Actual source code: ex27.c

      1: static char help[] = "Poisson Problem in 2d and 3d with simplicial finite elements in both primal and mixed form.\n\
      2: We solve the Poisson problem in a rectangular\n\
      3: domain, using a parallel unstructured mesh (DMPLEX) to discretize it.\n\
      4: This example solves mixed form equation to get the flux field to calculate flux norm. We then use that for adaptive mesh refinement. \n\n\n";

      6: /*
      7: The primal (or original) Poisson problem, in the strong form, is given by,

      9: \begin{align}
     10:   - \nabla \cdot ( \nabla u ) = f
     11: \end{align}
     12: where $u$ is potential.

     14: The weak form of this equation is

     16: \begin{align}
     17:   < \nabla v, \nabla u > - < v, \nabla u \cdot \hat{n} >_\Gamma - < v, f > = 0
     18: \end{align}

     20: The mixed Poisson problem, in the strong form, is given by,

     22: \begin{align}
     23:   q - \nabla u &= 0 \\
     24:   - \nabla \cdot q &= f
     25: \end{align}
     26: where $u$ is the potential and $q$ is the flux.

     28: The weak form of this equation is

     30: \begin{align}
     31:   < t, q > + < \nabla \cdot t, u > - < t \cdot \hat{n}, u >_\Gamma &= 0 \\
     32:   <v, \nabla \cdot q> - < v, f > &= 0
     33: \end{align}

     35: We solve both primal and mixed problem and calculate the error in the flux norm, namely || e || = || q^m_h - \nabla u^p_h ||. Here superscript 'm' represents field from mixed form and 'p' represents field from the primal form.

     37: The following boundary conditions are prescribed.

     39: Primal problem:
     40: \begin{align}
     41:   u = u_0                    on \Gamma_D
     42:   \nabla u \cdot \hat{n} = g on \Gamma_N
     43: \end{align}

     45: Mixed problem:
     46: \begin{align}
     47:   u = u_0             on \Gamma_D
     48:   q \cdot \hat{n} = g on \Gamma_N
     49: \end{align}
     50:         __________\Gamma_D_____________
     51:         |                              |
     52:         |                              |
     53:         |                              |
     54: \Gamma_N                               \Gamma_N
     55:         |                              |
     56:         |                              |
     57:         |                              |
     58:         |_________\Gamma_D_____________|

     60: To visualize the automated adaptation

     62:   -dm_adapt_pre_view draw -dm_adapt_view draw -draw_pause -1 -geometry 0,0,1024,1024

     64: and to compare with a naice gradient estimator use

     66:   -adaptor_type gradient

     68: To see a sequence of adaptations

     70:   -snes_adapt_sequence 8 -adaptor_monitor_error draw::draw_lg
     71:   -dm_adapt_pre_view draw -dm_adapt_iter_view draw -dm_adapt_view draw -draw_pause 1 -geometry 0,0,1024,1024

     73: To get a better view of the by-hand process, use

     75:   -dm_view hdf5:${PWD}/mesh.h5
     76:   -primal_sol_vec_view hdf5:${PWD}/mesh.h5::append
     77:   -flux_error_vec_view hdf5:${PWD}/mesh.h5::append
     78:   -exact_error_vec_view hdf5:${PWD}/mesh.h5::append
     79:   -refine_vec_view hdf5:${PWD}/mesh.h5::append
     80:   -adapt_dm_view draw -draw_pause -1

     82: This is also possible with the automated path

     84:   -dm_view hdf5:${PWD}/mesh.h5
     85:   -adapt_primal_sol_vec_view hdf5:${PWD}/mesh.h5::append
     86:   -adapt_error_vec_view hdf5:${PWD}/mesh.h5::append
     87:   -adapt_vec_view hdf5:${PWD}/mesh.h5::append
     88: */

     90: #include <petsc/private/petscfeimpl.h>
     91: #include <petscdmplex.h>
     92: #include <petscsnes.h>
     93: #include <petscdmadaptor.h>
     94: #include <petscds.h>
     95: #include <petscviewerhdf5.h>
     96: #include <petscbag.h>

     98: PETSC_EXTERN PetscErrorCode SetupMixed(DMAdaptor, DM);

    100: typedef enum {
    101:   SOL_QUADRATIC,
    102:   SOL_TRIG,
    103:   SOL_SENSOR,
    104:   SOL_UNKNOWN,
    105:   NUM_SOL_TYPE
    106: } SolType;
    107: const char *SolTypeNames[NUM_SOL_TYPE + 4] = {"quadratic", "trig", "sensor", "unknown", "SolType", "SOL_", NULL};

    109: typedef struct {
    110:   PetscBag  param;
    111:   SolType   solType;
    112:   PetscBool byHand;
    113:   PetscInt  numAdapt;
    114: } AppCtx;

    116: typedef struct {
    117:   PetscReal k;
    118: } Parameter;

    120: /* Exact solution: u = x^2 + y^2 */
    121: static PetscErrorCode quadratic_u(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, PetscCtx ctx)
    122: {
    123:   PetscInt d;

