Actual source code: ex15.c

petsc-master 2019-12-10
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  1: static const char help[] = "p-Bratu nonlinear PDE in 2d.\n\
  2: We solve the  p-Laplacian (nonlinear diffusion) combined with\n\
  3: the Bratu (solid fuel ignition) nonlinearity in a 2D rectangular\n\
  4: domain, using distributed arrays (DAs) to partition the parallel grid.\n\
  5: The command line options include:\n\
  6:   -p <2>: `p' in p-Laplacian term\n\
  7:   -epsilon <1e-05>: Strain-regularization in p-Laplacian\n\
  8:   -lambda <6>: Bratu parameter\n\
  9:   -blocks <bx,by>: number of coefficient interfaces in x and y direction\n\
 10:   -kappa <1e-3>: diffusivity in odd regions\n\
 11: \n";


The $p$-Bratu problem is a combination of the $p$-Laplacian (nonlinear diffusion) and the Brutu solid fuel ignition problem.
This problem is modeled by the partial differential equation

\begin{equation*}
-\nabla\cdot (\eta \nabla u) - \lambda \exp(u) = 0
\end{equation*}

on $\Omega = (-1,1)^2$ with closure

\begin{align*}
\eta(\gamma) &= (\epsilon^2 + \gamma)^{(p-2)/2} & \gamma &= \frac 1 2 |\nabla u|^2
\end{align*}

and boundary conditions $u = 0$ for $(x,y) \in \partial \Omega$

A 9-point finite difference stencil is used to discretize
the boundary value problem to obtain a nonlinear system of equations.
This would be a 5-point stencil if not for the $p$-Laplacian's nonlinearity.
 35: /*
 36:       mpiexec -n 2 ./ex15 -snes_monitor -ksp_monitor log_summary
 37: */

 39: /*
 40:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 41:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
 42:    file automatically includes:
 43:      petsc.h       - base PETSc routines   petscvec.h - vectors
 44:      petscsys.h    - system routines       petscmat.h - matrices
 45:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 46:      petscviewer.h - viewers               petscpc.h  - preconditioners
 47:      petscksp.h   - linear solvers
 48: */

 50:  #include <petscdm.h>
 51:  #include <petscdmda.h>
 52:  #include <petscsnes.h>

 54: typedef enum {JAC_BRATU,JAC_PICARD,JAC_STAR,JAC_NEWTON} JacType;
 55: static const char *const JacTypes[] = {"BRATU","PICARD","STAR","NEWTON","JacType","JAC_",0};

 57: /*
 58:    User-defined application context - contains data needed by the
 59:    application-provided call-back routines, FormJacobianLocal() and
 60:    FormFunctionLocal().
 61: */
 62: typedef struct {
 63:   PetscReal   lambda;         /* Bratu parameter */
 64:   PetscReal   p;              /* Exponent in p-Laplacian */
 65:   PetscReal   epsilon;        /* Regularization */
 66:   PetscReal   source;         /* Source term */
 67:   JacType     jtype;          /* What type of Jacobian to assemble */
 68:   PetscBool   picard;
 69:   PetscInt    blocks[2];
 70:   PetscReal   kappa;
 71:   PetscInt    initial;        /* initial conditions type */
 72: } AppCtx;

 74: /*
 75:    User-defined routines
 76: */
 77: static PetscErrorCode FormRHS(AppCtx*,DM,Vec);
 78: static PetscErrorCode FormInitialGuess(AppCtx*,DM,Vec);
 79: static PetscErrorCode FormFunctionLocal(DMDALocalInfo*,PetscScalar**,PetscScalar**,AppCtx*);
 80: static PetscErrorCode FormFunctionPicardLocal(DMDALocalInfo*,PetscScalar**,PetscScalar**,AppCtx*);
 81: static PetscErrorCode FormJacobianLocal(DMDALocalInfo*,PetscScalar**,Mat,Mat,AppCtx*);
 82: static PetscErrorCode NonlinearGS(SNES,Vec,Vec,void*);

 84: typedef struct _n_PreCheck *PreCheck;
 85: struct _n_PreCheck {
 86:   MPI_Comm    comm;
 87:   PetscReal   angle;
 88:   Vec         Ylast;
 89:   PetscViewer monitor;
 90: };
 91: PetscErrorCode PreCheckCreate(MPI_Comm,PreCheck*);
 92: PetscErrorCode PreCheckDestroy(PreCheck*);
 93: PetscErrorCode PreCheckFunction(SNESLineSearch,Vec,Vec,PetscBool*,void*);
 94: PetscErrorCode PreCheckSetFromOptions(PreCheck);

 96: int main(int argc,char **argv)
 97: {
 98:   SNES                snes;                    /* nonlinear solver */
 99:   Vec                 x,r,b;                   /* solution, residual, rhs vectors */
100:   Mat                 A,B;                     /* Jacobian and preconditioning matrices */
101:   AppCtx              user;                    /* user-defined work context */
102:   PetscInt            its;                     /* iterations for convergence */
103:   SNESConvergedReason reason;                  /* Check convergence */
104:   PetscBool           alloc_star;              /* Only allocate for the STAR stencil  */
105:   PetscBool           write_output;
106:   char                filename[PETSC_MAX_PATH_LEN] = "ex15.vts";
107:   PetscReal           bratu_lambda_max             = 6.81,bratu_lambda_min = 0.;
108:   DM                  da,dastar;               /* distributed array data structure */
109:   PreCheck            precheck = NULL;    /* precheck context for version in this file */
110:   PetscInt            use_precheck;      /* 0=none, 1=version in this file, 2=SNES-provided version */
111:   PetscReal           precheck_angle;    /* When manually setting the SNES-provided precheck function */
112:   PetscErrorCode      ierr;
113:   SNESLineSearch      linesearch;

115:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
116:      Initialize program
117:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

119:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;

121:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
122:      Initialize problem parameters
123:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
124:   user.lambda    = 0.0; user.p = 2.0; user.epsilon = 1e-5; user.source = 0.1; user.jtype = JAC_NEWTON;user.initial=-1;
125:   user.blocks[0] = 1; user.blocks[1] = 1; user.kappa = 1e-3;
126:   alloc_star     = PETSC_FALSE;
127:   use_precheck   = 0; precheck_angle = 10.;
128:   user.picard    = PETSC_FALSE;
129:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"p-Bratu options",__FILE__);
130:   {
131:     PetscInt two=2;
132:     PetscOptionsReal("-lambda","Bratu parameter","",user.lambda,&user.lambda,NULL);
133:     PetscOptionsReal("-p","Exponent `p' in p-Laplacian","",user.p,&user.p,NULL);
134:     PetscOptionsReal("-epsilon","Strain-regularization in p-Laplacian","",user.epsilon,&user.epsilon,NULL);
135:     PetscOptionsReal("-source","Constant source term","",user.source,&user.source,NULL);
136:     PetscOptionsEnum("-jtype","Jacobian approximation to assemble","",JacTypes,(PetscEnum)user.jtype,(PetscEnum*)&user.jtype,NULL);
137:     PetscOptionsName("-picard","Solve with defect-correction Picard iteration","",&user.picard);
138:     if (user.picard) {user.jtype = JAC_PICARD; user.p = 3;}
139:     PetscOptionsBool("-alloc_star","Allocate for STAR stencil (5-point)","",alloc_star,&alloc_star,NULL);
140:     PetscOptionsInt("-precheck","Use a pre-check correction intended for use with Picard iteration 1=this version, 2=library","",use_precheck,&use_precheck,NULL);
141:     PetscOptionsInt("-initial","Initial conditions type (-1: default, 0: zero-valued, 1: peaked guess)","",user.initial,&user.initial,NULL);
142:     if (use_precheck == 2) {    /* Using library version, get the angle */
143:       PetscOptionsReal("-precheck_angle","Angle in degrees between successive search directions necessary to activate step correction","",precheck_angle,&precheck_angle,NULL);
144:     }
145:     PetscOptionsIntArray("-blocks","number of coefficient interfaces in x and y direction","",user.blocks,&two,NULL);
146:     PetscOptionsReal("-kappa","diffusivity in odd regions","",user.kappa,&user.kappa,NULL);
147:     PetscOptionsString("-o","Output solution in vts format","",filename,filename,sizeof(filename),&write_output);
148:   }
149:   PetscOptionsEnd();
150:   if (user.lambda > bratu_lambda_max || user.lambda < bratu_lambda_min) {
151:     PetscPrintf(PETSC_COMM_WORLD,"WARNING: lambda %g out of range for p=2\n",(double)user.lambda);
152:   }

154:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155:      Create nonlinear solver context
156:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
157:   SNESCreate(PETSC_COMM_WORLD,&snes);

159:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
160:      Create distributed array (DMDA) to manage parallel grid and vectors
161:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
162:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,4,4,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);
163:   DMSetFromOptions(da);
164:   DMSetUp(da);
165:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&dastar);
166:   DMSetFromOptions(dastar);
167:   DMSetUp(dastar);

169:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
170:      Extract global vectors from DM; then duplicate for remaining
171:      vectors that are the same types
172:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
173:   DMCreateGlobalVector(da,&x);
174:   VecDuplicate(x,&r);
175:   VecDuplicate(x,&b);

177:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
178:      Create matrix data structure; set Jacobian evaluation routine

180:      Set Jacobian matrix data structure and default Jacobian evaluation
181:      routine. User can override with:
182:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
183:                 (unless user explicitly sets preconditioner)
184:      -snes_mf_operator : form preconditioning matrix as set by the user,
185:                          but use matrix-free approx for Jacobian-vector
186:                          products within Newton-Krylov method

188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
189:   /* B can be type of MATAIJ,MATBAIJ or MATSBAIJ */
190:   DMCreateMatrix(alloc_star ? dastar : da,&B);
191:   A    = B;

193:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194:      Set local function evaluation routine
195:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
196:   DMSetApplicationContext(da, &user);
197:   SNESSetDM(snes,da);
198:   if (user.picard) {
199:     /*
200:         This is not really right requiring the user to call SNESSetFunction/Jacobian but the DMDASNESSetPicardLocal() cannot access
201:         the SNES to set it
202:     */
203:     DMDASNESSetPicardLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionPicardLocal,
204:                                   (PetscErrorCode (*)(DMDALocalInfo*,void*,Mat,Mat,void*))FormJacobianLocal,&user);
205:     SNESSetFunction(snes,NULL,SNESPicardComputeFunction,&user);
206:     SNESSetJacobian(snes,NULL,NULL,SNESPicardComputeJacobian,&user);
207:   } else {
208:     DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,&user);
209:     DMDASNESSetJacobianLocal(da,(PetscErrorCode (*)(DMDALocalInfo*,void*,Mat,Mat,void*))FormJacobianLocal,&user);
210:   }


213:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
214:      Customize nonlinear solver; set runtime options
215:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
216:   SNESSetFromOptions(snes);
217:   SNESSetNGS(snes,NonlinearGS,&user);
218:   SNESGetLineSearch(snes, &linesearch);
219:   /* Set up the precheck context if requested */
220:   if (use_precheck == 1) {      /* Use the precheck routines in this file */
221:     PreCheckCreate(PETSC_COMM_WORLD,&precheck);
222:     PreCheckSetFromOptions(precheck);
223:     SNESLineSearchSetPreCheck(linesearch,PreCheckFunction,precheck);
224:   } else if (use_precheck == 2) { /* Use the version provided by the library */
225:     SNESLineSearchSetPreCheck(linesearch,SNESLineSearchPreCheckPicard,&precheck_angle);
226:   }

228:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
229:      Evaluate initial guess
230:      Note: The user should initialize the vector, x, with the initial guess
231:      for the nonlinear solver prior to calling SNESSolve().  In particular,
232:      to employ an initial guess of zero, the user should explicitly set
233:      this vector to zero by calling VecSet().
234:   */

236:   FormInitialGuess(&user,da,x);
237:   FormRHS(&user,da,b);

239:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
240:      Solve nonlinear system
241:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
242:   SNESSolve(snes,b,x);
243:   SNESGetIterationNumber(snes,&its);
244:   SNESGetConvergedReason(snes,&reason);

246:   PetscPrintf(PETSC_COMM_WORLD,"%s Number of nonlinear iterations = %D\n",SNESConvergedReasons[reason],its);

248:   if (write_output) {
249:     PetscViewer viewer;
250:     PetscViewerVTKOpen(PETSC_COMM_WORLD,filename,FILE_MODE_WRITE,&viewer);
251:     VecView(x,viewer);
252:     PetscViewerDestroy(&viewer);
253:   }

255:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
256:      Free work space.  All PETSc objects should be destroyed when they
257:      are no longer needed.
258:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

260:   if (A != B) {
261:     MatDestroy(&A);
262:   }
263:   MatDestroy(&B);
264:   VecDestroy(&x);
265:   VecDestroy(&r);
266:   VecDestroy(&b);
267:   SNESDestroy(&snes);
268:   DMDestroy(&da);
269:   DMDestroy(&dastar);
270:   PreCheckDestroy(&precheck);
271:   PetscFinalize();
272:   return ierr;
273: }

275: /* ------------------------------------------------------------------- */
276: /*
277:    FormInitialGuess - Forms initial approximation.

279:    Input Parameters:
280:    user - user-defined application context
281:    X - vector

283:    Output Parameter:
284:    X - vector
285:  */
286: static PetscErrorCode FormInitialGuess(AppCtx *user,DM da,Vec X)
287: {
288:   PetscInt       i,j,Mx,My,xs,ys,xm,ym;
290:   PetscReal      temp1,temp,hx,hy;
291:   PetscScalar    **x;

294:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

296:   hx    = 1.0/(PetscReal)(Mx-1);
297:   hy    = 1.0/(PetscReal)(My-1);
298:   temp1 = user->lambda / (user->lambda + 1.);

300:   /*
301:      Get a pointer to vector data.
302:        - For default PETSc vectors, VecGetArray() returns a pointer to
303:          the data array.  Otherwise, the routine is implementation dependent.
304:        - You MUST call VecRestoreArray() when you no longer need access to
305:          the array.
306:   */
307:   DMDAVecGetArray(da,X,&x);

309:   /*
310:      Get local grid boundaries (for 2-dimensional DA):
311:        xs, ys   - starting grid indices (no ghost points)
312:        xm, ym   - widths of local grid (no ghost points)

314:   */
315:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

317:   /*
318:      Compute initial guess over the locally owned part of the grid
319:   */
320:   for (j=ys; j<ys+ym; j++) {
321:     temp = (PetscReal)(PetscMin(j,My-j-1))*hy;
322:     for (i=xs; i<xs+xm; i++) {
323:       if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
324:         /* boundary conditions are all zero Dirichlet */
325:         x[j][i] = 0.0;
326:       } else {
327:         if (user->initial == -1) {
328:           if (user->lambda != 0) {
329:             x[j][i] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,temp));
330:           } else {
331:             /* The solution above is an exact solution for lambda=0, this avoids "accidentally" starting
332:              * with an exact solution. */
333:             const PetscReal
334:               xx = 2*(PetscReal)i/(Mx-1) - 1,
335:               yy = 2*(PetscReal)j/(My-1) - 1;
336:             x[j][i] = (1 - xx*xx) * (1-yy*yy) * xx * yy;
337:           }
338:         } else if (user->initial == 0) {
339:           x[j][i] = 0.;
340:         } else if (user->initial == 1) {
341:           const PetscReal
342:             xx = 2*(PetscReal)i/(Mx-1) - 1,
343:             yy = 2*(PetscReal)j/(My-1) - 1;
344:           x[j][i] = (1 - xx*xx) * (1-yy*yy) * xx * yy;
345:         } else {
346:           if (user->lambda != 0) {
347:             x[j][i] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,temp));
348:           } else {
349:             x[j][i] = 0.5*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,temp));
350:           }
351:         }
352:       }
353:     }
354:   }
355:   /*
356:      Restore vector
357:   */
358:   DMDAVecRestoreArray(da,X,&x);
359:   return(0);
360: }

362: /*
363:    FormRHS - Forms constant RHS for the problem.

