xref: /petsc/src/snes/tutorials/ex76.c (revision 30602db00db74b7e41a0c75e517aefe6711423f0)

static char help[] = "Low Mach Flow in 2d and 3d channels with finite elements.\n\
We solve the Low Mach flow problem in a rectangular\n\
domain, using a parallel unstructured mesh (DMPLEX) to discretize it.\n\n\n";

/*F
This Low Mach flow is a steady-state isoviscous Navier-Stokes flow. We discretize using the
finite element method on an unstructured mesh. The weak form equations are

\begin{align*}
    < q, \nabla\cdot u >                                                                                     = 0
    <v, u \cdot \nabla u> + < \nabla v, \nu (\nabla u + {\nabla u}^T) > - < \nabla\cdot v, p >  - < v, f  >  = 0
    < w, u \cdot \nabla T > - < \nabla w, \alpha \nabla T > - < w, Q >                                       = 0
\end{align*}

where $\nu$ is the kinematic viscosity and $\alpha$ is thermal diffusivity.

For visualization, use

  -dm_view hdf5:$PWD/sol.h5 -sol_vec_view hdf5:$PWD/sol.h5::append -exact_vec_view hdf5:$PWD/sol.h5::append
F*/

#include <petscdmplex.h>
#include <petscsnes.h>
#include <petscds.h>
#include <petscbag.h>

typedef enum {SOL_QUADRATIC, SOL_CUBIC, NUM_SOL_TYPES} SolType;
const char *solTypes[NUM_SOL_TYPES+1] = {"quadratic", "cubic",  "unknown"};

typedef struct {
  PetscReal nu;      /* Kinematic viscosity */
  PetscReal theta;   /* Angle of pipe wall to x-axis */
  PetscReal alpha;   /* Thermal diffusivity */
  PetscReal T_in;    /* Inlet temperature*/
} Parameter;

typedef struct {
  PetscBool showError;
  PetscBag  bag;
  SolType   solType;
} AppCtx;

static PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;
  for (d = 0; d < Nc; ++d) u[d] = 0.0;
  return 0;
}

static PetscErrorCode constant(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;
  for (d = 0; d < Nc; ++d) u[d] = 1.0;
  return 0;
}

/*
  CASE: quadratic
  In 2D we use exact solution:

    u = x^2 + y^2
    v = 2x^2 - 2xy
    p = x + y - 1
    T = x + y
    f = <2x^3 + 4x^2y - 2xy^2 -4\nu + 1,  4xy^2 + 2x^2y - 2y^3 -4\nu + 1>
    Q = 3x^2 + y^2 - 2xy

  so that

(1)  \nabla \cdot u  = 2x - 2x = 0

(2)  u \cdot \nabla u - \nu \Delta u + \nabla p - f
     = <2x^3 + 4x^2y -2xy^2, 4xy^2 + 2x^2y - 2y^3> -\nu <4, 4> + <1, 1> - <2x^3 + 4x^2y - 2xy^2 -4\nu + 1,  4xy^2 + 2x^2y - 2y^3 -         4\nu + 1>  = 0

(3) u \cdot \nabla T - \alpha \Delta T - Q = 3x^2 + y^2 - 2xy - \alpha*0 - 3x^2 - y^2 + 2xy = 0
*/

static PetscErrorCode quadratic_u(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *u, void *ctx)
{
  u[0] = X[0]*X[0] + X[1]*X[1];
  u[1] = 2.0*X[0]*X[0] - 2.0*X[0]*X[1];
  return 0;
}

static PetscErrorCode linear_p(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *p, void *ctx)
{
  p[0] = X[0] + X[1] - 1.0;
  return 0;
}

static PetscErrorCode linear_T(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *T, void *ctx)
{
  T[0] = X[0] + X[1];
  return 0;
}

static void f0_quadratic_v(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                   c, d;
  PetscInt                   Nc = dim;
  const PetscReal    nu = PetscRealPart(constants[0]);

  for (c=0; c<Nc; ++c) {
    for (d=0; d<dim; ++d) f0[c] += u[d]*u_x[c*dim+d];
  }
  f0[0] -= (2*X[0]*X[0]*X[0] + 4*X[0]*X[0]*X[1] - 2*X[0]*X[1]*X[1] - 4.0*nu + 1);
  f0[1] -= (4*X[0]*X[1]*X[1] + 2*X[0]*X[0]*X[1] - 2*X[1]*X[1]*X[1] - 4.0*nu + 1);
}

static void f0_quadratic_w(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;
  for (d = 0, f0[0] = 0; d < dim; ++d) f0[0] += u[uOff[0]+d]*u_x[uOff_x[2]+d];
  f0[0] -= (3*X[0]*X[0] + X[1]*X[1] - 2*X[0]*X[1]);
}


