xref: /petsc/src/ts/utils/dmplexts.c (revision ebcb266d3a667e450268eb330045d5d1ecf6fdff)
1 #include <petsc/private/dmpleximpl.h> /*I "petscdmplex.h" I*/
2 #include <petsc/private/tsimpl.h>     /*I "petscts.h" I*/
3 #include <petsc/private/snesimpl.h>
4 #include <petscds.h>
5 #include <petscfv.h>
6 
7 static PetscErrorCode DMTSConvertPlex(DM dm, DM *plex, PetscBool copy)
8 {
9   PetscBool      isPlex;
10   PetscErrorCode ierr;
11 
12   PetscFunctionBegin;
13   ierr = PetscObjectTypeCompare((PetscObject) dm, DMPLEX, &isPlex);CHKERRQ(ierr);
14   if (isPlex) {
15     *plex = dm;
16     ierr = PetscObjectReference((PetscObject) dm);CHKERRQ(ierr);
17   } else {
18     ierr = PetscObjectQuery((PetscObject) dm, "dm_plex", (PetscObject *) plex);CHKERRQ(ierr);
19     if (!*plex) {
20       ierr = DMConvert(dm,DMPLEX,plex);CHKERRQ(ierr);
21       ierr = PetscObjectCompose((PetscObject) dm, "dm_plex", (PetscObject) *plex);CHKERRQ(ierr);
22       if (copy) {
23         ierr = DMCopyDMTS(dm, *plex);CHKERRQ(ierr);
24         ierr = DMCopyDMSNES(dm, *plex);CHKERRQ(ierr);
25         ierr = DMCopyAuxiliaryVec(dm, *plex);CHKERRQ(ierr);
26       }
27     } else {
28       ierr = PetscObjectReference((PetscObject) *plex);CHKERRQ(ierr);
29     }
30   }
31   PetscFunctionReturn(0);
32 }
33 
34 /*@
35   DMPlexTSComputeRHSFunctionFVM - Form the local forcing F from the local input X using pointwise functions specified by the user
36 
37   Input Parameters:
38 + dm - The mesh
39 . t - The time
40 . locX  - Local solution
41 - user - The user context
42 
43   Output Parameter:
44 . F  - Global output vector
45 
46   Level: developer
47 
48 .seealso: DMPlexComputeJacobianActionFEM()
49 @*/
50 PetscErrorCode DMPlexTSComputeRHSFunctionFVM(DM dm, PetscReal time, Vec locX, Vec F, void *user)
51 {
52   Vec            locF;
53   IS             cellIS;
54   DM             plex;
55   PetscInt       depth;
56   PetscHashFormKey key = {NULL, 0, 0};
57   PetscErrorCode ierr;
58 
59   PetscFunctionBegin;
60   ierr = DMTSConvertPlex(dm,&plex,PETSC_TRUE);CHKERRQ(ierr);
61   ierr = DMPlexGetDepth(plex, &depth);CHKERRQ(ierr);
62   ierr = DMGetStratumIS(plex, "dim", depth, &cellIS);CHKERRQ(ierr);
63   if (!cellIS) {
64     ierr = DMGetStratumIS(plex, "depth", depth, &cellIS);CHKERRQ(ierr);
65   }
66   ierr = DMGetLocalVector(plex, &locF);CHKERRQ(ierr);
67   ierr = VecZeroEntries(locF);CHKERRQ(ierr);
68   ierr = DMPlexComputeResidual_Internal(plex, key, cellIS, time, locX, NULL, time, locF, user);CHKERRQ(ierr);
69   ierr = DMLocalToGlobalBegin(plex, locF, ADD_VALUES, F);CHKERRQ(ierr);
70   ierr = DMLocalToGlobalEnd(plex, locF, ADD_VALUES, F);CHKERRQ(ierr);
71   ierr = DMRestoreLocalVector(plex, &locF);CHKERRQ(ierr);
72   ierr = ISDestroy(&cellIS);CHKERRQ(ierr);
73   ierr = DMDestroy(&plex);CHKERRQ(ierr);
74   PetscFunctionReturn(0);
75 }
76 
77 /*@
78   DMPlexTSComputeBoundary - Insert the essential boundary values for the local input X and/or its time derivative X_t using pointwise functions specified by the user
79 
80   Input Parameters:
81 + dm - The mesh
82 . t - The time
83 . locX  - Local solution