365:    Input Parameters:
366:    user - user-defined application context
367:    B - RHS vector

369:    Output Parameter:
370:    B - vector
371:  */
372: static PetscErrorCode FormRHS(AppCtx *user,DM da,Vec B)
373: {
374:   PetscInt       i,j,Mx,My,xs,ys,xm,ym;
376:   PetscReal      hx,hy;
377:   PetscScalar    **b;

380:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

382:   hx    = 1.0/(PetscReal)(Mx-1);
383:   hy    = 1.0/(PetscReal)(My-1);
384:   DMDAVecGetArray(da,B,&b);
385:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);
386:   for (j=ys; j<ys+ym; j++) {
387:     for (i=xs; i<xs+xm; i++) {
388:       if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
389:         b[j][i] = 0.0;
390:       } else {
391:         b[j][i] = hx*hy*user->source;
392:       }
393:     }
394:   }
395:   DMDAVecRestoreArray(da,B,&b);
396:   return(0);
397: }

399: PETSC_STATIC_INLINE PetscReal kappa(const AppCtx *ctx,PetscReal x,PetscReal y)
400: {
401:   return (((PetscInt)(x*ctx->blocks[0])) + ((PetscInt)(y*ctx->blocks[1]))) % 2 ? ctx->kappa : 1.0;
402: }
403: /* p-Laplacian diffusivity */
404: PETSC_STATIC_INLINE PetscScalar eta(const AppCtx *ctx,PetscReal x,PetscReal y,PetscScalar ux,PetscScalar uy)
405: {
406:   return kappa(ctx,x,y) * PetscPowScalar(PetscSqr(ctx->epsilon)+0.5*(ux*ux + uy*uy),0.5*(ctx->p-2.));
407: }
408: PETSC_STATIC_INLINE PetscScalar deta(const AppCtx *ctx,PetscReal x,PetscReal y,PetscScalar ux,PetscScalar uy)
409: {
410:   return (ctx->p == 2)
411:          ? 0
412:          : kappa(ctx,x,y)*PetscPowScalar(PetscSqr(ctx->epsilon)+0.5*(ux*ux + uy*uy),0.5*(ctx->p-4)) * 0.5 * (ctx->p-2.);
413: }


416: /* ------------------------------------------------------------------- */
417: /*
418:    FormFunctionLocal - Evaluates nonlinear function, F(x).
419:  */
420: static PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,PetscScalar **x,PetscScalar **f,AppCtx *user)
421: {
422:   PetscReal      hx,hy,dhx,dhy,sc;
423:   PetscInt       i,j;
424:   PetscScalar    eu;


429:   hx     = 1.0/(PetscReal)(info->mx-1);
430:   hy     = 1.0/(PetscReal)(info->my-1);
431:   sc     = hx*hy*user->lambda;
432:   dhx    = 1/hx;
433:   dhy    = 1/hy;
434:   /*
435:      Compute function over the locally owned part of the grid
436:   */
437:   for (j=info->ys; j<info->ys+info->ym; j++) {
438:     for (i=info->xs; i<info->xs+info->xm; i++) {
439:       PetscReal xx = i*hx,yy = j*hy;
440:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
441:         f[j][i] = x[j][i];
442:       } else {
443:         const PetscScalar
444:           u    = x[j][i],
445:           ux_E = dhx*(x[j][i+1]-x[j][i]),
446:           uy_E = 0.25*dhy*(x[j+1][i]+x[j+1][i+1]-x[j-1][i]-x[j-1][i+1]),
447:           ux_W = dhx*(x[j][i]-x[j][i-1]),
448:           uy_W = 0.25*dhy*(x[j+1][i-1]+x[j+1][i]-x[j-1][i-1]-x[j-1][i]),
449:           ux_N = 0.25*dhx*(x[j][i+1]+x[j+1][i+1]-x[j][i-1]-x[j+1][i-1]),
450:           uy_N = dhy*(x[j+1][i]-x[j][i]),
451:           ux_S = 0.25*dhx*(x[j-1][i+1]+x[j][i+1]-x[j-1][i-1]-x[j][i-1]),
452:           uy_S = dhy*(x[j][i]-x[j-1][i]),
453:           e_E  = eta(user,xx,yy,ux_E,uy_E),
454:           e_W  = eta(user,xx,yy,ux_W,uy_W),
455:           e_N  = eta(user,xx,yy,ux_N,uy_N),
456:           e_S  = eta(user,xx,yy,ux_S,uy_S),
457:           uxx  = -hy * (e_E*ux_E - e_W*ux_W),
458:           uyy  = -hx * (e_N*uy_N - e_S*uy_S);
459:         if (sc) eu = PetscExpScalar(u);
460:         else    eu = 0.;
461:         /** For p=2, these terms decay to:
462:         * uxx = (2.0*u - x[j][i-1] - x[j][i+1])*hydhx
463:         * uyy = (2.0*u - x[j-1][i] - x[j+1][i])*hxdhy
464:         **/
465:         f[j][i] = uxx + uyy - sc*eu;
466:       }
467:     }
468:   }
469:   PetscLogFlops(info->xm*info->ym*(72.0));
470:   return(0);
471: }