/*
  CASE: cubic
  In 2D we use exact solution:

    u = x^3 + y^3
    v = 2x^3 - 3x^2y
    p = 3/2 x^2 + 3/2 y^2 - 1
    T = 1/2 x^2 + 1/2 y^2
    f = <3x^5 + 6x^3y^2 - 6x^2y^3 - \nu(6x + 6y), 6x^2y^3 + 3x^4y - 6xy^4 - \nu(12x - 6y) + 3y>
    Q = x^4 + xy^3 + 2x^3y - 3x^2y^2 - 2

  so that

  \nabla \cdot u = 3x^2 - 3x^2 = 0

  u \cdot \nabla u - \nu \Delta u + \nabla p - f
  = <3x^5 + 6x^3y^2 - 6x^2y^3, 6x^2y^3 + 3x^4y - 6xy^4> - \nu<6x + 6y, 12x - 6y> + <3x, 3y> - <3x^5 + 6x^3y^2 - 6x^2y^3 - \nu(6x + 6y), 6x^2y^3 + 3x^4y - 6xy^4 - \nu(12x - 6y) + 3y> = 0

  u \cdot \nabla T - \alpha\Delta T - Q = (x^3 + y^3) x + (2x^3 - 3x^2y) y - 2*\alpha - (x^4 + xy^3 + 2x^3y - 3x^2y^2 - 2)   = 0
*/

static PetscErrorCode cubic_u(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *u, void *ctx)
{
  u[0] = X[0]*X[0]*X[0] + X[1]*X[1]*X[1];
  u[1] = 2.0*X[0]*X[0]*X[0] - 3.0*X[0]*X[0]*X[1];
  return 0;
}

static PetscErrorCode quadratic_p(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *p, void *ctx)
{
  p[0] = 3.0*X[0]*X[0]/2.0 + 3.0*X[1]*X[1]/2.0 - 1.0;
  return 0;
}

static PetscErrorCode quadratic_T(PetscInt Dim, PetscReal time, const PetscReal X[], PetscInt Nf, PetscScalar *T, void *ctx)
{
  T[0] = X[0]*X[0]/2.0 + X[1]*X[1]/2.0;
  return 0;
}


static void f0_cubic_v(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                   c, d;
  PetscInt                   Nc = dim;
  const PetscReal    nu = PetscRealPart(constants[0]);

  for (c=0; c<Nc; ++c) {
    for (d=0; d<dim; ++d) f0[c] += u[d]*u_x[c*dim+d];
  }
  f0[0] -= (3*X[0]*X[0]*X[0]*X[0]*X[0] + 6*X[0]*X[0]*X[0]*X[1]*X[1] - 6*X[0]*X[0]*X[1]*X[1]*X[1] - (6*X[0]+6*X[1])*nu + 3*X[0]);
  f0[1] -= (6*X[0]*X[0]*X[1]*X[1]*X[1] + 3*X[0]*X[0]*X[0]*X[0]*X[1] - 6*X[0]*X[1]*X[1]*X[1]*X[1] - (12*X[0] - 6*X[1])*nu + 3*X[1]);
}

static void f0_cubic_w(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[])
{
  const PetscReal alpha = PetscRealPart(constants[1]);
  PetscInt        d;

  for (d = 0, f0[0] = 0; d < dim; ++d) f0[0] += u[uOff[0]+d]*u_x[uOff_x[2]+d];
  f0[0] -= (X[0]*X[0]*X[0]*X[0] + X[0]*X[1]*X[1]*X[1] + 2.0*X[0]*X[0]*X[0]*X[1] - 3.0*X[0]*X[0]*X[1]*X[1] - 2.0*alpha);
}