    125:   u[0] = 0.0;
    126:   for (d = 0; d < dim; ++d) u[0] += x[d] * x[d];
    127:   return PETSC_SUCCESS;
    128: }
    129: /* Exact solution: q = (2x, 2y) */
    130: static PetscErrorCode quadratic_q(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, PetscCtx ctx)
    131: {
    132:   PetscInt c;
    133:   for (c = 0; c < Nc; ++c) u[c] = 2.0 * x[c];
    134:   return PETSC_SUCCESS;
    135: }

    137: /* Exact solution: u = sin( n \pi x ) * sin( n \pi y ) */
    138: static PetscErrorCode trig_u(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, PetscCtx ctx)
    139: {
    140:   const PetscReal n = 5.5;

    142:   u[0] = 1.0;
    143:   for (PetscInt d = 0; d < dim; ++d) u[0] *= PetscSinReal(n * PETSC_PI * x[d]);
    144:   return PETSC_SUCCESS;
    145: }
    146: static PetscErrorCode trig_q(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, PetscCtx ctx)
    147: {
    148:   const PetscReal n = 5.5;

    150:   for (PetscInt c = 0; c < Nc; c++) u[c] = n * PETSC_PI * PetscCosReal(n * PETSC_PI * x[c]) * PetscSinReal(n * PETSC_PI * x[Nc - c - 1]);
    151:   return PETSC_SUCCESS;
    152: }

    154: /*
    155: Classic hyperbolic sensor function for testing multi-scale anisotropic mesh adaptation:

    157:   f:[-1, 1]x[-1, 1] \to R,
    158:     f(x, y) = sin(50xy)/100 if |xy| > 2\pi/50 else sin(50xy)

    160: (mapped to have domain [0,1] x [0,1] in this case).
    161: */
    162: static PetscErrorCode sensor_u(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], PetscCtx ctx)
    163: {
    164:   const PetscReal xref = 2. * x[0] - 1.;
    165:   const PetscReal yref = 2. * x[1] - 1.;
    166:   const PetscReal xy   = xref * yref;

    168:   u[0] = PetscSinReal(50. * xy);
    169:   if (PetscAbsReal(xy) > 2. * PETSC_PI / 50.) u[0] *= 0.01;

    171:   return PETSC_SUCCESS;
    172: }

    174: /* Flux is (cos(50xy) * 50y/100, cos(50xy) * 50x/100) if |xy| > 2\pi/50 else (cos(50xy) * 50y, cos(50xy) * 50x) */
    175: static PetscErrorCode sensor_q(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], PetscCtx ctx)
    176: {
    177:   const PetscReal xref = 2. * x[0] - 1.;
    178:   const PetscReal yref = 2. * x[1] - 1.;
    179:   const PetscReal xy   = xref * yref;

    181:   u[0] = 50. * yref * PetscCosReal(50. * xy) * 2.0;
    182:   u[1] = 50. * xref * PetscCosReal(50. * xy) * 2.0;
    183:   if (PetscAbsReal(xy) > 2. * PETSC_PI / 50.) {
    184:     u[0] *= 0.01;
    185:     u[1] *= 0.01;
    186:   }
    187:   return PETSC_SUCCESS;
    188: }

    190: /* We set up residuals and Jacobians for the primal problem. */
    191: static void f0_quadratic_primal(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[])
    192: {
    193:   f0[0] = 4.0;
    194: }

    196: static void f0_trig_primal(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[])
    197: {
    198:   const PetscReal n = 5.5;

    200:   f0[0] = -2.0 * PetscSqr(n * PETSC_PI) * PetscSinReal(n * PETSC_PI * x[0]) * PetscSinReal(n * PETSC_PI * x[1]);
    201: }

    203: static void f0_sensor_primal(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[])
    204: {
    205:   const PetscReal xref = 2. * x[0] - 1.;
    206:   const PetscReal yref = 2. * x[1] - 1.;
    207:   const PetscReal xy   = xref * yref;

    209:   f0[0] = -2500.0 * PetscSinReal(50. * xy) * (xref * xref + yref * yref) * 4.0;
    210:   if (PetscAbsReal(xy) > 2. * PETSC_PI / 50.) f0[0] *= 0.01;
    211: }

    213: static void f1_primal(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[])
    214: {
    215:   const PetscReal k = PetscRealPart(constants[0]);

    217:   for (PetscInt d = 0; d < dim; ++d) f1[d] = k * u_x[uOff_x[0] + d];
    218: }

    220: static void f0_quadratic_bd_primal(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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    221: {
    222:   const PetscReal k = PetscRealPart(constants[0]);
    223:   PetscScalar     flux;

    225:   PetscCallAbort(PETSC_COMM_SELF, quadratic_q(dim, t, x, dim, &flux, NULL));
    226:   for (PetscInt d = 0; d < dim; ++d) f0[d] = -k * flux * n[d];
    227: }

    229: static void f0_trig_bd_primal(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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    230: {
    231:   const PetscReal k = PetscRealPart(constants[0]);
    232:   PetscScalar     flux;

    234:   PetscCallAbort(PETSC_COMM_SELF, trig_q(dim, t, x, dim, &flux, NULL));
    235:   for (PetscInt d = 0; d < dim; ++d) f0[d] = -k * flux * n[d];
    236: }

    238: static void f0_sensor_bd_primal(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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    239: {
    240:   const PetscReal k = PetscRealPart(constants[0]);
    241:   PetscScalar     flux[2];