473: /*
474:     This is the opposite sign of the part of FormFunctionLocal that excludes the A(x) x part of the operation,
475:     that is FormFunction applies A(x) x - b(x) while this applies b(x) because for Picard we think of it as solving A(x) x = b(x)

477: */
478: static PetscErrorCode FormFunctionPicardLocal(DMDALocalInfo *info,PetscScalar **x,PetscScalar **f,AppCtx *user)
479: {
480:   PetscReal hx,hy,sc;
481:   PetscInt  i,j;

485:   hx     = 1.0/(PetscReal)(info->mx-1);
486:   hy     = 1.0/(PetscReal)(info->my-1);
487:   sc     = hx*hy*user->lambda;
488:   /*
489:      Compute function over the locally owned part of the grid
490:   */
491:   for (j=info->ys; j<info->ys+info->ym; j++) {
492:     for (i=info->xs; i<info->xs+info->xm; i++) {
493:       if (!(i == 0 || j == 0 || i == info->mx-1 || j == info->my-1)) {
494:         const PetscScalar u = x[j][i];
495:         f[j][i] = sc*PetscExpScalar(u);
496:       }
497:     }
498:   }
499:   PetscLogFlops(info->xm*info->ym*2.0);
500:   return(0);
501: }

503: /*
504:    FormJacobianLocal - Evaluates Jacobian matrix.
505: */
506: static PetscErrorCode FormJacobianLocal(DMDALocalInfo *info,PetscScalar **x,Mat J,Mat B,AppCtx *user)
507: {
509:   PetscInt       i,j;
510:   MatStencil     col[9],row;
511:   PetscScalar    v[9];
512:   PetscReal      hx,hy,hxdhy,hydhx,dhx,dhy,sc;

515:   hx    = 1.0/(PetscReal)(info->mx-1);
516:   hy    = 1.0/(PetscReal)(info->my-1);
517:   sc    = hx*hy*user->lambda;
518:   dhx   = 1/hx;
519:   dhy   = 1/hy;
520:   hxdhy = hx/hy;
521:   hydhx = hy/hx;

