0_view"));

  PetscInt           rhosize;
  PetscReal         *charges;
  const PetscScalar *rho_vals;
  PetscCall(DMSwarmGetField(sw, "charges", NULL, NULL, (void **)&charges));
  PetscCall(VecGetSize(rho0, &rhosize));
  PetscCall(VecGetArrayRead(rho0, &rho_vals));
  for (c = 0; c < rhosize; ++c) charges[c] = rho_vals[c];
  PetscCall(VecRestoreArrayRead(rho0, &rho_vals));
  PetscCall(DMSwarmRestoreField(sw, "charges", NULL, NULL, (void **)&charges));

  PetscCall(VecScale(rho, -1.0));

  PetscCall(VecViewFromOptions(rho0, NULL, "-rho0_view"));
  PetscCall(VecViewFromOptions(rho, NULL, "-rho_view"));
  PetscCall(DMRestoreGlobalVector(dm, &rho0));
  PetscCall(KSPDestroy(&ksp));
  PetscCall(MatDestroy(&M_p));
  PetscCall(MatDestroy(&M));

  PetscCall(DMGetGlobalVector(dm, &phi));
  PetscCall(PetscObjectSetName((PetscObject)phi, "potential"));
  PetscCall(VecSet(phi, 0.0));
  PetscCall(SNESSolve(snes, rho, phi));
  PetscCall(DMRestoreGlobalVector(dm, &rho));
  PetscCall(VecViewFromOptions(phi, NULL, "-phi_view"));

  PetscInt           phisize;
  PetscReal         *pot;
  const PetscScalar *phi_vals;
  PetscCall(DMSwarmGetField(sw, "potential", NULL, NULL, (void **)&pot));
  PetscCall(VecGetSize(phi, &phisize));
  PetscCall(VecGetArrayRead(phi, &phi_vals));
  for (c = 0; c < phisize; ++c) pot[c] = phi_vals[c];
  PetscCall(VecRestoreArrayRead(phi, &phi_vals));
  PetscCall(DMSwarmRestoreField(sw, "potential", NULL, NULL, (void **)&pot));

  PetscCall(DMGetLocalVector(dm, &locPhi));
  PetscCall(DMGlobalToLocalBegin(dm, phi, INSERT_VALUES, locPhi));
  PetscCall(DMGlobalToLocalEnd(dm, phi, INSERT_VALUES, locPhi));
  PetscCall(DMRestoreGlobalVector(dm, &phi));

  PetscCall(DMGetDS(dm, &ds));
  PetscCall(PetscDSGetDiscretization(ds, 0, (PetscObject *)&fe));
  PetscCall(DMSwarmSortGetAccess(sw));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));

  for (c = cStart; c < cEnd; ++c) {
    PetscTabulation tab;
    PetscScalar    *clPhi = NULL;
    PetscReal      *pcoord, *refcoord;
    PetscReal       v[3], J[9], invJ[9], detJ;
    PetscInt       *points;
    PetscInt        Ncp, cp;

    PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Ncp, &points));
    PetscCall(DMGetWorkArray(dm, Ncp * dim, MPIU_REAL, &pcoord));
    PetscCall(DMGetWorkArray(dm, Ncp * dim, MPIU_REAL, &refcoord));
    for (cp = 0; cp < Ncp; ++cp)
      for (d = 0; d < dim; ++d) pcoord[cp * dim + d] = coords[points[cp] * dim + d];
    PetscCall(DMPlexCoordinatesToReference(dm, c, Ncp, pcoord, refcoord));
    PetscCall(PetscFECreateTabulation(fe, 1, Ncp, refcoord, 1, &tab));
    PetscCall(DMPlexComputeCellGeometryFEM(dm, c, NULL, v, J, invJ, &detJ));
    PetscCall(DMPlexVecGetClosure(dm, NULL, locPhi, c, NULL, &clPhi));
    for (cp = 0; cp < Ncp; ++cp) {
      const PetscReal *basisDer = tab->T[1];
      const PetscInt   p        = points[cp];

      for (d = 0; d < dim; ++d) E[p * dim + d] = 0.;
      PetscCall(PetscFEGetQuadrature(fe, &q));
      PetscCall(PetscFEFreeInterpolateGradient_Static(fe, basisDer, clPhi, dim, invJ, NULL, cp, &E[p * dim]));
      for (d = 0; d < dim; ++d) {
        E[p * dim + d] *= -1.0;
        if (user->fake_1D && d > 0) E[p * dim + d] = 0;
      }
    }
    PetscCall(DMPlexVecRestoreClosure(dm, NULL, locPhi, c, NULL, &clPhi));
    PetscCall(DMRestoreWorkArray(dm, Ncp * dim, MPIU_REAL, &pcoord));
    PetscCall(DMRestoreWorkArray(dm, Ncp * dim, MPIU_REAL, &refcoord));
    PetscCall(PetscTabulationDestroy(&tab));
    PetscCall(PetscFree(points));
  }
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmSortRestoreAccess(sw));
  PetscCall(DMRestoreLocalVector(dm, &locPhi));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ComputeFieldAtParticles_Mixed(SNES snes, DM sw, PetscReal E[])
{
  AppCtx         *user;
  DM              dm, potential_dm;
  KSP             ksp;
  IS              potential_IS;
  PetscDS         ds;
  PetscFE         fe;
  PetscFEGeom     feGeometry;
  Mat             M_p, M;
  Vec             phi, locPhi, rho, f, temp_rho, rho0;
  PetscQuadrature q;
  PetscReal      *coords, *pot;
  PetscInt        dim, d, cStart, cEnd, c, Np, fields = 1;