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[])
{
  PetscInt d;
  for (d = 0, f0[0] = 0.0; d < dim; ++d) f0[0] += u_x[d*dim+d];
}

static void f1_v(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[])
{
  const PetscReal nu = PetscRealPart(constants[0]);
  const PetscInt  Nc = dim;
  PetscInt        c, d;

  for (c = 0; c < Nc; ++c) {
    for (d = 0; d < dim; ++d) {
      f1[c*dim+d] = nu*(u_x[c*dim+d] + u_x[d*dim+c]);
      //f1[c*dim+d] = nu*u_x[c*dim+d];
    }
    f1[c*dim+c] -= u[uOff[1]];
  }
}

static void f1_w(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[])
{
  const PetscReal alpha = PetscRealPart(constants[1]);
  PetscInt d;
  for (d = 0; d < dim; ++d) f1[d] = alpha*u_x[uOff_x[2]+d];
}

/*Jacobians*/
static void g1_qu(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[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g1[d*dim+d] = 1.0;
}

static void g0_vu(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[])
{
  const PetscInt  Nc = dim;
   PetscInt            c, d;

  for (c = 0; c < Nc; ++c) {
    for (d = 0; d < dim; ++d) {
      g0[c*Nc+d] = u_x[ c*Nc+d];
    }
  }
}

static void g1_vu(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[])
{
  PetscInt NcI = dim;
  PetscInt NcJ = dim;
  PetscInt c, d, e;

  for (c = 0; c < NcI; ++c) {
    for (d = 0; d < NcJ; ++d) {
      for (e = 0; e < dim; ++e) {
        if (c == d) {
          g1[(c*NcJ+d)*dim+e] = u[e];
        }
      }
    }
  }
}

static void g2_vp(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[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g2[d*dim+d] = -1.0;
}

static void g3_vu(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[])
{
   const PetscReal      nu = PetscRealPart(constants[0]);
   const PetscInt         Nc = dim;
   PetscInt                     c, d;

  for (c = 0; c < Nc; ++c) {
    for (d = 0; d < dim; ++d) {
      g3[((c*Nc+c)*dim+d)*dim+d] += nu; // gradU
      g3[((c*Nc+d)*dim+d)*dim+c] += nu; // gradU transpose
    }
  }
}

static void g0_wu(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[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g0[d] = u_x[uOff_x[2]+d];
}


static void g1_wT(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[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g1[d] = u[uOff[0]+d];
}

static void g3_wT(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[])
{
  const PetscReal alpha = PetscRealPart(constants[1]);
  PetscInt        d;

  for (d = 0; d < dim; ++d) g3[d*dim+d] = alpha;
}

static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
{
  PetscInt       sol;
  PetscErrorCode ierr;


  PetscFunctionBeginUser;
  options->solType   = SOL_QUADRATIC;
  options->showError = PETSC_FALSE;

  ierr = PetscOptionsBegin(comm, "", "Stokes Problem Options", "DMPLEX");CHKERRQ(ierr);
  sol = options->solType;
  ierr = PetscOptionsEList("-sol_type", "The solution type", "ex62.c", solTypes, NUM_SOL_TYPES, solTypes[options->solType], &sol, NULL);CHKERRQ(ierr);
  options->solType = (SolType) sol;
  ierr = PetscOptionsBool("-show_error", "Output the error for verification", "ex62.c", options->showError, &options->showError, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();
  PetscFunctionReturn(0);
}

static PetscErrorCode SetupParameters(AppCtx *user)
{
  PetscBag       bag;
  Parameter     *p;
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  /* setup PETSc parameter bag */
  ierr = PetscBagGetData(user->bag, (void **) &p);CHKERRQ(ierr);
  ierr = PetscBagSetName(user->bag, "par", "Poiseuille flow parameters");CHKERRQ(ierr);
  bag  = user->bag;
  ierr = PetscBagRegisterReal(bag, &p->nu,    1.0,   "nu",      "Kinematic viscosity");CHKERRQ(ierr);
  ierr = PetscBagRegisterReal(bag, &p->alpha, 1.0,   "alpha",   "Thermal diffusivity");CHKERRQ(ierr);
  ierr = PetscBagRegisterReal(bag, &p->theta, 0.0,   "theta",   "Angle of pipe wall to x-axis");CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetFromOptions(*dm);CHKERRQ(ierr);
  {
    Parameter   *param;
    Vec          coordinates;
    PetscScalar *coords;
    PetscReal    theta;
    PetscInt     cdim, N, bs, i;