    243:   PetscCallAbort(PETSC_COMM_SELF, sensor_q(dim, t, x, dim, flux, NULL));
    244:   for (PetscInt d = 0; d < dim; ++d) f0[d] = -k * flux[d] * n[d];
    245: }

    247: static void g3_primal_uu(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[])
    248: {
    249:   const PetscReal k = PetscRealPart(constants[0]);

    251:   for (PetscInt d = 0; d < dim; ++d) g3[d * dim + d] = k;
    252: }

    254: /* Now we set up the residuals and Jacobians mixed problem. */
    255: static void f0_mixed_quadratic_u(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[])
    256: {
    257:   f0[0] = 4.0;
    258:   for (PetscInt d = 0; d < dim; ++d) f0[0] += -u_x[uOff_x[0] + d * dim + d];
    259: }
    260: static void f0_mixed_trig_u(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[])
    261: {
    262:   PetscReal n = 5.5;

    264:   f0[0] = -2.0 * PetscSqr(n * PETSC_PI) * PetscSinReal(n * PETSC_PI * x[0]) * PetscSinReal(n * PETSC_PI * x[1]);
    265:   for (PetscInt d = 0; d < dim; ++d) f0[0] += -u_x[uOff_x[0] + d * dim + d];
    266: }
    267: static void f0_mixed_sensor_u(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[])
    268: {
    269:   const PetscReal xref = 2. * x[0] - 1.;
    270:   const PetscReal yref = 2. * x[1] - 1.;
    271:   const PetscReal xy   = xref * yref;

    273:   f0[0] = -2500.0 * PetscSinReal(50. * xy) * (xref * xref + yref * yref) * 4.0;
    274:   if (PetscAbsReal(xy) > 2. * PETSC_PI / 50.) f0[0] *= 0.01;
    275:   for (PetscInt d = 0; d < dim; ++d) f0[0] += -u_x[uOff_x[0] + d * dim + d];
    276: }

    278: static void f0_mixed_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[])
    279: {
    280:   for (PetscInt d = 0; d < dim; d++) f0[d] = u[uOff[0] + d];
    281: }

    283: static void f1_mixed_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[])
    284: {
    285:   const PetscReal k = PetscRealPart(constants[0]);

    287:   for (PetscInt d = 0; d < dim; d++) f1[d * dim + d] = k * u[uOff[1]];
    288: }

    290: static void f0_quadratic_bd_mixed_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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    291: {
    292:   const PetscReal k = PetscRealPart(constants[0]);
    293:   PetscScalar     potential;

    295:   PetscCallAbort(PETSC_COMM_SELF, quadratic_u(dim, t, x, dim, &potential, NULL));
    296:   for (PetscInt d = 0; d < dim; ++d) f0[d] = -k * potential * n[d];
    297: }

    299: static void f0_trig_bd_mixed_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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    300: {
    301:   const PetscReal k = PetscRealPart(constants[0]);
    302:   PetscScalar     potential;

    304:   PetscCallAbort(PETSC_COMM_SELF, trig_u(dim, t, x, dim, &potential, NULL));
    305:   for (PetscInt d = 0; d < dim; ++d) f0[d * dim + d] = -k * potential * n[d];
    306: }

    308: static void f0_sensor_bd_mixed_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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    309: {
    310:   const PetscReal k = PetscRealPart(constants[0]);
    311:   PetscScalar     potential;

    313:   PetscCallAbort(PETSC_COMM_SELF, sensor_u(dim, t, x, dim, &potential, NULL));
    314:   for (PetscInt d = 0; d < dim; ++d) f0[d * dim + d] = -k * potential * n[d];
    315: }

    317: /* <v, \nabla\cdot q> */
    318: static void g1_mixed_uq(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[])
    319: {
    320:   for (PetscInt d = 0; d < dim; d++) g1[d * dim + d] = -1.0;
    321: }

    323: /* < t, q> */
    324: static void g0_mixed_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[])
    325: {
    326:   for (PetscInt d = 0; d < dim; d++) g0[d * dim + d] = 1.0;
    327: }

    329: /* <\nabla\cdot t, u> = <\nabla t, Iu> */
    330: static void g2_mixed_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 g2[])
    331: {
    332:   const PetscReal k = PetscRealPart(constants[0]);

    334:   for (PetscInt d = 0; d < dim; d++) g2[d * dim + d] = k;
    335: }

    337: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *user)
    338: {
    339:   PetscFunctionBeginUser;
    340:   PetscOptionsBegin(comm, "", "Flux norm error in primal poisson problem Options", "DMPLEX");
    341:   user->byHand   = PETSC_TRUE;
    342:   user->numAdapt = 1;
    343:   user->solType  = SOL_QUADRATIC;

    345:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-by_hand", &user->byHand, NULL));
    346:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-num_adapt", &user->numAdapt, NULL));
    347:   PetscCall(PetscOptionsEnum("-sol_type", "Type of exact solution", "ex27.c", SolTypeNames, (PetscEnum)user->solType, (PetscEnum *)&user->solType, NULL));
    348:   PetscOptionsEnd();
    349:   PetscFunctionReturn(PETSC_SUCCESS);
    350: }

    352: static PetscErrorCode SetupParameters(PetscBag bag, AppCtx *user)
    353: {
    354:   Parameter *param;