84 . locX_t - Local solution time derivative, or NULL
85 - user - The user context
86 
87   Level: developer
88 
89 .seealso: DMPlexComputeJacobianActionFEM()
90 @*/
91 PetscErrorCode DMPlexTSComputeBoundary(DM dm, PetscReal time, Vec locX, Vec locX_t, void *user)
92 {
93   DM             plex;
94   Vec            faceGeometryFVM = NULL;
95   PetscInt       Nf, f;
96   PetscErrorCode ierr;
97 
98   PetscFunctionBegin;
99   ierr = DMTSConvertPlex(dm, &plex, PETSC_TRUE);CHKERRQ(ierr);
100   ierr = DMGetNumFields(plex, &Nf);CHKERRQ(ierr);
101   if (!locX_t) {
102     /* This is the RHS part */
103     for (f = 0; f < Nf; f++) {
104       PetscObject  obj;
105       PetscClassId id;
106 
107       ierr = DMGetField(plex, f, NULL, &obj);CHKERRQ(ierr);
108       ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
109       if (id == PETSCFV_CLASSID) {
110         ierr = DMPlexGetGeometryFVM(plex, &faceGeometryFVM, NULL, NULL);CHKERRQ(ierr);
111         break;
112       }
113     }
114   }
115   ierr = DMPlexInsertBoundaryValues(plex, PETSC_TRUE, locX, time, faceGeometryFVM, NULL, NULL);CHKERRQ(ierr);
116   ierr = DMPlexInsertTimeDerivativeBoundaryValues(plex, PETSC_TRUE, locX_t, time, faceGeometryFVM, NULL, NULL);CHKERRQ(ierr);
117   ierr = DMDestroy(&plex);CHKERRQ(ierr);
118   PetscFunctionReturn(0);
119 }
120 
121 /*@
122   DMPlexTSComputeIFunctionFEM - Form the local residual F from the local input X using pointwise functions specified by the user
123 
124   Input Parameters:
125 + dm - The mesh
126 . t - The time
127 . locX  - Local solution
128 . locX_t - Local solution time derivative, or NULL
129 - user - The user context
130 
131   Output Parameter:
132 . locF  - Local output vector
133 
134   Level: developer
135 
136 .seealso: DMPlexComputeJacobianActionFEM()
137 @*/
138 PetscErrorCode DMPlexTSComputeIFunctionFEM(DM dm, PetscReal time, Vec locX, Vec locX_t, Vec locF, void *user)
139 {
140   DM             plex;
141   IS             allcellIS;
142   PetscInt       Nds, s;
143   PetscErrorCode ierr;
144 
145   PetscFunctionBegin;
146   ierr = DMTSConvertPlex(dm, &plex, PETSC_TRUE);CHKERRQ(ierr);
147   ierr = DMPlexGetAllCells_Internal(plex, &allcellIS);CHKERRQ(ierr);
148   ierr = DMGetNumDS(dm, &Nds);CHKERRQ(ierr);
149   for (s = 0; s < Nds; ++s) {
150     PetscDS          ds;
151     IS               cellIS;
152     PetscHashFormKey key;
153 
154     ierr = DMGetRegionNumDS(dm, s, &key.label, NULL, &ds);CHKERRQ(ierr);
155     key.value = 0;
156     key.field = 0;
157     if (!key.label) {
158       ierr = PetscObjectReference((PetscObject) allcellIS);CHKERRQ(ierr);
159       cellIS = allcellIS;
160     } else {
161       IS pointIS;
162 
163       key.value = 1;
164       ierr = DMLabelGetStratumIS(key.label, key.value, &pointIS);CHKERRQ(ierr);
165       ierr = ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS);CHKERRQ(ierr);
166       ierr = ISDestroy(&pointIS);CHKERRQ(ierr);
167     }
168     ierr = DMPlexComputeResidual_Internal(plex, key, cellIS, time, locX, locX_t, time, locF, user);CHKERRQ(ierr);
169     ierr = ISDestroy(&cellIS);CHKERRQ(ierr);
170   }
171   ierr = ISDestroy(&allcellIS);CHKERRQ(ierr);