523:   /*
524:      Compute entries for the locally owned part of the Jacobian.
525:       - PETSc parallel matrix formats are partitioned by
526:         contiguous chunks of rows across the processors.
527:       - Each processor needs to insert only elements that it owns
528:         locally (but any non-local elements will be sent to the
529:         appropriate processor during matrix assembly).
530:       - Here, we set all entries for a particular row at once.
531:   */
532:   for (j=info->ys; j<info->ys+info->ym; j++) {
533:     for (i=info->xs; i<info->xs+info->xm; i++) {
534:       PetscReal xx = i*hx,yy = j*hy;
535:       row.j = j; row.i = i;
536:       /* boundary points */
537:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
538:         v[0] = 1.0;
539:         MatSetValuesStencil(B,1,&row,1,&row,v,INSERT_VALUES);
540:       } else {
541:         /* interior grid points */
542:         const PetscScalar
543:           ux_E     = dhx*(x[j][i+1]-x[j][i]),
544:           uy_E     = 0.25*dhy*(x[j+1][i]+x[j+1][i+1]-x[j-1][i]-x[j-1][i+1]),
545:           ux_W     = dhx*(x[j][i]-x[j][i-1]),
546:           uy_W     = 0.25*dhy*(x[j+1][i-1]+x[j+1][i]-x[j-1][i-1]-x[j-1][i]),
547:           ux_N     = 0.25*dhx*(x[j][i+1]+x[j+1][i+1]-x[j][i-1]-x[j+1][i-1]),
548:           uy_N     = dhy*(x[j+1][i]-x[j][i]),
549:           ux_S     = 0.25*dhx*(x[j-1][i+1]+x[j][i+1]-x[j-1][i-1]-x[j][i-1]),
550:           uy_S     = dhy*(x[j][i]-x[j-1][i]),
551:           u        = x[j][i],
552:           e_E      = eta(user,xx,yy,ux_E,uy_E),
553:           e_W      = eta(user,xx,yy,ux_W,uy_W),
554:           e_N      = eta(user,xx,yy,ux_N,uy_N),
555:           e_S      = eta(user,xx,yy,ux_S,uy_S),
556:           de_E     = deta(user,xx,yy,ux_E,uy_E),
557:           de_W     = deta(user,xx,yy,ux_W,uy_W),
558:           de_N     = deta(user,xx,yy,ux_N,uy_N),
559:           de_S     = deta(user,xx,yy,ux_S,uy_S),
560:           skew_E   = de_E*ux_E*uy_E,
561:           skew_W   = de_W*ux_W*uy_W,
562:           skew_N   = de_N*ux_N*uy_N,
563:           skew_S   = de_S*ux_S*uy_S,
564:           cross_EW = 0.25*(skew_E - skew_W),
565:           cross_NS = 0.25*(skew_N - skew_S),
566:           newt_E   = e_E+de_E*PetscSqr(ux_E),
567:           newt_W   = e_W+de_W*PetscSqr(ux_W),
568:           newt_N   = e_N+de_N*PetscSqr(uy_N),
569:           newt_S   = e_S+de_S*PetscSqr(uy_S);
570:         /* interior grid points */
571:         switch (user->jtype) {
572:         case JAC_BRATU:
573:           /* Jacobian from p=2 */
574:           v[0] = -hxdhy;                                           col[0].j = j-1;   col[0].i = i;
575:           v[1] = -hydhx;                                           col[1].j = j;     col[1].i = i-1;
576:           v[2] = 2.0*(hydhx + hxdhy) - sc*PetscExpScalar(u);       col[2].j = row.j; col[2].i = row.i;
577:           v[3] = -hydhx;                                           col[3].j = j;     col[3].i = i+1;
578:           v[4] = -hxdhy;                                           col[4].j = j+1;   col[4].i = i;
579:           MatSetValuesStencil(B,1,&row,5,col,v,INSERT_VALUES);
580:           break;
581:         case JAC_PICARD:
582:           /* Jacobian arising from Picard linearization */
583:           v[0] = -hxdhy*e_S;                                           col[0].j = j-1;   col[0].i = i;
584:           v[1] = -hydhx*e_W;                                           col[1].j = j;     col[1].i = i-1;
585:           v[2] = (e_W+e_E)*hydhx + (e_S+e_N)*hxdhy;                    col[2].j = row.j; col[2].i = row.i;
586:           v[3] = -hydhx*e_E;                                           col[3].j = j;     col[3].i = i+1;
587:           v[4] = -hxdhy*e_N;                                           col[4].j = j+1;   col[4].i = i;
588:           MatSetValuesStencil(B,1,&row,5,col,v,INSERT_VALUES);
589:           break;
590:         case JAC_STAR:
591:           /* Full Jacobian, but only a star stencil */
592:           col[0].j = j-1; col[0].i = i;
593:           col[1].j = j;   col[1].i = i-1;
594:           col[2].j = j;   col[2].i = i;
595:           col[3].j = j;   col[3].i = i+1;
596:           col[4].j = j+1; col[4].i = i;
597:           v[0]     = -hxdhy*newt_S + cross_EW;
598:           v[1]     = -hydhx*newt_W + cross_NS;
599:           v[2]     = hxdhy*(newt_N + newt_S) + hydhx*(newt_E + newt_W) - sc*PetscExpScalar(u);
600:           v[3]     = -hydhx*newt_E - cross_NS;
601:           v[4]     = -hxdhy*newt_N - cross_EW;
602:           MatSetValuesStencil(B,1,&row,5,col,v,INSERT_VALUES);
603:           break;
604:         case JAC_NEWTON:
605:           /** The Jacobian is
606:           *
607:           * -div [ eta (grad u) + deta (grad u0 . grad u) grad u0 ] - (eE u0) u
608:           *
609:           **/
610:           col[0].j = j-1; col[0].i = i-1;
611:           col[1].j = j-1; col[1].i = i;
612:           col[2].j = j-1; col[2].i = i+1;
613:           col[3].j = j;   col[3].i = i-1;
614:           col[4].j = j;   col[4].i = i;
615:           col[5].j = j;   col[5].i = i+1;
616:           col[6].j = j+1; col[6].i = i-1;
617:           col[7].j = j+1; col[7].i = i;
618:           col[8].j = j+1; col[8].i = i+1;
619:           v[0]     = -0.25*(skew_S + skew_W);
620:           v[1]     = -hxdhy*newt_S + cross_EW;
621:           v[2]     =  0.25*(skew_S + skew_E);
622:           v[3]     = -hydhx*newt_W + cross_NS;
623:           v[4]     = hxdhy*(newt_N + newt_S) + hydhx*(newt_E + newt_W) - sc*PetscExpScalar(u);
624:           v[5]     = -hydhx*newt_E - cross_NS;
625:           v[6]     =  0.25*(skew_N + skew_W);
626:           v[7]     = -hxdhy*newt_N - cross_EW;
627:           v[8]     = -0.25*(skew_N + skew_E);
628:           MatSetValuesStencil(B,1,&row,9,col,v,INSERT_VALUES);
629:           break;
630:         default:
631:           SETERRQ1(PetscObjectComm((PetscObject)info->da),PETSC_ERR_SUP,"Jacobian type %d not implemented",user->jtype);
632:         }
633:       }
634:     }
635:   }

637:   /*
638:      Assemble matrix, using the 2-step process:
639:        MatAssemblyBegin(), MatAssemblyEnd().
640:   */
641:   MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
642:   MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);

644:   if (J != B) {
645:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
646:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
647:   }

649:   /*
650:      Tell the matrix we will never add a new nonzero location to the
651:      matrix. If we do, it will generate an error.
652:   */
653:   if (user->jtype == JAC_NEWTON) {
654:     PetscLogFlops(info->xm*info->ym*(131.0));
655:   }
656:   MatSetOption(B,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
657:   return(0);
658: }

660: /***********************************************************
661:  * PreCheck implementation
662:  ***********************************************************/
663: PetscErrorCode PreCheckSetFromOptions(PreCheck precheck)
664: {
666:   PetscBool      flg;