    356:   PetscFunctionBeginUser;
    357:   PetscCall(PetscBagGetData(bag, &param));
    358:   PetscCall(PetscBagSetName(bag, "par", "Poisson parameters"));
    359:   PetscCall(PetscBagRegisterReal(bag, &param->k, 1.0, "k", "Thermal conductivity"));
    360:   PetscCall(PetscBagSetFromOptions(bag));
    361:   PetscFunctionReturn(PETSC_SUCCESS);
    362: }

    364: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
    365: {
    366:   PetscFunctionBeginUser;
    367:   PetscCall(DMCreate(comm, dm));
    368:   PetscCall(DMSetType(*dm, DMPLEX));
    369:   PetscCall(DMSetFromOptions(*dm));
    370:   PetscCall(DMSetApplicationContext(*dm, &user));
    371:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
    372:   PetscFunctionReturn(PETSC_SUCCESS);
    373: }

    375: static PetscErrorCode SetupPrimalProblem(DM dm, AppCtx *user)
    376: {
    377:   PetscDS       ds;
    378:   DMLabel       label;
    379:   PetscInt      id, bd;
    380:   Parameter    *param;
    381:   PetscWeakForm wf;

    383:   PetscFunctionBeginUser;
    384:   PetscCall(DMGetDS(dm, &ds));
    385:   PetscCall(DMGetLabel(dm, "marker", &label));

    387:   PetscCall(PetscDSSetJacobian(ds, 0, 0, NULL, NULL, NULL, g3_primal_uu));

    389:   switch (user->solType) {
    390:   case SOL_QUADRATIC:
    391:     PetscCall(PetscDSSetResidual(ds, 0, f0_quadratic_primal, f1_primal));

    393:     id = 1;
    394:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "bottom wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_u, NULL, user, NULL));

    396:     id = 2;
    397:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "right wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    398:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    399:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_quadratic_bd_primal, 0, NULL));

    401:     id = 3;
    402:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "top wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_u, NULL, user, NULL));

    404:     id = 4;
    405:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "left wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    406:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    407:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_quadratic_bd_primal, 0, NULL));

    409:     PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_u, user));
    410:     break;
    411:   case SOL_TRIG:
    412:     PetscCall(PetscDSSetResidual(ds, 0, f0_trig_primal, f1_primal));

    414:     id = 1;
    415:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "bottom wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)trig_u, NULL, user, NULL));

    417:     id = 2;
    418:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "right wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    419:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    420:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_trig_bd_primal, 0, NULL));

    422:     id = 3;
    423:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "top wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)trig_u, NULL, user, NULL));

    425:     id = 4;
    426:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "left wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    427:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    428:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_trig_bd_primal, 0, NULL));

    430:     PetscCall(PetscDSSetExactSolution(ds, 0, trig_u, user));
    431:     break;
    432:   case SOL_SENSOR:
    433:     PetscCall(PetscDSSetResidual(ds, 0, f0_sensor_primal, f1_primal));

    435:     id = 1;
    436:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "bottom wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)sensor_u, NULL, user, NULL));

    438:     id = 2;
    439:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "right wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    440:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    441:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_sensor_bd_primal, 0, NULL));

    443:     id = 3;
    444:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "top wall primal potential", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)sensor_u, NULL, user, NULL));

    446:     id = 4;
    447:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "left wall flux", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    448:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    449:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_sensor_bd_primal, 0, NULL));

    451:     PetscCall(PetscDSSetExactSolution(ds, 0, sensor_u, user));
    452:     break;
    453:   default:
    454:     SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Invalid exact solution type %s", SolTypeNames[PetscMin(user->solType, SOL_UNKNOWN)]);
    455:   }

    457:   /* Setup constants */
    458:   {
    459:     PetscCall(PetscBagGetData(user->param, &param));
    460:     PetscScalar constants[1];

    462:     constants[0] = param->k;

    464:     PetscCall(PetscDSSetConstants(ds, 1, constants));
    465:   }
    466:   PetscFunctionReturn(PETSC_SUCCESS);
    467: }

    469: static PetscErrorCode SetupPrimalDiscretization(DM dm, AppCtx *user)
    470: {
    471:   DM             cdm = dm;
    472:   PetscFE        fe[1];
    473:   DMPolytopeType ct;
    474:   PetscInt       dim, cStart;
    475:   MPI_Comm       comm;

    477:   PetscFunctionBeginUser;
    478:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
    479:   PetscCall(DMGetDimension(dm, &dim));

    481:   /* Create finite element */
    482:   PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, NULL));
    483:   PetscCall(DMPlexGetCellType(dm, cStart, &ct));
    484:   PetscCall(PetscFECreateByCell(comm, dim, 1, ct, "primal_potential_", PETSC_DEFAULT, &fe[0]));
    485:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "primal potential"));

    487:   /* Set discretization and boundary conditions for each mesh */
    488:   PetscCall(DMSetField(dm, 0, NULL, (PetscObject)fe[0]));
    489:   PetscCall(DMCreateDS(dm));
    490:   while (cdm) {
    491:     PetscCall(DMCopyDisc(dm, cdm));
    492:     PetscCall(DMGetCoarseDM(cdm, &cdm));
    493:   }

    495:   PetscCall(PetscFEDestroy(&fe[0]));
    496:   PetscFunctionReturn(PETSC_SUCCESS);
    497: }