172   ierr = DMDestroy(&plex);CHKERRQ(ierr);
173   PetscFunctionReturn(0);
174 }
175 
176 /*@
177   DMPlexTSComputeIJacobianFEM - Form the local Jacobian J from the local input X using pointwise functions specified by the user
178 
179   Input Parameters:
180 + dm - The mesh
181 . t - The time
182 . locX  - Local solution
183 . locX_t - Local solution time derivative, or NULL
184 . X_tshift - The multiplicative parameter for dF/du_t
185 - user - The user context
186 
187   Output Parameter:
188 . locF  - Local output vector
189 
190   Level: developer
191 
192 .seealso: DMPlexComputeJacobianActionFEM()
193 @*/
194 PetscErrorCode DMPlexTSComputeIJacobianFEM(DM dm, PetscReal time, Vec locX, Vec locX_t, PetscReal X_tShift, Mat Jac, Mat JacP, void *user)
195 {
196   DM             plex;
197   IS             allcellIS;
198   PetscBool      hasJac, hasPrec;
199   PetscInt       Nds, s;
200   PetscErrorCode ierr;
201 
202   PetscFunctionBegin;
203   ierr = DMTSConvertPlex(dm, &plex, PETSC_TRUE);CHKERRQ(ierr);
204   ierr = DMPlexGetAllCells_Internal(plex, &allcellIS);CHKERRQ(ierr);
205   ierr = DMGetNumDS(dm, &Nds);CHKERRQ(ierr);
206   for (s = 0; s < Nds; ++s) {
207     PetscDS          ds;
208     IS               cellIS;
209     PetscHashFormKey key;
210 
211     ierr = DMGetRegionNumDS(dm, s, &key.label, NULL, &ds);CHKERRQ(ierr);
212     key.value = 0;
213     key.field = 0;
214     if (!key.label) {
215       ierr = PetscObjectReference((PetscObject) allcellIS);CHKERRQ(ierr);
216       cellIS = allcellIS;
217     } else {
218       IS pointIS;
219 
220       key.value = 1;
221       ierr = DMLabelGetStratumIS(key.label, key.value, &pointIS);CHKERRQ(ierr);
222       ierr = ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS);CHKERRQ(ierr);
223       ierr = ISDestroy(&pointIS);CHKERRQ(ierr);
224     }
225     if (!s) {
226       ierr = PetscDSHasJacobian(ds, &hasJac);CHKERRQ(ierr);
227       ierr = PetscDSHasJacobianPreconditioner(ds, &hasPrec);CHKERRQ(ierr);
228       if (hasJac && hasPrec) {ierr = MatZeroEntries(Jac);CHKERRQ(ierr);}
229       ierr = MatZeroEntries(JacP);CHKERRQ(ierr);
230     }
231     ierr = DMPlexComputeJacobian_Internal(plex, key, cellIS, time, X_tShift, locX, locX_t, Jac, JacP, user);CHKERRQ(ierr);
232     ierr = ISDestroy(&cellIS);CHKERRQ(ierr);
233   }
234   ierr = ISDestroy(&allcellIS);CHKERRQ(ierr);
235   ierr = DMDestroy(&plex);CHKERRQ(ierr);
236   PetscFunctionReturn(0);
237 }
238 
239 /*@C
240   DMTSCheckResidual - Check the residual of the exact solution
241 
242   Input Parameters:
243 + ts  - the TS object
244 . dm  - the DM
245 . t   - the time
246 . u   - a DM vector
247 . u_t - a DM vector
248 - tol - A tolerance for the check, or -1 to print the results instead
249 
250   Output Parameters:
251 . residual - The residual norm of the exact solution, or NULL
252 
253   Level: developer
254 
255 .seealso: DNTSCheckFromOptions(), DMTSCheckJacobian(), DNSNESCheckFromOptions(), DMSNESCheckDiscretization(), DMSNESCheckJacobian()
256 @*/
257 PetscErrorCode DMTSCheckResidual(TS ts, DM dm, PetscReal t, Vec u, Vec u_t, PetscReal tol, PetscReal *residual)