669:   PetscOptionsBegin(precheck->comm,NULL,"PreCheck Options","none");
670:   PetscOptionsReal("-precheck_angle","Angle in degrees between successive search directions necessary to activate step correction","",precheck->angle,&precheck->angle,NULL);
671:   flg  = PETSC_FALSE;
672:   PetscOptionsBool("-precheck_monitor","Monitor choices made by precheck routine","",flg,&flg,NULL);
673:   if (flg) {
674:     PetscViewerASCIIOpen(precheck->comm,"stdout",&precheck->monitor);
675:   }
676:   PetscOptionsEnd();
677:   return(0);
678: }

680: /*
681:   Compare the direction of the current and previous step, modify the current step accordingly
682: */
683: PetscErrorCode PreCheckFunction(SNESLineSearch linesearch,Vec X,Vec Y,PetscBool *changed, void *ctx)
684: {
686:   PreCheck       precheck;
687:   Vec            Ylast;
688:   PetscScalar    dot;
689:   PetscInt       iter;
690:   PetscReal      ynorm,ylastnorm,theta,angle_radians;
691:   SNES           snes;

694:   SNESLineSearchGetSNES(linesearch, &snes);
695:   precheck = (PreCheck)ctx;
696:   if (!precheck->Ylast) {VecDuplicate(Y,&precheck->Ylast);}
697:   Ylast = precheck->Ylast;
698:   SNESGetIterationNumber(snes,&iter);
699:   if (iter < 1) {
700:     VecCopy(Y,Ylast);
701:     *changed = PETSC_FALSE;
702:     return(0);
703:   }

705:   VecDot(Y,Ylast,&dot);
706:   VecNorm(Y,NORM_2,&ynorm);
707:   VecNorm(Ylast,NORM_2,&ylastnorm);
708:   /* Compute the angle between the vectors Y and Ylast, clip to keep inside the domain of acos() */
709:   theta         = PetscAcosReal((PetscReal)PetscClipInterval(PetscAbsScalar(dot) / (ynorm * ylastnorm),-1.0,1.0));
710:   angle_radians = precheck->angle * PETSC_PI / 180.;
711:   if (PetscAbsReal(theta) < angle_radians || PetscAbsReal(theta - PETSC_PI) < angle_radians) {
712:     /* Modify the step Y */
713:     PetscReal alpha,ydiffnorm;
714:     VecAXPY(Ylast,-1.0,Y);
715:     VecNorm(Ylast,NORM_2,&ydiffnorm);
716:     alpha = ylastnorm / ydiffnorm;
717:     VecCopy(Y,Ylast);
718:     VecScale(Y,alpha);
719:     if (precheck->monitor) {
720:       PetscViewerASCIIPrintf(precheck->monitor,"Angle %E degrees less than threshold %g, corrected step by alpha=%g\n",(double)(theta*180./PETSC_PI),(double)precheck->angle,(double)alpha);
721:     }
722:   } else {
723:     VecCopy(Y,Ylast);
724:     *changed = PETSC_FALSE;
725:     if (precheck->monitor) {
726:       PetscViewerASCIIPrintf(precheck->monitor,"Angle %E degrees exceeds threshold %g, no correction applied\n",(double)(theta*180./PETSC_PI),(double)precheck->angle);
727:     }
728:   }
729:   return(0);
730: }

732: PetscErrorCode PreCheckDestroy(PreCheck *precheck)
733: {

737:   if (!*precheck) return(0);
738:   VecDestroy(&(*precheck)->Ylast);
739:   PetscViewerDestroy(&(*precheck)->monitor);
740:   PetscFree(*precheck);
741:   return(0);
742: }

744: PetscErrorCode PreCheckCreate(MPI_Comm comm,PreCheck *precheck)
745: {

749:   PetscNew(precheck);

751:   (*precheck)->comm  = comm;
752:   (*precheck)->angle = 10.;     /* only active if angle is less than 10 degrees */
753:   return(0);
754: }

756: /*
757:       Applies some sweeps on nonlinear Gauss-Seidel on each process

759:  */
760: PetscErrorCode NonlinearGS(SNES snes,Vec X, Vec B, void *ctx)
761: {
762:   PetscInt       i,j,k,xs,ys,xm,ym,its,tot_its,sweeps,l,m;
764:   PetscReal      hx,hy,hxdhy,hydhx,dhx,dhy,sc;
765:   PetscScalar    **x,**b,bij,F,F0=0,J,y,u,eu;
766:   PetscReal      atol,rtol,stol;
767:   DM             da;
768:   AppCtx         *user = (AppCtx*)ctx;
769:   Vec            localX,localB;
770:   DMDALocalInfo  info;

773:   SNESGetDM(snes,&da);
774:   DMDAGetLocalInfo(da,&info);

776:   hx     = 1.0/(PetscReal)(info.mx-1);
777:   hy     = 1.0/(PetscReal)(info.my-1);
778:   sc     = hx*hy*user->lambda;
779:   dhx    = 1/hx;
780:   dhy    = 1/hy;
781:   hxdhy  = hx/hy;
782:   hydhx  = hy/hx;

