Actual source code: ex69.c


  2: #include <petscdt.h>
  3: #include <petscdraw.h>
  4: #include <petscviewer.h>
  5: #include <petscksp.h>
  6: #include <petscdmda.h>

  8: /*
  9:       Solves -Laplacian u = f,  u(-1) = u(1) = 0 with multiple spectral elements

 11:       Uses DMDA to manage the parallelization of the elements

 13:       This is not intended to be highly optimized in either memory usage or time, but strifes for simplicity.

 15: */

 17: typedef struct {
 18:   PetscInt   n;       /* number of nodes */
 19:   PetscReal *nodes;   /* GLL nodes */
 20:   PetscReal *weights; /* GLL weights */
 21: } PetscGLL;

 23: PetscErrorCode ComputeSolution(DM da, PetscGLL *gll, Vec u)
 24: {
 25:   PetscInt     j, xs, xn;
 26:   PetscScalar *uu, *xx;
 27:   PetscReal    xd;
 28:   Vec          x;

 30:   PetscFunctionBegin;
 31:   PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xn, NULL, NULL));
 32:   PetscCall(DMGetCoordinates(da, &x));
 33:   PetscCall(DMDAVecGetArray(da, x, &xx));
 34:   PetscCall(DMDAVecGetArray(da, u, &uu));
 35:   /* loop over local nodes */
 36:   for (j = xs; j < xs + xn; j++) {
 37:     xd    = xx[j];
 38:     uu[j] = (xd * xd - 1.0) * PetscCosReal(5. * PETSC_PI * xd);
 39:   }
 40:   PetscCall(DMDAVecRestoreArray(da, x, &xx));
 41:   PetscCall(DMDAVecRestoreArray(da, u, &uu));
 42:   PetscFunctionReturn(PETSC_SUCCESS);
 43: }

 45: /*
 46:       Evaluates \integral_{-1}^{1} f*v_i  where v_i is the ith basis polynomial via the GLL nodes and weights, since the v_i
 47:       basis function is zero at all nodes except the ith one the integral is simply the weight_i * f(node_i)
 48: */
 49: PetscErrorCode ComputeRhs(DM da, PetscGLL *gll, Vec b)
 50: {
 51:   PetscInt     i, j, xs, xn, n = gll->n;
 52:   PetscScalar *bb, *xx;
 53:   PetscReal    xd;
 54:   Vec          blocal, xlocal;

 56:   PetscFunctionBegin;
 57:   PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xn, NULL, NULL));
 58:   xs = xs / (n - 1);
 59:   xn = xn / (n - 1);
 60:   PetscCall(DMGetLocalVector(da, &blocal));
 61:   PetscCall(VecZeroEntries(blocal));
 62:   PetscCall(DMDAVecGetArray(da, blocal, &bb));
 63:   PetscCall(DMGetCoordinatesLocal(da, &xlocal));
 64:   PetscCall(DMDAVecGetArray(da, xlocal, &xx));
 65:   /* loop over local spectral elements */
 66:   for (j = xs; j < xs + xn; j++) {
 67:     /* loop over GLL points in each element */
 68:     for (i = 0; i < n; i++) {
 69:       xd = xx[j * (n - 1) + i];
 70:       bb[j * (n - 1) + i] += -gll->weights[i] * (-20. * PETSC_PI * xd * PetscSinReal(5. * PETSC_PI * xd) + (2. - (5. * PETSC_PI) * (5. * PETSC_PI) * (xd * xd - 1.)) * PetscCosReal(5. * PETSC_PI * xd));
 71:     }
 72:   }
 73:   PetscCall(DMDAVecRestoreArray(da, xlocal, &xx));
 74:   PetscCall(DMDAVecRestoreArray(da, blocal, &bb));
 75:   PetscCall(VecZeroEntries(b));
 76:   PetscCall(DMLocalToGlobalBegin(da, blocal, ADD_VALUES, b));
 77:   PetscCall(DMLocalToGlobalEnd(da, blocal, ADD_VALUES, b));
 78:   PetscCall(DMRestoreLocalVector(da, &blocal));
 79:   PetscFunctionReturn(PETSC_SUCCESS);
 80: }

 82: /*
 83:      Run with -build_twosided allreduce -pc_type bjacobi -sub_pc_type lu -q 16 -ksp_rtol 1.e-34 (or 1.e-14 for double precision)

 85:      -q <q> number of spectral elements to use
 86:      -N <N> maximum number of GLL points per element

 88: */
 89: int main(int argc, char **args)
 90: {
 91:   PetscGLL      gll;
 92:   PetscInt      N = 80, n, q = 8, xs, xn, j, l;
 93:   PetscReal   **A;
 94:   Mat           K;
 95:   KSP           ksp;
 96:   PC            pc;
 97:   Vec           x, b;
 98:   PetscInt     *rows;
 99:   PetscReal     norm, xc, yc, h;
100:   PetscScalar  *f;
101:   PetscDraw     draw;
102:   PetscDrawLG   lg;
103:   PetscDrawAxis axis;
104:   DM            da;
105:   PetscMPIInt   rank, size;

107:   PetscFunctionBeginUser;
108:   PetscCall(PetscInitialize(&argc, &args, NULL, NULL));
109:   PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD, &rank));
110:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
111:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-N", &N, NULL));
112:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-q", &q, NULL));

114:   PetscCall(PetscDrawCreate(PETSC_COMM_WORLD, NULL, "Log(Error norm) vs Number of GLL points", 0, 0, 500, 500, &draw));
115:   PetscCall(PetscDrawSetFromOptions(draw));
116:   PetscCall(PetscDrawLGCreate(draw, 1, &lg));
117:   PetscCall(PetscDrawLGSetUseMarkers(lg, PETSC_TRUE));
118:   PetscCall(PetscDrawLGGetAxis(lg, &axis));
119:   PetscCall(PetscDrawAxisSetLabels(axis, NULL, "Number of GLL points", "Log(Error Norm)"));