  PetscFunctionBegin;
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMGetApplicationContext(sw, &user));

  /* Create the charges rho */
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(DMGetGlobalVector(dm, &rho));
  PetscCall(PetscObjectSetName((PetscObject)rho, "rho"));

  PetscCall(DMCreateSubDM(dm, 1, &fields, &potential_IS, &potential_dm));
  PetscCall(DMCreateMassMatrix(sw, potential_dm, &M_p));
  PetscCall(DMCreateMassMatrix(potential_dm, potential_dm, &M));
  PetscCall(MatViewFromOptions(M_p, NULL, "-mp_view"));
  PetscCall(MatViewFromOptions(M, NULL, "-m_view"));
  PetscCall(DMGetGlobalVector(potential_dm, &temp_rho));
  PetscCall(PetscObjectSetName((PetscObject)temp_rho, "Mf"));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "w_q", &f));
  PetscCall(PetscObjectSetName((PetscObject)f, "particle weight"));
  PetscCall(VecViewFromOptions(f, NULL, "-weights_view"));
  PetscCall(MatMultTranspose(M_p, f, temp_rho));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "w_q", &f));
  PetscCall(DMGetGlobalVector(potential_dm, &rho0));
  PetscCall(PetscObjectSetName((PetscObject)rho0, "Charge density (rho0) from Mixed Compute"));

  PetscCall(KSPCreate(PetscObjectComm((PetscObject)dm), &ksp));
  PetscCall(KSPSetOptionsPrefix(ksp, "em_proj"));
  PetscCall(KSPSetOperators(ksp, M, M));
  PetscCall(KSPSetFromOptions(ksp));
  PetscCall(KSPSolve(ksp, temp_rho, rho0));
  PetscCall(VecViewFromOptions(rho0, NULL, "-rho0_view"));

  PetscInt           rhosize;
  PetscReal         *charges;
  const PetscScalar *rho_vals;
  Parameter         *param;
  PetscCall(PetscBagGetData(user->bag, (void **)&param));
  PetscCall(DMSwarmGetField(sw, "charges", NULL, NULL, (void **)&charges));
  PetscCall(VecGetSize(rho0, &rhosize));

  /* Integral over reference element is size 1.  Reference element area is 4.  Scale rho0 by 1/4 because the basis function is 1/4 */
  PetscCall(VecScale(rho0, 0.25));
  PetscCall(VecGetArrayRead(rho0, &rho_vals));
  for (c = 0; c < rhosize; ++c) charges[c] = rho_vals[c];
  PetscCall(VecRestoreArrayRead(rho0, &rho_vals));
  PetscCall(DMSwarmRestoreField(sw, "charges", NULL, NULL, (void **)&charges));

  PetscCall(VecISCopy(rho, potential_IS, SCATTER_FORWARD, temp_rho));
  PetscCall(VecScale(rho, 0.25));
  PetscCall(VecViewFromOptions(rho0, NULL, "-rho0_view"));
  PetscCall(VecViewFromOptions(temp_rho, NULL, "-temprho_view"));
  PetscCall(VecViewFromOptions(rho, NULL, "-rho_view"));
  PetscCall(DMRestoreGlobalVector(potential_dm, &temp_rho));
  PetscCall(DMRestoreGlobalVector(potential_dm, &rho0));

  PetscCall(MatDestroy(&M_p));
  PetscCall(MatDestroy(&M));
  PetscCall(KSPDestroy(&ksp));
  PetscCall(DMDestroy(&potential_dm));
  PetscCall(ISDestroy(&potential_IS));

  PetscCall(DMGetGlobalVector(dm, &phi));
  PetscCall(PetscObjectSetName((PetscObject)phi, "potential"));
  PetscCall(VecSet(phi, 0.0));
  PetscCall(SNESSolve(snes, rho, phi));
  PetscCall(DMRestoreGlobalVector(dm, &rho));

  PetscInt           phisize;
  const PetscScalar *phi_vals;
  PetscCall(DMSwarmGetField(sw, "potential", NULL, NULL, (void **)&pot));
  PetscCall(VecGetSize(phi, &phisize));
  PetscCall(VecViewFromOptions(phi, NULL, "-phi_view"));
  PetscCall(VecGetArrayRead(phi, &phi_vals));
  for (c = 0; c < phisize; ++c) pot[c] = phi_vals[c];
  PetscCall(VecRestoreArrayRead(phi, &phi_vals));
  PetscCall(DMSwarmRestoreField(sw, "potential", NULL, NULL, (void **)&pot));

  PetscCall(DMGetLocalVector(dm, &locPhi));
  PetscCall(DMGlobalToLocalBegin(dm, phi, INSERT_VALUES, locPhi));
  PetscCall(DMGlobalToLocalEnd(dm, phi, INSERT_VALUES, locPhi));
  PetscCall(DMRestoreGlobalVector(dm, &phi));

  PetscCall(DMGetDS(dm, &ds));
  PetscCall(PetscDSGetDiscretization(ds, 0, (PetscObject *)&fe));
  PetscCall(DMSwarmSortGetAccess(sw));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
  PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(PetscFEGetQuadrature(fe, &q));
  PetscCall(PetscFECreateCellGeometry(fe, q, &feGeometry));
  for (c = cStart; c < cEnd; ++c) {
    PetscTabulation tab;
    PetscScalar    *clPhi = NULL;
    PetscReal      *pcoord, *refcoord;
    PetscInt       *points;
    PetscInt        Ncp, cp;