    499: static PetscErrorCode SetupMixedProblem(DM dm, AppCtx *user)
    500: {
    501:   PetscDS       ds;
    502:   DMLabel       label;
    503:   PetscInt      id, bd;
    504:   Parameter    *param;
    505:   PetscWeakForm wf;

    507:   PetscFunctionBeginUser;
    508:   PetscCall(DMGetDS(dm, &ds));
    509:   PetscCall(DMGetLabel(dm, "marker", &label));

    511:   /* Residual terms */
    512:   PetscCall(PetscDSSetResidual(ds, 0, f0_mixed_q, f1_mixed_q));

    514:   PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_mixed_qq, NULL, NULL, NULL));
    515:   PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_mixed_qu, NULL));

    517:   PetscCall(PetscDSSetJacobian(ds, 1, 0, NULL, g1_mixed_uq, NULL, NULL));

    519:   switch (user->solType) {
    520:   case SOL_QUADRATIC:
    521:     PetscCall(PetscDSSetResidual(ds, 1, f0_mixed_quadratic_u, NULL));

    523:     id = 1;
    524:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral bottom wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    525:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    526:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_quadratic_bd_mixed_q, 0, NULL));

    528:     id = 2;
    529:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "right wall flux", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_q, NULL, user, NULL));

    531:     id = 4;
    532:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "left wall flux", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_q, NULL, user, NULL));

    534:     id = 3;
    535:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral top wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    536:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    537:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_quadratic_bd_mixed_q, 0, NULL));

    539:     PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_q, user));
    540:     PetscCall(PetscDSSetExactSolution(ds, 1, quadratic_u, user));
    541:     break;
    542:   case SOL_TRIG:
    543:     PetscCall(PetscDSSetResidual(ds, 1, f0_mixed_trig_u, NULL));

    545:     id = 1;
    546:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral bottom wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    547:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    548:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_trig_bd_mixed_q, 0, NULL));

    550:     id = 2;
    551:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "right wall flux", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)trig_q, NULL, user, NULL));

    553:     id = 4;
    554:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "left wall flux", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)trig_q, NULL, user, NULL));

    556:     id = 3;
    557:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral top wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    558:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    559:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_trig_bd_mixed_q, 0, NULL));

    561:     PetscCall(PetscDSSetExactSolution(ds, 0, trig_q, user));
    562:     PetscCall(PetscDSSetExactSolution(ds, 1, trig_u, user));
    563:     break;
    564:   case SOL_SENSOR:
    565:     PetscCall(PetscDSSetResidual(ds, 1, f0_mixed_sensor_u, NULL));

    567:     id = 1;
    568:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral bottom wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    569:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    570:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_sensor_bd_mixed_q, 0, NULL));

    572:     id = 2;
    573:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "right wall flux", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)sensor_q, NULL, user, NULL));

    575:     id = 4;
    576:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "left wall flux", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)sensor_q, NULL, user, NULL));

    578:     id = 3;
    579:     PetscCall(DMAddBoundary(dm, DM_BC_NATURAL, "Dirichlet Bd Integral top wall", label, 1, &id, 0, 0, NULL, NULL, NULL, user, &bd));
    580:     PetscCall(PetscDSGetBoundary(ds, bd, &wf, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
    581:     PetscCall(PetscWeakFormSetIndexBdResidual(wf, label, id, 0, 0, 0, f0_sensor_bd_mixed_q, 0, NULL));

    583:     PetscCall(PetscDSSetExactSolution(ds, 0, sensor_q, user));
    584:     PetscCall(PetscDSSetExactSolution(ds, 1, sensor_u, user));
    585:     break;
    586:   default:
    587:     SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Invalid exact solution type %s", SolTypeNames[PetscMin(user->solType, SOL_UNKNOWN)]);
    588:   }

    590:   /* Setup constants */
    591:   {
    592:     PetscCall(PetscBagGetData(user->param, &param));
    593:     PetscScalar constants[1];

    595:     constants[0] = param->k;

    597:     PetscCall(PetscDSSetConstants(ds, 1, constants));
    598:   }
    599:   PetscFunctionReturn(PETSC_SUCCESS);
    600: }

    602: static PetscErrorCode SetupMixedDiscretization(DM dm, AppCtx *user)
    603: {
    604:   DM             cdm = dm;
    605:   PetscFE        fe[2];
    606:   DMPolytopeType ct;
    607:   PetscInt       dim, cStart;
    608:   MPI_Comm       comm;

    610:   PetscFunctionBeginUser;
    611:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
    612:   PetscCall(DMGetDimension(dm, &dim));

    614:   /* Create finite element */
    615:   PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, NULL));
    616:   PetscCall(DMPlexGetCellType(dm, cStart, &ct));
    617:   PetscCall(PetscFECreateByCell(comm, dim, dim, ct, "mixed_flux_", PETSC_DEFAULT, &fe[0]));
    618:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "mixed flux"));
    619:   /* NOTE:  Set the same quadrature order as the primal problem here or use the command line option. */

    621:   PetscCall(PetscFECreateByCell(comm, dim, 1, ct, "mixed_potential_", PETSC_DEFAULT, &fe[1]));
    622:   PetscCall(PetscFECopyQuadrature(fe[0], fe[1]));
    623:   PetscCall(PetscObjectSetName((PetscObject)fe[1], "mixed potential"));