    ierr = DMGetCoordinateDim(*dm, &cdim);CHKERRQ(ierr);
    ierr = DMGetCoordinates(*dm, &coordinates);CHKERRQ(ierr);
    ierr = VecGetLocalSize(coordinates, &N);CHKERRQ(ierr);
    ierr = VecGetBlockSize(coordinates, &bs);CHKERRQ(ierr);
    if (bs != cdim) SETERRQ2(comm, PETSC_ERR_ARG_WRONG, "Invalid coordinate blocksize %D != embedding dimension %D", bs, cdim);
    ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
    ierr = PetscBagGetData(user->bag, (void **) &param);CHKERRQ(ierr);
    theta = param->theta;
    for (i = 0; i < N; i += cdim) {
      PetscScalar x = coords[i+0];
      PetscScalar y = coords[i+1];

      coords[i+0] = PetscCosReal(theta)*x - PetscSinReal(theta)*y;
      coords[i+1] = PetscSinReal(theta)*x + PetscCosReal(theta)*y;
    }
    ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
    ierr = DMSetCoordinates(*dm, coordinates);CHKERRQ(ierr);
  }
  ierr = DMViewFromOptions(*dm, NULL, "-dm_view");CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode SetupProblem(DM dm, AppCtx *user)
{
  PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
  PetscDS          prob;
  DMLabel          label;
  Parameter       *ctx;
  PetscInt         id;
  PetscErrorCode   ierr;

  PetscFunctionBeginUser;
  ierr = DMGetLabel(dm, "marker", &label);CHKERRQ(ierr);
  ierr = DMGetDS(dm, &prob);CHKERRQ(ierr);
  switch(user->solType){
  case SOL_QUADRATIC:
    ierr = PetscDSSetResidual(prob, 0, f0_quadratic_v, f1_v);CHKERRQ(ierr);
    ierr = PetscDSSetResidual(prob, 2, f0_quadratic_w, f1_w);CHKERRQ(ierr);

    exactFuncs[0] = quadratic_u;
    exactFuncs[1] = linear_p;
    exactFuncs[2] = linear_T;
    break;
  case SOL_CUBIC:
    ierr = PetscDSSetResidual(prob, 0, f0_cubic_v, f1_v);CHKERRQ(ierr);
    ierr = PetscDSSetResidual(prob, 2, f0_cubic_w, f1_w);CHKERRQ(ierr);

    exactFuncs[0] = cubic_u;
    exactFuncs[1] = quadratic_p;
    exactFuncs[2] = quadratic_T;
    break;
   default: SETERRQ2(PetscObjectComm((PetscObject) prob), PETSC_ERR_ARG_WRONG, "Unsupported solution type: %s (%D)", solTypes[PetscMin(user->solType, NUM_SOL_TYPES)], user->solType);
  }

  ierr = PetscDSSetResidual(prob, 1, f0_q, NULL);CHKERRQ(ierr);

  ierr = PetscDSSetJacobian(prob, 0, 0, g0_vu, g1_vu,  NULL,  g3_vu);CHKERRQ(ierr);
  ierr = PetscDSSetJacobian(prob, 0, 1, NULL, NULL,  g2_vp, NULL);CHKERRQ(ierr);
  ierr = PetscDSSetJacobian(prob, 1, 0, NULL, g1_qu, NULL,  NULL);CHKERRQ(ierr);
  ierr = PetscDSSetJacobian(prob, 2, 0, g0_wu, NULL, NULL,  NULL);CHKERRQ(ierr);
  ierr = PetscDSSetJacobian(prob, 2, 2, NULL, g1_wT, NULL,  g3_wT);CHKERRQ(ierr);
  /* Setup constants */
  {
    Parameter  *param;
    PetscScalar constants[3];

    ierr = PetscBagGetData(user->bag, (void **) &param);CHKERRQ(ierr);