    PetscCall(DMSwarmSortGetPointsPerCell(sw, c, &Ncp, &points));
    PetscCall(DMGetWorkArray(dm, Ncp * dim, MPIU_REAL, &pcoord));
    PetscCall(DMGetWorkArray(dm, Ncp * dim, MPIU_REAL, &refcoord));
    for (cp = 0; cp < Ncp; ++cp)
      for (d = 0; d < dim; ++d) pcoord[cp * dim + d] = coords[points[cp] * dim + d];
    PetscCall(DMPlexCoordinatesToReference(dm, c, Ncp, pcoord, refcoord));
    PetscCall(PetscFECreateTabulation(fe, 1, Ncp, refcoord, 1, &tab));
    PetscCall(DMPlexComputeCellGeometryFEM(dm, c, q, feGeometry.v, feGeometry.J, feGeometry.invJ, feGeometry.detJ));
    PetscCall(DMPlexVecGetClosure(dm, NULL, locPhi, c, NULL, &clPhi));

    for (cp = 0; cp < Ncp; ++cp) {
      const PetscInt p = points[cp];

      for (d = 0; d < dim; ++d) E[p * dim + d] = 0.;
      PetscCall(PetscFEInterpolateAtPoints_Static(fe, tab, clPhi, &feGeometry, cp, &E[p * dim]));
      PetscCall(PetscFEPushforward(fe, &feGeometry, 1, &E[p * dim]));
      for (d = 0; d < dim; ++d) {
        E[p * dim + d] *= -2.0;
        if (user->fake_1D && d > 0) E[p * dim + d] = 0;
      }
    }
    PetscCall(DMPlexVecRestoreClosure(dm, NULL, locPhi, c, NULL, &clPhi));
    PetscCall(DMRestoreWorkArray(dm, Ncp * dim, MPIU_REAL, &pcoord));
    PetscCall(DMRestoreWorkArray(dm, Ncp * dim, MPIU_REAL, &refcoord));
    PetscCall(PetscTabulationDestroy(&tab));
    PetscCall(PetscFree(points));
  }
  PetscCall(PetscFEDestroyCellGeometry(fe, &feGeometry));
  PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmSortRestoreAccess(sw));
  PetscCall(DMRestoreLocalVector(dm, &locPhi));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ComputeFieldAtParticles(SNES snes, DM sw, PetscReal E[])
{
  AppCtx  *ctx;
  PetscInt dim, Np;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  PetscValidHeaderSpecific(sw, DM_CLASSID, 2);
  PetscAssertPointer(E, 3);
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMGetApplicationContext(sw, &ctx));
  PetscCall(PetscArrayzero(E, Np * dim));

  switch (ctx->em) {
  case EM_PRIMAL:
    PetscCall(ComputeFieldAtParticles_Primal(snes, sw, E));
    break;
  case EM_COULOMB:
    PetscCall(ComputeFieldAtParticles_Coulomb(snes, sw, E));
    break;
  case EM_MIXED:
    PetscCall(ComputeFieldAtParticles_Mixed(snes, sw, E));
    break;
  case EM_NONE:
    break;
  default:
    SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "No solver for electrostatic model %s", EMTypes[ctx->em]);
  }

  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec U, Vec G, void *ctx)
{
  DM                 sw;
  SNES               snes = ((AppCtx *)ctx)->snes;
  const PetscReal   *coords, *vel;
  const PetscScalar *u;
  PetscScalar       *g;
  PetscReal         *E, m_p = 1., q_p = -1.;
  PetscInt           dim, d, Np, p;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(DMSwarmGetField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(VecGetArrayRead(U, &u));
  PetscCall(VecGetArray(G, &g));

  PetscCall(ComputeFieldAtParticles(snes, sw, E));

  Np /= 2 * dim;
  for (p = 0; p < Np; ++p) {
    for (d = 0; d < dim; ++d) {
      g[(p * 2 + 0) * dim + d] = u[(p * 2 + 1) * dim + d];
      g[(p * 2 + 1) * dim + d] = q_p * E[p * dim + d] / m_p;
    }
  }
  PetscCall(DMSwarmRestoreField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmRestoreField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(DMSwarmRestoreField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(VecRestoreArray(G, &g));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* J_{ij} = dF_i/dx_j
   J_p = (  0   1)
         (-w^2  0)
   TODO Now there is another term with w^2 from the electric field. I think we will need to invert the operator.
        Perhaps we can approximate the Jacobian using only the cellwise P-P gradient from Coulomb
*/
static PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec U, Mat J, Mat P, void *ctx)
{
  DM               sw;
  const PetscReal *coords, *vel;
  PetscInt         dim, d, Np, p, rStart;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(MatGetOwnershipRange(J, &rStart, NULL));
  PetscCall(DMSwarmGetField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  Np /= 2 * dim;
  for (p = 0; p < Np; ++p) {
    const PetscReal x0      = coords[p * dim + 0];
    const PetscReal vy0     = vel[p * dim + 1];
    const PetscReal omega   = vy0 / x0;
    PetscScalar     vals[4] = {0., 1., -PetscSqr(omega), 0.};