258 {
259   MPI_Comm       comm;
260   Vec            r;
261   PetscReal      res;
262   PetscErrorCode ierr;
263 
264   PetscFunctionBegin;
265   PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
266   PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
267   PetscValidHeaderSpecific(u, VEC_CLASSID, 3);
268   if (residual) PetscValidRealPointer(residual, 5);
269   ierr = PetscObjectGetComm((PetscObject) ts, &comm);CHKERRQ(ierr);
270   ierr = DMComputeExactSolution(dm, t, u, u_t);CHKERRQ(ierr);
271   ierr = VecDuplicate(u, &r);CHKERRQ(ierr);
272   ierr = TSComputeIFunction(ts, t, u, u_t, r, PETSC_FALSE);CHKERRQ(ierr);
273   ierr = VecNorm(r, NORM_2, &res);CHKERRQ(ierr);
274   if (tol >= 0.0) {
275     if (res > tol) SETERRQ2(comm, PETSC_ERR_ARG_WRONG, "L_2 Residual %g exceeds tolerance %g", (double) res, (double) tol);
276   } else if (residual) {
277     *residual = res;
278   } else {
279     ierr = PetscPrintf(comm, "L_2 Residual: %g\n", (double)res);CHKERRQ(ierr);
280     ierr = VecChop(r, 1.0e-10);CHKERRQ(ierr);
281     ierr = PetscObjectCompose((PetscObject) r, "__Vec_bc_zero__", (PetscObject) dm);CHKERRQ(ierr);
282     ierr = PetscObjectSetName((PetscObject) r, "Initial Residual");CHKERRQ(ierr);
283     ierr = PetscObjectSetOptionsPrefix((PetscObject)r,"res_");CHKERRQ(ierr);
284     ierr = VecViewFromOptions(r, NULL, "-vec_view");CHKERRQ(ierr);
285     ierr = PetscObjectCompose((PetscObject) r, "__Vec_bc_zero__", NULL);CHKERRQ(ierr);
286   }
287   ierr = VecDestroy(&r);CHKERRQ(ierr);
288   PetscFunctionReturn(0);
289 }
290 
291 /*@C
292   DMTSCheckJacobian - Check the Jacobian of the exact solution against the residual using the Taylor Test
293 
294   Input Parameters:
295 + ts  - the TS object
296 . dm  - the DM
297 . t   - the time
298 . u   - a DM vector
299 . u_t - a DM vector
300 - tol - A tolerance for the check, or -1 to print the results instead
301 
302   Output Parameters:
303 + isLinear - Flag indicaing that the function looks linear, or NULL
304 - convRate - The rate of convergence of the linear model, or NULL
305 
306   Level: developer
307 
308 .seealso: DNTSCheckFromOptions(), DMTSCheckResidual(), DNSNESCheckFromOptions(), DMSNESCheckDiscretization(), DMSNESCheckResidual()
309 @*/
310 PetscErrorCode DMTSCheckJacobian(TS ts, DM dm, PetscReal t, Vec u, Vec u_t, PetscReal tol, PetscBool *isLinear, PetscReal *convRate)
311 {
312   MPI_Comm       comm;
313   PetscDS        ds;
314   Mat            J, M;
315   MatNullSpace   nullspace;
316   PetscReal      dt, shift, slope, intercept;
317   PetscBool      hasJac, hasPrec, isLin = PETSC_FALSE;
318   PetscErrorCode ierr;
319 
320   PetscFunctionBegin;
321   PetscValidHeaderSpecific(ts, TS_CLASSID, 1);
322   PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
323   PetscValidHeaderSpecific(u, VEC_CLASSID, 3);
324   if (isLinear) PetscValidBoolPointer(isLinear, 5);
325   if (convRate) PetscValidRealPointer(convRate, 5);
326   ierr = PetscObjectGetComm((PetscObject) ts, &comm);CHKERRQ(ierr);
327   ierr = DMComputeExactSolution(dm, t, u, u_t);CHKERRQ(ierr);
328   /* Create and view matrices */