    constants[0] = param->nu;
    constants[1] = param->alpha;
    constants[2] = param->theta;
    ierr = PetscDSSetConstants(prob, 3, constants);CHKERRQ(ierr);
  }
  /* Setup Boundary Conditions */
  ierr = PetscBagGetData(user->bag, (void **) &ctx);CHKERRQ(ierr);
  id   = 3;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "top wall velocity",    label, 1, &id, 0, 0, NULL, (void (*)(void)) exactFuncs[0], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 1;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "bottom wall velocity", label, 1, &id, 0, 0, NULL, (void (*)(void)) exactFuncs[0], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 2;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "right wall velocity",  label, 1, &id, 0, 0, NULL, (void (*)(void)) exactFuncs[0], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 4;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "left wall velocity",   label, 1, &id, 0, 0, NULL, (void (*)(void)) exactFuncs[0], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 3;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "top wall temp",    label, 1, &id, 2, 0, NULL, (void (*)(void)) exactFuncs[2], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 1;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "bottom wall temp", label, 1, &id, 2, 0, NULL, (void (*)(void)) exactFuncs[2], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 2;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "right wall temp",  label, 1, &id, 2, 0, NULL, (void (*)(void)) exactFuncs[2], NULL, ctx, NULL);CHKERRQ(ierr);
  id   = 4;
  ierr = PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "left wall temp",   label, 1, &id, 2, 0, NULL, (void (*)(void)) exactFuncs[2], NULL, ctx, NULL);CHKERRQ(ierr);

  /*setup exact solution.*/
  ierr = PetscDSSetExactSolution(prob, 0, exactFuncs[0], ctx);CHKERRQ(ierr);
  ierr = PetscDSSetExactSolution(prob, 1, exactFuncs[1], ctx);CHKERRQ(ierr);
  ierr = PetscDSSetExactSolution(prob, 2, exactFuncs[2], ctx);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
{
  DM              cdm   = dm;
  PetscFE         fe[3];
  Parameter      *param;
  MPI_Comm        comm;
  PetscInt        dim;
  PetscBool       simplex;
  PetscErrorCode  ierr;

  PetscFunctionBeginUser;
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMPlexIsSimplex(dm, &simplex);CHKERRQ(ierr);
  /* Create finite element */
  ierr = PetscObjectGetComm((PetscObject) dm, &comm);CHKERRQ(ierr);
  ierr = PetscFECreateDefault(comm, dim, dim, simplex, "vel_", PETSC_DEFAULT, &fe[0]);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) fe[0], "velocity");CHKERRQ(ierr);

  ierr = PetscFECreateDefault(comm, dim, 1, simplex, "pres_", PETSC_DEFAULT, &fe[1]);CHKERRQ(ierr);
  ierr = PetscFECopyQuadrature(fe[0], fe[1]);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) fe[1], "pressure");CHKERRQ(ierr);

  ierr = PetscFECreateDefault(comm, dim, 1, simplex, "temp_", PETSC_DEFAULT, &fe[2]);CHKERRQ(ierr);
  ierr = PetscFECopyQuadrature(fe[0], fe[2]);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) fe[2], "temperature");CHKERRQ(ierr);

  /* Set discretization and boundary conditions for each mesh */
  ierr = DMSetField(dm, 0, NULL, (PetscObject) fe[0]);CHKERRQ(ierr);
  ierr = DMSetField(dm, 1, NULL, (PetscObject) fe[1]);CHKERRQ(ierr);
  ierr = DMSetField(dm, 2, NULL, (PetscObject) fe[2]);CHKERRQ(ierr);
  ierr = DMCreateDS(dm);CHKERRQ(ierr);
  ierr = SetupProblem(dm, user);CHKERRQ(ierr);
  ierr = PetscBagGetData(user->bag, (void **) &param);CHKERRQ(ierr);
  while (cdm) {
    ierr = DMCopyDisc(dm, cdm);CHKERRQ(ierr);
    ierr = DMPlexCreateBasisRotation(cdm, param->theta, 0.0, 0.0);CHKERRQ(ierr);
    ierr = DMGetCoarseDM(cdm, &cdm);CHKERRQ(ierr);
  }
  ierr = PetscFEDestroy(&fe[0]);CHKERRQ(ierr);
  ierr = PetscFEDestroy(&fe[1]);CHKERRQ(ierr);
  ierr = PetscFEDestroy(&fe[2]);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode CreatePressureNullSpace(DM dm, PetscInt ofield, PetscInt nfield, MatNullSpace *nullSpace)
{
  Vec              vec;
  PetscErrorCode (*funcs[3])(PetscInt, PetscReal, const PetscReal[], PetscInt, PetscScalar *, void *) = {zero, zero, zero};
  PetscErrorCode   ierr;