    for (d = 0; d < dim; ++d) {
      const PetscInt rows[2] = {(p * 2 + 0) * dim + d + rStart, (p * 2 + 1) * dim + d + rStart};
      PetscCall(MatSetValues(J, 2, rows, 2, rows, vals, INSERT_VALUES));
    }
  }
  PetscCall(DMSwarmRestoreField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmRestoreField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode RHSFunctionX(TS ts, PetscReal t, Vec V, Vec Xres, void *ctx)
{
  AppCtx            *user = (AppCtx *)ctx;
  DM                 sw;
  const PetscScalar *v;
  PetscScalar       *xres;
  PetscInt           Np, p, d, dim;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(VecGetLocalSize(Xres, &Np));
  PetscCall(VecGetArrayRead(V, &v));
  PetscCall(VecGetArray(Xres, &xres));
  Np /= dim;
  for (p = 0; p < Np; ++p) {
    for (d = 0; d < dim; ++d) {
      xres[p * dim + d] = v[p * dim + d];
      if (user->fake_1D && d > 0) xres[p * dim + d] = 0;
    }
  }
  PetscCall(VecRestoreArrayRead(V, &v));
  PetscCall(VecRestoreArray(Xres, &xres));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode RHSFunctionV(TS ts, PetscReal t, Vec X, Vec Vres, void *ctx)
{
  DM                 sw;
  AppCtx            *user = (AppCtx *)ctx;
  SNES               snes = ((AppCtx *)ctx)->snes;
  const PetscScalar *x;
  const PetscReal   *coords, *vel;
  PetscReal         *E, m_p, q_p;
  PetscScalar       *vres;
  PetscInt           Np, p, dim, d;
  Parameter         *param;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(DMSwarmGetField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscCall(PetscBagGetData(user->bag, (void **)&param));
  m_p = user->masses[0] * param->m0;
  q_p = user->charges[0] * param->q0;
  PetscCall(VecGetLocalSize(Vres, &Np));
  PetscCall(VecGetArrayRead(X, &x));
  PetscCall(VecGetArray(Vres, &vres));
  PetscCheck(dim == 2, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Dimension must be 2");
  PetscCall(ComputeFieldAtParticles(snes, sw, E));

  Np /= dim;
  for (p = 0; p < Np; ++p) {
    for (d = 0; d < dim; ++d) {
      vres[p * dim + d] = q_p * E[p * dim + d] / m_p;
      if (user->fake_1D && d > 0) vres[p * dim + d] = 0.;
    }
  }
  PetscCall(VecRestoreArrayRead(X, &x));
  PetscCall(VecRestoreArray(Vres, &vres));
  PetscCall(DMSwarmRestoreField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmRestoreField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(DMSwarmRestoreField(sw, "E_field", NULL, NULL, (void **)&E));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreateSolution(TS ts)
{
  DM       sw;
  Vec      u;
  PetscInt dim, Np;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &u));
  PetscCall(VecSetBlockSize(u, dim));
  PetscCall(VecSetSizes(u, 2 * Np * dim, PETSC_DECIDE));
  PetscCall(VecSetUp(u));
  PetscCall(TSSetSolution(ts, u));
  PetscCall(VecDestroy(&u));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SetProblem(TS ts)
{
  AppCtx *user;
  DM      sw;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetApplicationContext(sw, (void **)&user));
  // Define unified system for (X, V)
  {
    Mat      J;
    PetscInt dim, Np;

    PetscCall(DMGetDimension(sw, &dim));
    PetscCall(DMSwarmGetLocalSize(sw, &Np));
    PetscCall(MatCreate(PETSC_COMM_WORLD, &J));
    PetscCall(MatSetSizes(J, 2 * Np * dim, 2 * Np * dim, PETSC_DECIDE, PETSC_DECIDE));
    PetscCall(MatSetBlockSize(J, 2 * dim));
    PetscCall(MatSetFromOptions(J));
    PetscCall(MatSetUp(J));
    PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, user));
    PetscCall(TSSetRHSJacobian(ts, J, J, RHSJacobian, user));
    PetscCall(MatDestroy(&J));
  }
  /* Define split system for X and V */
  {
    Vec             u;
    IS              isx, isv, istmp;
    const PetscInt *idx;
    PetscInt        dim, Np, rstart;

    PetscCall(TSGetSolution(ts, &u));
    PetscCall(DMGetDimension(sw, &dim));
    PetscCall(DMSwarmGetLocalSize(sw, &Np));
    PetscCall(VecGetOwnershipRange(u, &rstart, NULL));
    PetscCall(ISCreateStride(PETSC_COMM_WORLD, Np, (rstart / dim) + 0, 2, &istmp));
    PetscCall(ISGetIndices(istmp, &idx));
    PetscCall(ISCreateBlock(PETSC_COMM_WORLD, dim, Np, idx, PETSC_COPY_VALUES, &isx));
    PetscCall(ISRestoreIndices(istmp, &idx));
    PetscCall(ISDestroy(&istmp));
    PetscCall(ISCreateStride(PETSC_COMM_WORLD, Np, (rstart / dim) + 1, 2, &istmp));
    PetscCall(ISGetIndices(istmp, &idx));
    PetscCall(ISCreateBlock(PETSC_COMM_WORLD, dim, Np, idx, PETSC_COPY_VALUES, &isv));
    PetscCall(ISRestoreIndices(istmp, &idx));
    PetscCall(ISDestroy(&istmp));
    PetscCall(TSRHSSplitSetIS(ts, "position", isx));
    PetscCall(TSRHSSplitSetIS(ts, "momentum", isv));
    PetscCall(ISDestroy(&isx));
    PetscCall(ISDestroy(&isv));
    PetscCall(TSRHSSplitSetRHSFunction(ts, "position", NULL, RHSFunctionX, user));
    PetscCall(TSRHSSplitSetRHSFunction(ts, "momentum", NULL, RHSFunctionV, user));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DMSwarmTSRedistribute(TS ts)
{
  DM        sw;
  Vec       u;
  PetscReal t, maxt, dt;
  PetscInt  n, maxn;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(TSGetTime(ts, &t));
  PetscCall(TSGetMaxTime(ts, &maxt));
  PetscCall(TSGetTimeStep(ts, &dt));
  PetscCall(TSGetStepNumber(ts, &n));
  PetscCall(TSGetMaxSteps(ts, &maxn));