    625:   /* Set discretization and boundary conditions for each mesh */
    626:   PetscCall(DMSetField(dm, 0, NULL, (PetscObject)fe[0]));
    627:   PetscCall(DMSetField(dm, 1, NULL, (PetscObject)fe[1]));
    628:   PetscCall(DMCreateDS(dm));
    629:   while (cdm) {
    630:     PetscCall(DMCopyDisc(dm, cdm));
    631:     PetscCall(DMGetCoarseDM(cdm, &cdm));
    632:   }
    633:   PetscCall(PetscFEDestroy(&fe[0]));
    634:   PetscCall(PetscFEDestroy(&fe[1]));
    635:   PetscFunctionReturn(PETSC_SUCCESS);
    636: }

    638: PetscErrorCode SetupMixed(DMAdaptor adaptor, DM mdm)
    639: {
    640:   AppCtx *ctx;

    642:   PetscFunctionBeginUser;
    643:   PetscCall(DMGetApplicationContext(mdm, &ctx));
    644:   PetscCall(SetupMixedDiscretization(mdm, ctx));
    645:   PetscCall(SetupMixedProblem(mdm, ctx));
    646:   PetscFunctionReturn(PETSC_SUCCESS);
    647: }

    649: int main(int argc, char **argv)
    650: {
    651:   DM        dm, mdm;               /* problem specification */
    652:   SNES      snes, msnes;           /* nonlinear solvers */
    653:   Vec       u, mu;                 /* solution vectors */
    654:   Vec       fluxError, exactError; /* Element wise error vector */
    655:   PetscReal fluxNorm, exactNorm;   /* Flux error norm */
    656:   AppCtx    user;                  /* user-defined work context */

    658:   PetscFunctionBeginUser;
    659:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    660:   PetscCall(PetscBagCreate(PETSC_COMM_WORLD, sizeof(Parameter), &user.param));
    661:   PetscCall(SetupParameters(user.param, &user));
    662:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));

    664:   // Set up and solve primal problem
    665:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
    666:   PetscCall(DMSetApplicationContext(dm, &user));
    667:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
    668:   PetscCall(SNESSetDM(snes, dm));

    670:   PetscCall(SetupPrimalDiscretization(dm, &user));
    671:   PetscCall(SetupPrimalProblem(dm, &user));
    672:   PetscCall(DMCreateGlobalVector(dm, &u));
    673:   PetscCall(VecSet(u, 0.0));
    674:   PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user));
    675:   PetscCall(SNESSetFromOptions(snes));
    676:   PetscCall(DMSNESCheckFromOptions(snes, u));

    678:   for (PetscInt a = 0; a < user.numAdapt; ++a) {
    679:     if (a > 0) {
    680:       char prefix[16];

    682:       PetscCall(PetscSNPrintf(prefix, 16, "a%d_", (int)a));
    683:       PetscCall(SNESSetOptionsPrefix(snes, prefix));
    684:     }
    685:     PetscCall(SNESSolve(snes, NULL, u));

    687:     // Needed if you allow SNES to refine
    688:     PetscCall(SNESGetSolution(snes, &u));
    689:     PetscCall(VecGetDM(u, &dm));
    690:   }

    692:   PetscCall(PetscObjectSetName((PetscObject)u, "Primal Solution "));
    693:   PetscCall(VecViewFromOptions(u, NULL, "-primal_sol_vec_view"));

    695:   if (user.byHand) {
    696:     // Set up and solve mixed problem
    697:     PetscCall(DMClone(dm, &mdm));
    698:     PetscCall(SNESCreate(PETSC_COMM_WORLD, &msnes));
    699:     PetscCall(SNESSetOptionsPrefix(msnes, "mixed_"));
    700:     PetscCall(SNESSetDM(msnes, mdm));

    702:     PetscCall(SetupMixedDiscretization(mdm, &user));
    703:     PetscCall(SetupMixedProblem(mdm, &user));
    704:     PetscCall(DMCreateGlobalVector(mdm, &mu));
    705:     PetscCall(VecSet(mu, 0.0));
    706:     PetscCall(DMPlexSetSNESLocalFEM(mdm, PETSC_FALSE, &user));
    707:     PetscCall(SNESSetFromOptions(msnes));

    709:     PetscCall(DMSNESCheckFromOptions(msnes, mu));
    710:     PetscCall(SNESSolve(msnes, NULL, mu));
    711:     PetscCall(PetscObjectSetName((PetscObject)mu, "Mixed Solution "));
    712:     PetscCall(VecViewFromOptions(mu, NULL, "-mixed_sol_vec_view"));

    714:     // Create the error space of piecewise constants
    715:     DM             dmErr;
    716:     PetscFE        feErr;
    717:     DMPolytopeType ct;
    718:     PetscInt       dim, cStart;

    720:     PetscCall(DMClone(dm, &dmErr));
    721:     PetscCall(DMGetDimension(dmErr, &dim));
    722:     PetscCall(DMPlexGetHeightStratum(dmErr, 0, &cStart, NULL));
    723:     PetscCall(DMPlexGetCellType(dmErr, cStart, &ct));
    724:     PetscCall(PetscFECreateLagrangeByCell(PETSC_COMM_SELF, dim, 1, ct, 0, PETSC_DEFAULT, &feErr));
    725:     PetscCall(PetscObjectSetName((PetscObject)feErr, "Error"));
    726:     PetscCall(DMSetField(dmErr, 0, NULL, (PetscObject)feErr));
    727:     PetscCall(PetscFEDestroy(&feErr));
    728:     PetscCall(DMCreateDS(dmErr));
    729:     PetscCall(DMViewFromOptions(dmErr, NULL, "-dmerr_view"));