  PetscFunctionBeginUser;
  if (ofield != 1) SETERRQ1(PetscObjectComm((PetscObject) dm), PETSC_ERR_ARG_WRONG, "Nullspace must be for pressure field at index 1, not %D", ofield);
  funcs[nfield] = constant;
  ierr = DMCreateGlobalVector(dm, &vec);CHKERRQ(ierr);
  ierr = DMProjectFunction(dm, 0.0, funcs, NULL, INSERT_ALL_VALUES, vec);CHKERRQ(ierr);
  ierr = VecNormalize(vec, NULL);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) vec, "Pressure Null Space");CHKERRQ(ierr);
  ierr = VecViewFromOptions(vec, NULL, "-pressure_nullspace_view");CHKERRQ(ierr);
  ierr = MatNullSpaceCreate(PetscObjectComm((PetscObject) dm), PETSC_FALSE, 1, &vec, nullSpace);CHKERRQ(ierr);
  ierr = VecDestroy(&vec);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

int main(int argc, char **argv)
{
  SNES            snes;                 /* nonlinear solver */
  DM              dm;                   /* problem definition */
  Vec             u, r;                 /* solution, residual vectors */
  AppCtx          user;                 /* user-defined work context */
  PetscErrorCode  ierr;

  ierr = PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr;
  ierr = ProcessOptions(PETSC_COMM_WORLD, &user);CHKERRQ(ierr);
  ierr = PetscBagCreate(PETSC_COMM_WORLD, sizeof(Parameter), &user.bag);CHKERRQ(ierr);
  ierr = SetupParameters(&user);CHKERRQ(ierr);
  ierr = SNESCreate(PETSC_COMM_WORLD, &snes);CHKERRQ(ierr);
  ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr);
  ierr = SNESSetDM(snes, dm);CHKERRQ(ierr);
  ierr = DMSetApplicationContext(dm, &user);CHKERRQ(ierr);
  /* Setup problem */
  ierr = SetupDiscretization(dm, &user);CHKERRQ(ierr);
  ierr = DMPlexCreateClosureIndex(dm, NULL);CHKERRQ(ierr);

  ierr = DMCreateGlobalVector(dm, &u);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) u, "Solution");CHKERRQ(ierr);
  ierr = VecDuplicate(u, &r);CHKERRQ(ierr);

  ierr = DMSetNullSpaceConstructor(dm, 1, CreatePressureNullSpace);CHKERRQ(ierr);
  ierr = DMPlexSetSNESLocalFEM(dm,&user,&user,&user);CHKERRQ(ierr);

  ierr = SNESSetFromOptions(snes);CHKERRQ(ierr);
  {
    PetscDS          ds;
    PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    void            *ctxs[3];

    ierr = DMGetDS(dm, &ds);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 0, &exactFuncs[0], &ctxs[0]);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 1, &exactFuncs[1], &ctxs[1]);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 2, &exactFuncs[2], &ctxs[2]);CHKERRQ(ierr);
    ierr = DMProjectFunction(dm, 0.0, exactFuncs, ctxs, INSERT_ALL_VALUES, u);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject) u, "Exact Solution");CHKERRQ(ierr);
    ierr = VecViewFromOptions(u, NULL, "-exact_vec_view");CHKERRQ(ierr);
  }
  ierr = DMSNESCheckFromOptions(snes, u);CHKERRQ(ierr);
  ierr = VecSet(u, 0.0);CHKERRQ(ierr);
  ierr = SNESSolve(snes, NULL, u);CHKERRQ(ierr);

  if (user.showError) {
    PetscDS          ds;
    Vec              r;
    PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    void            *ctxs[3];