  PetscCall(TSReset(ts));
  PetscCall(TSSetDM(ts, sw));
  PetscCall(TSSetFromOptions(ts));
  PetscCall(TSSetTime(ts, t));
  PetscCall(TSSetMaxTime(ts, maxt));
  PetscCall(TSSetTimeStep(ts, dt));
  PetscCall(TSSetStepNumber(ts, n));
  PetscCall(TSSetMaxSteps(ts, maxn));

  PetscCall(CreateSolution(ts));
  PetscCall(SetProblem(ts));
  PetscCall(TSGetSolution(ts, &u));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  InitializeSolveAndSwarm - Set the solution values to the swarm coordinates and velocities, and also possibly set the initial values.

  Input Parameters:
+ ts         - The TS
- useInitial - Flag to also set the initial conditions to the current coordinates and velocities and setup the problem

  Output Parameter:
. u - The initialized solution vector

  Level: advanced

.seealso: InitializeSolve()
*/
static PetscErrorCode InitializeSolveAndSwarm(TS ts, PetscBool useInitial)
{
  DM       sw;
  Vec      u, gc, gv, gc0, gv0;
  IS       isx, isv;
  PetscInt dim;
  AppCtx  *user;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetApplicationContext(sw, &user));
  PetscCall(DMGetDimension(sw, &dim));
  if (useInitial) {
    PetscReal v0[2] = {1., 0.};
    if (user->perturbed_weights) {
      PetscCall(InitializeParticles_PerturbedWeights(sw, user));
    } else {
      PetscCall(DMSwarmComputeLocalSizeFromOptions(sw));
      PetscCall(DMSwarmInitializeCoordinates(sw));
      if (user->fake_1D) {
        PetscCall(InitializeVelocites_Fake1D(sw, user));
      } else {
        PetscCall(DMSwarmInitializeVelocitiesFromOptions(sw, v0));
      }
    }
    PetscCall(DMSwarmMigrate(sw, PETSC_TRUE));
    PetscCall(DMSwarmTSRedistribute(ts));
  }
  PetscCall(TSGetSolution(ts, &u));
  PetscCall(TSRHSSplitGetIS(ts, "position", &isx));
  PetscCall(TSRHSSplitGetIS(ts, "momentum", &isv));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "initCoordinates", &gc0));
  if (useInitial) PetscCall(VecCopy(gc, gc0));
  PetscCall(VecISCopy(u, isx, SCATTER_FORWARD, gc));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "initCoordinates", &gc0));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "velocity", &gv));
  PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "initVelocity", &gv0));
  if (useInitial) PetscCall(VecCopy(gv, gv0));
  PetscCall(VecISCopy(u, isv, SCATTER_FORWARD, gv));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "velocity", &gv));
  PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "initVelocity", &gv0));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode InitializeSolve(TS ts, Vec u)
{
  PetscFunctionBegin;
  PetscCall(TSSetSolution(ts, u));
  PetscCall(InitializeSolveAndSwarm(ts, PETSC_TRUE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode ComputeError(TS ts, Vec U, Vec E)
{
  MPI_Comm           comm;
  DM                 sw;
  AppCtx            *user;
  const PetscScalar *u;
  const PetscReal   *coords, *vel;
  PetscScalar       *e;
  PetscReal          t;
  PetscInt           dim, Np, p;

  PetscFunctionBeginUser;
  PetscCall(PetscObjectGetComm((PetscObject)ts, &comm));
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetApplicationContext(sw, &user));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(TSGetSolveTime(ts, &t));
  PetscCall(VecGetArray(E, &e));
  PetscCall(VecGetArrayRead(U, &u));
  PetscCall(DMSwarmGetLocalSize(sw, &Np));
  PetscCall(DMSwarmGetField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmGetField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  Np /= 2 * dim;
  for (p = 0; p < Np; ++p) {
    /* TODO generalize initial conditions and project into plane instead of assuming x-y */
    const PetscReal    r0    = DMPlex_NormD_Internal(dim, &coords[p * dim]);
    const PetscReal    th0   = PetscAtan2Real(coords[p * dim + 1], coords[p * dim + 0]);
    const PetscReal    v0    = DMPlex_NormD_Internal(dim, &vel[p * dim]);
    const PetscReal    omega = v0 / r0;
    const PetscReal    ct    = PetscCosReal(omega * t + th0);
    const PetscReal    st    = PetscSinReal(omega * t + th0);
    const PetscScalar *x     = &u[(p * 2 + 0) * dim];
    const PetscScalar *v     = &u[(p * 2 + 1) * dim];
    const PetscReal    xe[3] = {r0 * ct, r0 * st, 0.0};
    const PetscReal    ve[3] = {-v0 * st, v0 * ct, 0.0};
    PetscInt           d;