329   ierr = TSGetTimeStep(ts, &dt);CHKERRQ(ierr);
330   shift = 1.0/dt;
331   ierr = DMCreateMatrix(dm, &J);CHKERRQ(ierr);
332   ierr = DMGetDS(dm, &ds);CHKERRQ(ierr);
333   ierr = PetscDSHasJacobian(ds, &hasJac);CHKERRQ(ierr);
334   ierr = PetscDSHasJacobianPreconditioner(ds, &hasPrec);CHKERRQ(ierr);
335   if (hasJac && hasPrec) {
336     ierr = DMCreateMatrix(dm, &M);CHKERRQ(ierr);
337     ierr = TSComputeIJacobian(ts, t, u, u_t, shift, J, M, PETSC_FALSE);CHKERRQ(ierr);
338     ierr = PetscObjectSetName((PetscObject) M, "Preconditioning Matrix");CHKERRQ(ierr);
339     ierr = PetscObjectSetOptionsPrefix((PetscObject) M, "jacpre_");CHKERRQ(ierr);
340     ierr = MatViewFromOptions(M, NULL, "-mat_view");CHKERRQ(ierr);
341     ierr = MatDestroy(&M);CHKERRQ(ierr);
342   } else {
343     ierr = TSComputeIJacobian(ts, t, u, u_t, shift, J, J, PETSC_FALSE);CHKERRQ(ierr);
344   }
345   ierr = PetscObjectSetName((PetscObject) J, "Jacobian");CHKERRQ(ierr);
346   ierr = PetscObjectSetOptionsPrefix((PetscObject) J, "jac_");CHKERRQ(ierr);
347   ierr = MatViewFromOptions(J, NULL, "-mat_view");CHKERRQ(ierr);
348   /* Check nullspace */
349   ierr = MatGetNullSpace(J, &nullspace);CHKERRQ(ierr);
350   if (nullspace) {
351     PetscBool isNull;
352     ierr = MatNullSpaceTest(nullspace, J, &isNull);CHKERRQ(ierr);
353     if (!isNull) SETERRQ(comm, PETSC_ERR_PLIB, "The null space calculated for the system operator is invalid.");
354   }
355   /* Taylor test */
356   {
357     PetscRandom rand;
358     Vec         du, uhat, uhat_t, r, rhat, df;
359     PetscReal   h;
360     PetscReal  *es, *hs, *errors;
361     PetscReal   hMax = 1.0, hMin = 1e-6, hMult = 0.1;
362     PetscInt    Nv, v;
363 
364     /* Choose a perturbation direction */
365     ierr = PetscRandomCreate(comm, &rand);CHKERRQ(ierr);
366     ierr = VecDuplicate(u, &du);CHKERRQ(ierr);
367     ierr = VecSetRandom(du, rand);CHKERRQ(ierr);
368     ierr = PetscRandomDestroy(&rand);CHKERRQ(ierr);
369     ierr = VecDuplicate(u, &df);CHKERRQ(ierr);
370     ierr = MatMult(J, du, df);CHKERRQ(ierr);
371     /* Evaluate residual at u, F(u), save in vector r */
372     ierr = VecDuplicate(u, &r);CHKERRQ(ierr);
373     ierr = TSComputeIFunction(ts, t, u, u_t, r, PETSC_FALSE);CHKERRQ(ierr);
374     /* Look at the convergence of our Taylor approximation as we approach u */
375     for (h = hMax, Nv = 0; h >= hMin; h *= hMult, ++Nv);
376     ierr = PetscCalloc3(Nv, &es, Nv, &hs, Nv, &errors);CHKERRQ(ierr);
377     ierr = VecDuplicate(u, &uhat);CHKERRQ(ierr);
378     ierr = VecDuplicate(u, &uhat_t);CHKERRQ(ierr);
379     ierr = VecDuplicate(u, &rhat);CHKERRQ(ierr);
380     for (h = hMax, Nv = 0; h >= hMin; h *= hMult, ++Nv) {
381       ierr = VecWAXPY(uhat, h, du, u);CHKERRQ(ierr);
382       ierr = VecWAXPY(uhat_t, h*shift, du, u_t);CHKERRQ(ierr);
383       /* F(\hat u, \hat u_t) \approx F(u, u_t) + J(u, u_t) (uhat - u) + J_t(u, u_t) (uhat_t - u_t) = F(u) + h * J(u) du + h * shift * J_t(u) du = F(u) + h F' du */
384       ierr = TSComputeIFunction(ts, t, uhat, uhat_t, rhat, PETSC_FALSE);CHKERRQ(ierr);
385       ierr = VecAXPBYPCZ(rhat, -1.0, -h, 1.0, r, df);CHKERRQ(ierr);