    731:     // Compute the flux norm
    732:     PetscCall(DMGetGlobalVector(dmErr, &fluxError));
    733:     PetscCall(PetscObjectSetName((PetscObject)fluxError, "Flux Error"));
    734:     PetscCall(DMGetGlobalVector(dmErr, &exactError));
    735:     PetscCall(PetscObjectSetName((PetscObject)exactError, "Analytical Error"));
    736:     PetscCall(DMPlexComputeL2FluxDiffVec(u, 0, mu, 0, fluxError));
    737:     {
    738:       PetscDS             ds;
    739:       PetscSimplePointFn *func[2] = {NULL, NULL};
    740:       void               *ctx[2]  = {NULL, NULL};

    742:       PetscCall(DMGetDS(mdm, &ds));
    743:       PetscCall(PetscDSGetExactSolution(ds, 0, &func[0], &ctx[0]));
    744:       PetscCall(DMPlexComputeL2DiffVec(mdm, 0.0, func, ctx, mu, exactError));
    745:     }
    746:     PetscCall(VecNorm(fluxError, NORM_2, &fluxNorm));
    747:     PetscCall(VecNorm(exactError, NORM_2, &exactNorm));
    748:     PetscCall(PetscPrintf(PETSC_COMM_SELF, "Flux error norm = %g\t Exact flux error norm = %g\n", (double)fluxNorm, (double)exactNorm));
    749:     PetscCall(VecViewFromOptions(fluxError, NULL, "-flux_error_vec_view"));
    750:     PetscCall(VecViewFromOptions(exactError, NULL, "-exact_error_vec_view"));

    752:     // Adaptive refinement based on calculated error
    753:     DM        rdm;
    754:     VecTagger refineTag;
    755:     DMLabel   adaptLabel;
    756:     IS        refineIS;
    757:     Vec       ref;

    759:     PetscCall(DMLabelCreate(PETSC_COMM_WORLD, "adapt", &adaptLabel));
    760:     PetscCall(DMLabelSetDefaultValue(adaptLabel, DM_ADAPT_COARSEN));
    761:     PetscCall(VecTaggerCreate(PETSC_COMM_WORLD, &refineTag));
    762:     PetscCall(VecTaggerSetFromOptions(refineTag));
    763:     PetscCall(VecTaggerSetUp(refineTag));
    764:     PetscCall(PetscObjectViewFromOptions((PetscObject)refineTag, NULL, "-tag_view"));

    766:     PetscCall(VecTaggerComputeIS(refineTag, fluxError, &refineIS, NULL));
    767:     PetscCall(VecTaggerDestroy(&refineTag));
    768:     PetscCall(DMLabelSetStratumIS(adaptLabel, DM_ADAPT_REFINE, refineIS));
    769:     PetscCall(ISViewFromOptions(refineIS, NULL, "-refine_is_view"));
    770:     PetscCall(ISDestroy(&refineIS));

    772:     PetscCall(DMPlexCreateLabelField(dm, adaptLabel, &ref));
    773:     PetscCall(VecViewFromOptions(ref, NULL, "-refine_vec_view"));
    774:     PetscCall(VecDestroy(&ref));

    776:     // Mark adaptation phase with prefix ref_
    777:     PetscCall(PetscObjectSetOptionsPrefix((PetscObject)dm, "adapt_"));
    778:     PetscCall(DMAdaptLabel(dm, adaptLabel, &rdm));
    779:     PetscCall(PetscObjectSetOptionsPrefix((PetscObject)dm, NULL));
    780:     PetscCall(PetscObjectSetName((PetscObject)rdm, "Adaptively Refined DM"));
    781:     PetscCall(DMViewFromOptions(rdm, NULL, "-adapt_dm_view"));
    782:     PetscCall(DMDestroy(&rdm));
    783:     PetscCall(DMLabelDestroy(&adaptLabel));

    785:     // Destroy the error structures
    786:     PetscCall(DMRestoreGlobalVector(dmErr, &fluxError));
    787:     PetscCall(DMRestoreGlobalVector(dmErr, &exactError));
    788:     PetscCall(DMDestroy(&dmErr));

    790:     // Destroy the mixed structures
    791:     PetscCall(VecDestroy(&mu));
    792:     PetscCall(DMDestroy(&mdm));
    793:     PetscCall(SNESDestroy(&msnes));
    794:   }

    796:   // Destroy the primal structures
    797:   PetscCall(VecDestroy(&u));
    798:   PetscCall(DMDestroy(&dm));
    799:   PetscCall(SNESDestroy(&snes));
    800:   PetscCall(PetscBagDestroy(&user.param));
    801:   PetscCall(PetscFinalize());
    802:   return 0;
    803: }