    for (d = 0; d < dim; ++d) {
      e[(p * 2 + 0) * dim + d] = x[d] - xe[d];
      e[(p * 2 + 1) * dim + d] = v[d] - ve[d];
    }
    if (user->error) {
      const PetscReal en   = 0.5 * DMPlex_DotRealD_Internal(dim, v, v);
      const PetscReal exen = 0.5 * PetscSqr(v0);
      PetscCall(PetscPrintf(comm, "t %.4g: p%" PetscInt_FMT " error [%.2g %.2g] sol [(%.6lf %.6lf) (%.6lf %.6lf)] exact [(%.6lf %.6lf) (%.6lf %.6lf)] energy/exact energy %g / %g (%.10lf%%)\n", (double)t, p, (double)DMPlex_NormD_Internal(dim, &e[(p * 2 + 0) * dim]), (double)DMPlex_NormD_Internal(dim, &e[(p * 2 + 1) * dim]), (double)x[0], (double)x[1], (double)v[0], (double)v[1], (double)xe[0], (double)xe[1], (double)ve[0], (double)ve[1], (double)en, (double)exen, (double)(PetscAbsReal(exen - en) * 100. / exen)));
    }
  }
  PetscCall(DMSwarmRestoreField(sw, "initCoordinates", NULL, NULL, (void **)&coords));
  PetscCall(DMSwarmRestoreField(sw, "initVelocity", NULL, NULL, (void **)&vel));
  PetscCall(VecRestoreArrayRead(U, &u));
  PetscCall(VecRestoreArray(E, &e));
  PetscFunctionReturn(PETSC_SUCCESS);
}

#if 0
static PetscErrorCode EnergyMonitor(TS ts, PetscInt step, PetscReal t, Vec U, void *ctx)
{
  const PetscInt     ostep = ((AppCtx *)ctx)->ostep;
  const EMType       em    = ((AppCtx *)ctx)->em;
  DM                 sw;
  const PetscScalar *u;
  PetscReal         *coords, *E;
  PetscReal          enKin = 0., enEM = 0.;
  PetscInt           dim, d, Np, p, q;

  PetscFunctionBeginUser;
  if (step % ostep == 0) {
    PetscCall(TSGetDM(ts, &sw));
    PetscCall(DMGetDimension(sw, &dim));
    PetscCall(VecGetArrayRead(U, &u));
    PetscCall(DMSwarmGetLocalSize(sw, &Np));
    Np /= 2 * dim;
    PetscCall(DMSwarmGetField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
    PetscCall(DMSwarmGetField(sw, "E_field", NULL, NULL, (void **)&E));
    if (!step) PetscCall(PetscPrintf(PetscObjectComm((PetscObject)ts), "Time     Step Part     Energy\n"));
    for (p = 0; p < Np; ++p) {
      const PetscReal v2     = DMPlex_DotRealD_Internal(dim, &u[(p * 2 + 1) * dim], &u[(p * 2 + 1) * dim]);
      PetscReal      *pcoord = &coords[p * dim];

      PetscCall(PetscSynchronizedPrintf(PetscObjectComm((PetscObject)ts), "%.6lf %4D %5D %10.4lf\n", t, step, p, (double)0.5 * v2));
      enKin += 0.5 * v2;
      if (em == EM_NONE) {
        continue;
      } else if (em == EM_COULOMB) {
        for (q = p + 1; q < Np; ++q) {
          PetscReal *qcoord = &coords[q * dim];
          PetscReal  rpq[3], r;
          for (d = 0; d < dim; ++d) rpq[d] = pcoord[d] - qcoord[d];
          r = DMPlex_NormD_Internal(dim, rpq);
          enEM += 1. / r;
        }
      } else if (em == EM_PRIMAL) {
        for (d = 0; d < dim; ++d) enEM += E[p * dim + d];
      }
    }
    PetscCall(PetscSynchronizedPrintf(PetscObjectComm((PetscObject)ts), "%.6lf %4" PetscInt_FMT " 2\t    %10.4lf\n", t, step, (double)enKin));
    PetscCall(PetscSynchronizedPrintf(PetscObjectComm((PetscObject)ts), "%.6lf %4" PetscInt_FMT " 3\t    %10.4lf\n", t, step, (double)enEM));
    PetscCall(PetscSynchronizedPrintf(PetscObjectComm((PetscObject)ts), "%.6lf %4" PetscInt_FMT " 4\t    %10.4lf\n", t, step, (double)enKin + enEM));
    PetscCall(DMSwarmRestoreField(sw, DMSwarmPICField_coor, NULL, NULL, (void **)&coords));
    PetscCall(DMSwarmRestoreField(sw, "E_field", NULL, NULL, (void **)&E));
    PetscCall(PetscSynchronizedFlush(PetscObjectComm((PetscObject)ts), NULL));
    PetscCall(VecRestoreArrayRead(U, &u));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}
#endif

static PetscErrorCode MigrateParticles(TS ts)
{
  DM sw;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMViewFromOptions(sw, NULL, "-migrate_view_pre"));
  {
    Vec u, gc, gv;
    IS  isx, isv;