121:   for (n = 4; n < N; n += 2) {
122:     /*
123:        da contains the information about the parallel layout of the elements
124:     */
125:     PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, q * (n - 1) + 1, 1, 1, NULL, &da));
126:     PetscCall(DMSetFromOptions(da));
127:     PetscCall(DMSetUp(da));
128:     PetscCall(DMDAGetInfo(da, NULL, &q, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
129:     q = (q - 1) / (n - 1); /* number of spectral elements */

131:     /*
132:        gll simply contains the GLL node and weight values
133:     */
134:     PetscCall(PetscMalloc2(n, &gll.nodes, n, &gll.weights));
135:     PetscCall(PetscDTGaussLobattoLegendreQuadrature(n, PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA, gll.nodes, gll.weights));
136:     gll.n = n;
137:     PetscCall(DMDASetGLLCoordinates(da, gll.n, gll.nodes));

139:     /*
140:        Creates the element stiffness matrix for the given gll
141:     */
142:     PetscCall(PetscGaussLobattoLegendreElementLaplacianCreate(gll.n, gll.nodes, gll.weights, &A));

144:     /*
145:       Scale the element stiffness and weights by the size of the element
146:     */
147:     h = 2.0 / q;
148:     for (j = 0; j < n; j++) {
149:       gll.weights[j] *= .5 * h;
150:       for (l = 0; l < n; l++) A[j][l] = 2. * A[j][l] / h;
151:     }

153:     /*
154:         Create the global stiffness matrix and add the element stiffness for each local element
155:     */
156:     PetscCall(DMCreateMatrix(da, &K));
157:     PetscCall(MatSetOption(K, MAT_NEW_NONZERO_ALLOCATION_ERR, PETSC_FALSE));
158:     PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xn, NULL, NULL));
159:     xs = xs / (n - 1);
160:     xn = xn / (n - 1);
161:     PetscCall(PetscMalloc1(n, &rows));
162:     /*
163:         loop over local elements
164:     */
165:     for (j = xs; j < xs + xn; j++) {
166:       for (l = 0; l < n; l++) rows[l] = j * (n - 1) + l;
167:       PetscCall(MatSetValues(K, n, rows, n, rows, &A[0][0], ADD_VALUES));
168:     }
169:     PetscCall(MatAssemblyBegin(K, MAT_FINAL_ASSEMBLY));
170:     PetscCall(MatAssemblyEnd(K, MAT_FINAL_ASSEMBLY));

172:     PetscCall(MatCreateVecs(K, &x, &b));
173:     PetscCall(ComputeRhs(da, &gll, b));

175:     /*
176:         Replace the first and last rows/columns of the matrix with the identity to obtain the zero Dirichlet boundary conditions
177:     */
178:     rows[0] = 0;
179:     rows[1] = q * (n - 1);
180:     PetscCall(MatZeroRowsColumns(K, 2, rows, 1.0, x, b));
181:     PetscCall(PetscFree(rows));

183:     PetscCall(KSPCreate(PETSC_COMM_WORLD, &ksp));
184:     PetscCall(KSPSetOperators(ksp, K, K));
185:     PetscCall(KSPGetPC(ksp, &pc));
186:     PetscCall(PCSetType(pc, PCLU));
187:     PetscCall(KSPSetFromOptions(ksp));
188:     PetscCall(KSPSolve(ksp, b, x));

190:     /* compute the error to the continium problem */
191:     PetscCall(ComputeSolution(da, &gll, b));
192:     PetscCall(VecAXPY(x, -1.0, b));

194:     /* compute the L^2 norm of the error */
195:     PetscCall(VecGetArray(x, &f));
196:     PetscCall(PetscGaussLobattoLegendreIntegrate(gll.n, gll.nodes, gll.weights, f, &norm));
197:     PetscCall(VecRestoreArray(x, &f));
198:     norm = PetscSqrtReal(norm);
199:     PetscCall(PetscViewerASCIIPrintf(PETSC_VIEWER_STDOUT_WORLD, "L^2 norm of the error %" PetscInt_FMT " %g\n", n, (double)norm));
200:     PetscCheck(n <= 10 || norm <= 1.e-8, PETSC_COMM_WORLD, PETSC_ERR_PLIB, "Slower convergence than expected");
201:     xc = (PetscReal)n;
202:     yc = PetscLog10Real(norm);
203:     PetscCall(PetscDrawLGAddPoint(lg, &xc, &yc));
204:     PetscCall(PetscDrawLGDraw(lg));

206:     PetscCall(VecDestroy(&b));
207:     PetscCall(VecDestroy(&x));
208:     PetscCall(KSPDestroy(&ksp));
209:     PetscCall(MatDestroy(&K));
210:     PetscCall(PetscGaussLobattoLegendreElementLaplacianDestroy(gll.n, gll.nodes, gll.weights, &A));
211:     PetscCall(PetscFree2(gll.nodes, gll.weights));
212:     PetscCall(DMDestroy(&da));
213:   }
214:   PetscCall(PetscDrawLGDestroy(&lg));
215:   PetscCall(PetscDrawDestroy(&draw));
216:   PetscCall(PetscFinalize());
217:   return 0;
218: }

220: /*TEST

222:   build:
223:       requires: !complex

225:    test:
226:      requires: !single

228:    test:
229:      suffix: 2
230:      nsize: 2
231:      requires: superlu_dist

233: TEST*/