    805: /*TEST

    807:   # Tests using the explicit code above
    808:   testset:
    809:     suffix: 2d_p2_rt0p0_byhand
    810:     requires: triangle
    811:     args: -dm_plex_box_faces 1,1 -dm_plex_box_lower 0,0 -dm_plex_box_upper 1,1 -dm_refine 3 \
    812:           -primal_potential_petscspace_degree 2 \
    813:           -mixed_potential_petscdualspace_lagrange_continuity 0 \
    814:           -mixed_flux_petscspace_type ptrimmed \
    815:           -mixed_flux_petscspace_components 2 \
    816:           -mixed_flux_petscspace_ptrimmed_form_degree -1 \
    817:           -mixed_flux_petscdualspace_order 1 \
    818:           -mixed_flux_petscdualspace_form_degree -1 \
    819:           -mixed_flux_petscdualspace_lagrange_trimmed true \
    820:           -mixed_flux_petscfe_default_quadrature_order 2 \
    821:           -vec_tagger_type cdf -vec_tagger_box 0.9,1.0 \
    822:             -tag_view \
    823:             -adapt_dm_adaptor cellrefiner -adapt_dm_plex_transform_type refine_sbr \
    824:           -dmsnes_check 0.001 -mixed_dmsnes_check 0.001 -pc_type jacobi -mixed_pc_type jacobi
    825:     test:
    826:       suffix: quadratic
    827:       args: -sol_type quadratic
    828:     test:
    829:       suffix: trig
    830:       args: -sol_type trig
    831:     test:
    832:       suffix: sensor
    833:       args: -sol_type sensor

    835:   # Tests using the embedded adaptor in SNES
    836:   testset:
    837:     suffix: 2d_p2_rt0p0
    838:     requires: triangle defined(PETSC_HAVE_EXECUTABLE_EXPORT)
    839:     args: -dm_plex_box_faces 1,1 -dm_plex_box_lower 0,0 -dm_plex_box_upper 1,1 -dm_refine 3 \
    840:           -primal_potential_petscspace_degree 2 \
    841:           -mixed_potential_petscdualspace_lagrange_continuity 0 \
    842:           -mixed_flux_petscspace_type ptrimmed \
    843:           -mixed_flux_petscspace_components 2 \
    844:           -mixed_flux_petscspace_ptrimmed_form_degree -1 \
    845:           -mixed_flux_petscdualspace_order 1 \
    846:           -mixed_flux_petscdualspace_form_degree -1 \
    847:           -mixed_flux_petscdualspace_lagrange_trimmed true \
    848:           -mixed_flux_petscfe_default_quadrature_order 2 \
    849:           -by_hand 0 \
    850:           -refine_vec_tagger_type cdf -refine_vec_tagger_box 0.9,1.0 \
    851:             -snes_adapt_view \
    852:             -adapt_dm_adaptor cellrefiner -adapt_dm_plex_transform_type refine_sbr \
    853:             -adaptor_criterion label -adaptor_type flux -adaptor_mixed_setup_function SetupMixed \
    854:           -snes_adapt_sequence 1 -pc_type jacobi -mixed_pc_type jacobi
    855:     test:
    856:       suffix: quadratic
    857:       args: -sol_type quadratic -adaptor_monitor_error
    858:     test:
    859:       suffix: trig
    860:       args: -sol_type trig -adaptor_monitor_error
    861:     test:
    862:       suffix: sensor
    863:       args: -sol_type sensor

    865:   # Tests using multiple adaptor loops
    866:   testset:
    867:     suffix: 2d_p2_rt0p0_a2
    868:     requires: triangle defined(PETSC_HAVE_EXECUTABLE_EXPORT)
    869:     args: -dm_plex_box_faces 1,1 -dm_plex_box_lower 0,0 -dm_plex_box_upper 1,1 -dm_refine 3 \
    870:           -primal_potential_petscspace_degree 2 \
    871:           -mixed_potential_petscdualspace_lagrange_continuity 0 \
    872:           -mixed_flux_petscspace_type ptrimmed \
    873:           -mixed_flux_petscspace_components 2 \
    874:           -mixed_flux_petscspace_ptrimmed_form_degree -1 \
    875:           -mixed_flux_petscdualspace_order 1 \
    876:           -mixed_flux_petscdualspace_form_degree -1 \
    877:           -mixed_flux_petscdualspace_lagrange_trimmed true \
    878:           -mixed_flux_petscfe_default_quadrature_order 2 \
    879:           -by_hand 0 \
    880:           -num_adapt 2 \
    881:           -refine_vec_tagger_type cdf -refine_vec_tagger_box 0.9,1.0 \
    882:             -snes_adapt_view \
    883:             -adapt_dm_adaptor cellrefiner -adapt_dm_plex_transform_type refine_sbr \
    884:             -adaptor_criterion label -adaptor_type gradient -adaptor_mixed_setup_function SetupMixed \
    885:           -snes_adapt_sequence 2 -pc_type jacobi \
    886:           -a1_refine_vec_tagger_type cdf -a1_refine_vec_tagger_box 0.9,1.0 \
    887:             -a1_snes_adapt_view \
    888:             -a1_adaptor_criterion label -a1_adaptor_type flux -a1_adaptor_mixed_setup_function SetupMixed \
    889:           -a1_snes_adapt_sequence 1 -a1_pc_type jacobi -a1_mixed_pc_type jacobi
    890:     test:
    891:       suffix: sensor
    892:       args: -sol_type sensor

    894: TEST*/