    PetscCall(TSGetSolution(ts, &u));
    PetscCall(TSRHSSplitGetIS(ts, "position", &isx));
    PetscCall(TSRHSSplitGetIS(ts, "momentum", &isv));
    PetscCall(DMSwarmCreateGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
    PetscCall(VecISCopy(u, isx, SCATTER_REVERSE, gc));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
    PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "velocity", &gv));
    PetscCall(VecISCopy(u, isv, SCATTER_REVERSE, gv));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "velocity", &gv));
  }
  PetscCall(DMSwarmMigrate(sw, PETSC_TRUE));
  PetscCall(DMSwarmTSRedistribute(ts));
  PetscCall(InitializeSolveAndSwarm(ts, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode MigrateParticles_Periodic(TS ts)
{
  DM       sw, dm;
  PetscInt dim;

  PetscFunctionBeginUser;
  PetscCall(TSGetDM(ts, &sw));
  PetscCall(DMGetDimension(sw, &dim));
  PetscCall(DMViewFromOptions(sw, NULL, "-migrate_view_pre"));
  {
    Vec        u, position, momentum, gc, gv;
    IS         isx, isv;
    PetscReal *pos, *mom, *x, *v;
    PetscReal  lower_bound[3], upper_bound[3];
    PetscInt   p, d, Np;

    PetscCall(TSGetSolution(ts, &u));
    PetscCall(DMSwarmGetLocalSize(sw, &Np));
    PetscCall(DMSwarmGetCellDM(sw, &dm));
    PetscCall(DMGetBoundingBox(dm, lower_bound, upper_bound));
    PetscCall(TSRHSSplitGetIS(ts, "position", &isx));
    PetscCall(TSRHSSplitGetIS(ts, "momentum", &isv));
    PetscCall(VecGetSubVector(u, isx, &position));
    PetscCall(VecGetSubVector(u, isv, &momentum));
    PetscCall(VecGetArray(position, &pos));
    PetscCall(VecGetArray(momentum, &mom));

    PetscCall(DMSwarmCreateGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
    PetscCall(VecISCopy(u, isx, SCATTER_REVERSE, gc));
    PetscCall(DMSwarmCreateGlobalVectorFromField(sw, "velocity", &gv));
    PetscCall(VecISCopy(u, isv, SCATTER_REVERSE, gv));

    PetscCall(VecGetArray(gc, &x));
    PetscCall(VecGetArray(gv, &v));
    for (p = 0; p < Np; ++p) {
      for (d = 0; d < dim; ++d) {
        if (pos[p * dim + d] < lower_bound[d]) {
          x[p * dim + d] = pos[p * dim + d] + (upper_bound[d] - lower_bound[d]);
        } else if (pos[p * dim + d] > upper_bound[d]) {
          x[p * dim + d] = pos[p * dim + d] - (upper_bound[d] - lower_bound[d]);
        } else {
          x[p * dim + d] = pos[p * dim + d];
        }
        PetscCheck(x[p * dim + d] >= lower_bound[d] && x[p * dim + d] <= upper_bound[d], PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "p: %" PetscInt_FMT "x[%" PetscInt_FMT "] %g", p, d, (double)x[p * dim + d]);
        v[p * dim + d] = mom[p * dim + d];
      }
    }
    PetscCall(VecRestoreArray(gc, &x));
    PetscCall(VecRestoreArray(gv, &v));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, DMSwarmPICField_coor, &gc));
    PetscCall(DMSwarmDestroyGlobalVectorFromField(sw, "velocity", &gv));

    PetscCall(VecRestoreArray(position, &pos));
    PetscCall(VecRestoreArray(momentum, &mom));
    PetscCall(VecRestoreSubVector(u, isx, &position));
    PetscCall(VecRestoreSubVector(u, isv, &momentum));
  }
  PetscCall(DMSwarmMigrate(sw, PETSC_TRUE));
  PetscCall(DMSwarmTSRedistribute(ts));
  PetscCall(InitializeSolveAndSwarm(ts, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  DM     dm, sw;
  TS     ts;
  Vec    u;
  AppCtx user;

  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
  PetscCall(PetscBagCreate(PETSC_COMM_SELF, sizeof(Parameter), &user.bag));
  PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
  PetscCall(CreatePoisson(dm, &user));
  PetscCall(CreateSwarm(dm, &user, &sw));
  PetscCall(SetupParameters(PETSC_COMM_WORLD, &user));
  PetscCall(InitializeConstants(sw, &user));
  PetscCall(DMSetApplicationContext(sw, &user));

  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
  PetscCall(TSSetDM(ts, sw));
  PetscCall(TSSetMaxTime(ts, 0.1));
  PetscCall(TSSetTimeStep(ts, 0.00001));
  PetscCall(TSSetMaxSteps(ts, 100));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));

  if (user.efield_monitor) PetscCall(TSMonitorSet(ts, MonitorEField, &user, NULL));
  if (user.initial_monitor) PetscCall(TSMonitorSet(ts, MonitorInitialConditions, &user, NULL));
  if (user.monitor_positions) PetscCall(TSMonitorSet(ts, MonitorPositions_2D, &user, NULL));
  if (user.poisson_monitor) PetscCall(TSMonitorSet(ts, MonitorPoisson, &user, NULL));

  PetscCall(TSSetFromOptions(ts));
  PetscReal dt;
  PetscInt  maxn;
  PetscCall(TSGetTimeStep(ts, &dt));
  PetscCall(TSGetMaxSteps(ts, &maxn));
  user.steps    = maxn;
  user.stepSize = dt;
  PetscCall(SetupContext(dm, sw, &user));