    ierr = DMGetDS(dm, &ds);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 0, &exactFuncs[0], &ctxs[0]);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 1, &exactFuncs[1], &ctxs[1]);CHKERRQ(ierr);
    ierr = PetscDSGetExactSolution(ds, 2, &exactFuncs[2], &ctxs[2]);CHKERRQ(ierr);
    ierr = DMGetGlobalVector(dm, &r);CHKERRQ(ierr);
    ierr = DMProjectFunction(dm, 0.0, exactFuncs, ctxs, INSERT_ALL_VALUES, r);CHKERRQ(ierr);
    ierr = VecAXPY(r, -1.0, u);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject) r, "Solution Error");CHKERRQ(ierr);
    ierr = VecViewFromOptions(r, NULL, "-error_vec_view");CHKERRQ(ierr);
    ierr = DMRestoreGlobalVector(dm, &r);CHKERRQ(ierr);
  }
  ierr = PetscObjectSetName((PetscObject) u, "Numerical Solution");CHKERRQ(ierr);
  ierr = VecViewFromOptions(u, NULL, "-sol_vec_view");CHKERRQ(ierr);

  ierr = VecDestroy(&u);CHKERRQ(ierr);
  ierr = VecDestroy(&r);CHKERRQ(ierr);
  ierr = DMDestroy(&dm);CHKERRQ(ierr);
  ierr = SNESDestroy(&snes);CHKERRQ(ierr);
  ierr = PetscBagDestroy(&user.bag);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

/*TEST

  test:
    suffix: 2d_tri_p2_p1_p1
    requires: triangle !single
    args: -dm_plex_separate_marker -sol_type quadratic -dm_refine 0 \
      -vel_petscspace_degree 2 -pres_petscspace_degree 1 -temp_petscspace_degree 1 \
      -dmsnes_check .001 -snes_error_if_not_converged \
      -ksp_type fgmres -ksp_gmres_restart 10 -ksp_rtol 1.0e-9 -ksp_error_if_not_converged \
      -pc_type fieldsplit -pc_fieldsplit_0_fields 0,2 -pc_fieldsplit_1_fields 1 -pc_fieldsplit_type schur -pc_fieldsplit_schur_factorization_type full \
        -fieldsplit_0_pc_type lu \
        -fieldsplit_pressure_ksp_rtol 1e-10 -fieldsplit_pressure_pc_type jacobi

  test:
    # Using -dm_refine 2 -convest_num_refine 3 gives L_2 convergence rate: [2.9, 2.3, 1.9]
    suffix: 2d_tri_p2_p1_p1_conv
    requires: triangle !single
    args: -dm_plex_separate_marker -sol_type cubic -dm_refine 0 \
      -vel_petscspace_degree 2 -pres_petscspace_degree 1 -temp_petscspace_degree 1 \
      -snes_error_if_not_converged -snes_convergence_test correct_pressure -snes_convergence_estimate -convest_num_refine 1 \
      -ksp_type fgmres -ksp_gmres_restart 10 -ksp_rtol 1.0e-9 -ksp_error_if_not_converged \
      -pc_type fieldsplit -pc_fieldsplit_0_fields 0,2 -pc_fieldsplit_1_fields 1 -pc_fieldsplit_type schur -pc_fieldsplit_schur_factorization_type full \
        -fieldsplit_0_pc_type lu \
        -fieldsplit_pressure_ksp_rtol 1e-10 -fieldsplit_pressure_pc_type jacobi

  test:
    suffix: 2d_tri_p3_p2_p2
    requires: triangle !single
    args: -dm_plex_separate_marker -sol_type cubic -dm_refine 0 \
      -vel_petscspace_degree 3 -pres_petscspace_degree 2 -temp_petscspace_degree 2 \
      -dmsnes_check .001 -snes_error_if_not_converged \
      -ksp_type fgmres -ksp_gmres_restart 10 -ksp_rtol 1.0e-9 -ksp_error_if_not_converged \
      -pc_type fieldsplit -pc_fieldsplit_0_fields 0,2 -pc_fieldsplit_1_fields 1 -pc_fieldsplit_type schur -pc_fieldsplit_schur_factorization_type full \
        -fieldsplit_0_pc_type lu \
        -fieldsplit_pressure_ksp_rtol 1e-10 -fieldsplit_pressure_pc_type jacobi

TEST*/