386       ierr = VecNorm(rhat, NORM_2, &errors[Nv]);CHKERRQ(ierr);
387 
388       es[Nv] = PetscLog10Real(errors[Nv]);
389       hs[Nv] = PetscLog10Real(h);
390     }
391     ierr = VecDestroy(&uhat);CHKERRQ(ierr);
392     ierr = VecDestroy(&uhat_t);CHKERRQ(ierr);
393     ierr = VecDestroy(&rhat);CHKERRQ(ierr);
394     ierr = VecDestroy(&df);CHKERRQ(ierr);
395     ierr = VecDestroy(&r);CHKERRQ(ierr);
396     ierr = VecDestroy(&du);CHKERRQ(ierr);
397     for (v = 0; v < Nv; ++v) {
398       if ((tol >= 0) && (errors[v] > tol)) break;
399       else if (errors[v] > PETSC_SMALL)    break;
400     }
401     if (v == Nv) isLin = PETSC_TRUE;
402     ierr = PetscLinearRegression(Nv, hs, es, &slope, &intercept);CHKERRQ(ierr);
403     ierr = PetscFree3(es, hs, errors);CHKERRQ(ierr);
404     /* Slope should be about 2 */
405     if (tol >= 0) {
406       if (!isLin && PetscAbsReal(2 - slope) > tol) SETERRQ1(comm, PETSC_ERR_ARG_WRONG, "Taylor approximation convergence rate should be 2, not %0.2f", (double) slope);
407     } else if (isLinear || convRate) {
408       if (isLinear) *isLinear = isLin;
409       if (convRate) *convRate = slope;
410     } else {
411       if (!isLin) {ierr = PetscPrintf(comm, "Taylor approximation converging at order %3.2f\n", (double) slope);CHKERRQ(ierr);}
412       else        {ierr = PetscPrintf(comm, "Function appears to be linear\n");CHKERRQ(ierr);}
413     }
414   }
415   ierr = MatDestroy(&J);CHKERRQ(ierr);
416   PetscFunctionReturn(0);
417 }
418 
419 /*@C
420   DMTSCheckFromOptions - Check the residual and Jacobian functions using the exact solution by outputting some diagnostic information
421 
422   Input Parameters:
423 + ts - the TS object
424 - u  - representative TS vector
425 
426   Note: The user must call PetscDSSetExactSolution() beforehand
427 
428   Level: developer
429 @*/
430 PetscErrorCode DMTSCheckFromOptions(TS ts, Vec u)
431 {
432   DM             dm;
433   SNES           snes;
434   Vec            sol, u_t;
435   PetscReal      t;
436   PetscBool      check;
437   PetscErrorCode ierr;
438 
439   PetscFunctionBegin;
440   ierr = PetscOptionsHasName(((PetscObject)ts)->options,((PetscObject)ts)->prefix, "-dmts_check", &check);CHKERRQ(ierr);
441   if (!check) PetscFunctionReturn(0);
442   ierr = VecDuplicate(u, &sol);CHKERRQ(ierr);
443   ierr = VecCopy(u, sol);CHKERRQ(ierr);
444   ierr = TSSetSolution(ts, u);CHKERRQ(ierr);
445   ierr = TSGetDM(ts, &dm);CHKERRQ(ierr);
446   ierr = TSSetUp(ts);CHKERRQ(ierr);
447   ierr = TSGetSNES(ts, &snes);CHKERRQ(ierr);
448   ierr = SNESSetSolution(snes, u);CHKERRQ(ierr);
449 
450   ierr = TSGetTime(ts, &t);CHKERRQ(ierr);
451   ierr = DMSNESCheckDiscretization(snes, dm, t, sol, -1.0, NULL);CHKERRQ(ierr);
452   ierr = DMGetGlobalVector(dm, &u_t);CHKERRQ(ierr);
453   ierr = DMTSCheckResidual(ts, dm, t, sol, u_t, -1.0, NULL);CHKERRQ(ierr);
454   ierr = DMTSCheckJacobian(ts, dm, t, sol, u_t, -1.0, NULL, NULL);CHKERRQ(ierr);
455   ierr = DMRestoreGlobalVector(dm, &u_t);CHKERRQ(ierr);
456 
457   ierr = VecDestroy(&sol);CHKERRQ(ierr);
458   PetscFunctionReturn(0);
459 }
460