  PetscCall(DMSwarmVectorDefineField(sw, "velocity"));
  PetscCall(TSSetComputeInitialCondition(ts, InitializeSolve));
  PetscCall(TSSetComputeExactError(ts, ComputeError));
  if (user.periodic) {
    PetscCall(TSSetPostStep(ts, MigrateParticles_Periodic));
  } else {
    PetscCall(TSSetPostStep(ts, MigrateParticles));
  }
  PetscCall(CreateSolution(ts));
  PetscCall(TSGetSolution(ts, &u));
  PetscCall(TSComputeInitialCondition(ts, u));

  PetscCall(TSSolve(ts, NULL));

  PetscCall(SNESDestroy(&user.snes));
  PetscCall(TSDestroy(&ts));
  PetscCall(DMDestroy(&sw));
  PetscCall(DMDestroy(&dm));
  PetscCall(DestroyContext(&user));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

   build:
     requires: double !complex

    # Recommend -draw_size 500,500
   testset:
     args: -dm_plex_dim 2 -fake_1D -dm_plex_simplex 0 -dm_plex_box_faces 20,1 -dm_plex_box_lower 0,-1 -dm_plex_box_upper 12.5664,1 \
           -dm_swarm_coordinate_density constant -dm_swarm_num_particles 100 \
           -dm_plex_box_bd periodic,none -periodic -ts_type basicsymplectic -ts_basicsymplectic_type 1\
           -dm_view -output_step 50 -sigma 1.0e-8 -timeScale 2.0e-14\
           -ts_monitor_sp_swarm -ts_monitor_sp_swarm_retain 0 -ts_monitor_sp_swarm_phase 0
     test:
       suffix: none_1d
       args: -em_type none -error
     test:
       suffix: coulomb_1d
       args: -em_type coulomb

   # For verification, we use
   # -dm_plex_box_faces 100,1 -vdm_plex_box_faces 8000 -dm_swarm_num_particles 800000
   # -ts_monitor_sp_swarm_multi_species 0 -ts_monitor_sp_swarm_retain 0 -ts_monitor_sp_swarm_phase 1 -draw_size 500,500
   testset:
     args: -dm_plex_dim 2 -dm_plex_box_bd periodic,none -dm_plex_simplex 0 -dm_plex_box_faces 10,1 -dm_plex_box_lower 0,-0.5 -dm_plex_box_upper 12.5664,0.5\
           -ts_dt 0.03 -ts_max_time 500 -ts_max_steps 500 -ts_type basicsymplectic -ts_basicsymplectic_type 1\
           -em_snes_atol 1.e-12 -em_snes_error_if_not_converged -em_ksp_error_if_not_converged\
           -dm_swarm_num_species 1 -dm_swarm_num_particles 100 -dm_view\
           -vdm_plex_dim 1 -vdm_plex_box_lower -10 -vdm_plex_box_upper 10 -vdm_plex_simplex 0 -vdm_plex_box_faces 10\
           -output_step 1 -fake_1D -perturbed_weights -periodic -cosine_coefficients 0.01,0.5 -charges -1.0,1.0 -total_weight 1.0
     test:
       suffix: uniform_equilibrium_1d
       args: -cosine_coefficients 0.0,0.5 -em_type primal -petscspace_degree 1 -em_pc_type svd
     test:
       suffix: uniform_primal_1d
       args: -em_type primal -petscspace_degree 1 -em_pc_type svd
     test:
       suffix: uniform_none_1d
       args: -em_type none
     test:
       suffix: uniform_mixed_1d
       args: -em_type mixed\
             -ksp_rtol 1e-10\
             -em_ksp_type preonly\
             -em_ksp_error_if_not_converged\
             -em_snes_error_if_not_converged\
             -em_pc_type fieldsplit\
             -em_fieldsplit_field_pc_type lu \
             -em_fieldsplit_potential_pc_type svd\
             -em_pc_fieldsplit_type schur\
             -em_pc_fieldsplit_schur_fact_type full\
             -em_pc_fieldsplit_schur_precondition full\
             -potential_petscspace_degree 0 \
             -potential_petscdualspace_lagrange_use_moments \
             -potential_petscdualspace_lagrange_moment_order 2 \
             -field_petscspace_degree 2\
             -field_petscfe_default_quadrature_order 1\
             -field_petscspace_type sum \
             -field_petscspace_variables 2 \
             -field_petscspace_components 2 \
             -field_petscspace_sum_spaces 2 \
             -field_petscspace_sum_concatenate true \
             -field_sumcomp_0_petscspace_variables 2 \
             -field_sumcomp_0_petscspace_type tensor \
             -field_sumcomp_0_petscspace_tensor_spaces 2 \
             -field_sumcomp_0_petscspace_tensor_uniform false \
             -field_sumcomp_0_tensorcomp_0_petscspace_degree 1 \
             -field_sumcomp_0_tensorcomp_1_petscspace_degree 0 \
             -field_sumcomp_1_petscspace_variables 2 \
             -field_sumcomp_1_petscspace_type tensor \
             -field_sumcomp_1_petscspace_tensor_spaces 2 \
             -field_sumcomp_1_petscspace_tensor_uniform false \
             -field_sumcomp_1_tensorcomp_0_petscspace_degree 0 \
             -field_sumcomp_1_tensorcomp_1_petscspace_degree 1 \
             -field_petscdualspace_form_degree -1 \
             -field_petscdualspace_order 1 \
             -field_petscdualspace_lagrange_trimmed true \
             -ksp_gmres_restart 500

TEST*/