784:   tot_its = 0;
785:   SNESNGSGetSweeps(snes,&sweeps);
786:   SNESNGSGetTolerances(snes,&atol,&rtol,&stol,&its);
787:   DMGetLocalVector(da,&localX);
788:   if (B) {
789:     DMGetLocalVector(da,&localB);
790:   }
791:   if (B) {
792:     DMGlobalToLocalBegin(da,B,INSERT_VALUES,localB);
793:     DMGlobalToLocalEnd(da,B,INSERT_VALUES,localB);
794:   }
795:   if (B) DMDAVecGetArrayRead(da,localB,&b);
796:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
797:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
798:   DMDAVecGetArray(da,localX,&x);
799:   for (l=0; l<sweeps; l++) {
800:     /*
801:      Get local grid boundaries (for 2-dimensional DMDA):
802:      xs, ys   - starting grid indices (no ghost points)
803:      xm, ym   - widths of local grid (no ghost points)
804:      */
805:     DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);
806:     for (m=0; m<2; m++) {
807:       for (j=ys; j<ys+ym; j++) {
808:         for (i=xs+(m+j)%2; i<xs+xm; i+=2) {
809:           PetscReal xx = i*hx,yy = j*hy;
810:           if (B) bij = b[j][i];
811:           else   bij = 0.;

813:           if (i == 0 || j == 0 || i == info.mx-1 || j == info.my-1) {
814:             /* boundary conditions are all zero Dirichlet */
815:             x[j][i] = 0.0 + bij;
816:           } else {
817:             const PetscScalar
818:               u_E = x[j][i+1],
819:               u_W = x[j][i-1],
820:               u_N = x[j+1][i],
821:               u_S = x[j-1][i];
822:             const PetscScalar
823:               uy_E   = 0.25*dhy*(x[j+1][i]+x[j+1][i+1]-x[j-1][i]-x[j-1][i+1]),
824:               uy_W   = 0.25*dhy*(x[j+1][i-1]+x[j+1][i]-x[j-1][i-1]-x[j-1][i]),
825:               ux_N   = 0.25*dhx*(x[j][i+1]+x[j+1][i+1]-x[j][i-1]-x[j+1][i-1]),
826:               ux_S   = 0.25*dhx*(x[j-1][i+1]+x[j][i+1]-x[j-1][i-1]-x[j][i-1]);
827:             u = x[j][i];
828:             for (k=0; k<its; k++) {
829:               const PetscScalar
830:                 ux_E   = dhx*(u_E-u),
831:                 ux_W   = dhx*(u-u_W),
832:                 uy_N   = dhy*(u_N-u),
833:                 uy_S   = dhy*(u-u_S),
834:                 e_E    = eta(user,xx,yy,ux_E,uy_E),
835:                 e_W    = eta(user,xx,yy,ux_W,uy_W),
836:                 e_N    = eta(user,xx,yy,ux_N,uy_N),
837:                 e_S    = eta(user,xx,yy,ux_S,uy_S),
838:                 de_E   = deta(user,xx,yy,ux_E,uy_E),
839:                 de_W   = deta(user,xx,yy,ux_W,uy_W),
840:                 de_N   = deta(user,xx,yy,ux_N,uy_N),
841:                 de_S   = deta(user,xx,yy,ux_S,uy_S),
842:                 newt_E = e_E+de_E*PetscSqr(ux_E),
843:                 newt_W = e_W+de_W*PetscSqr(ux_W),
844:                 newt_N = e_N+de_N*PetscSqr(uy_N),
845:                 newt_S = e_S+de_S*PetscSqr(uy_S),
846:                 uxx    = -hy * (e_E*ux_E - e_W*ux_W),
847:                 uyy    = -hx * (e_N*uy_N - e_S*uy_S);

849:               if (sc) eu = PetscExpScalar(u);
850:               else    eu = 0;

852:               F = uxx + uyy - sc*eu - bij;
853:               if (k == 0) F0 = F;
854:               J  = hxdhy*(newt_N + newt_S) + hydhx*(newt_E + newt_W) - sc*eu;
855:               y  = F/J;
856:               u -= y;
857:               tot_its++;
858:               if (atol > PetscAbsReal(PetscRealPart(F)) ||
859:                   rtol*PetscAbsReal(PetscRealPart(F0)) > PetscAbsReal(PetscRealPart(F)) ||
860:                   stol*PetscAbsReal(PetscRealPart(u)) > PetscAbsReal(PetscRealPart(y))) {
861:                 break;
862:               }
863:             }
864:             x[j][i] = u;
865:           }
866:         }
867:       }
868:     }
869:     /*
870: x     Restore vector
871:      */
872:   }
873:   DMDAVecRestoreArray(da,localX,&x);
874:   DMLocalToGlobalBegin(da,localX,INSERT_VALUES,X);
875:   DMLocalToGlobalEnd(da,localX,INSERT_VALUES,X);
876:   PetscLogFlops(tot_its*(118.0));
877:   DMRestoreLocalVector(da,&localX);
878:   if (B) {
879:     DMDAVecRestoreArrayRead(da,localB,&b);
880:     DMRestoreLocalVector(da,&localB);
881:   }
882:   return(0);
883: }


886: /*TEST

888:    test:
889:       nsize: 2
890:       args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype NEWTON
891:       requires: !single

893:    test:
894:       suffix: 2
895:       nsize: 2
896:       args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype PICARD -precheck 1
897:       requires: !single

899:    test:
900:       suffix: 3
901:       nsize: 2
902:       args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype PICARD -picard -precheck 1
903:       requires: !single

905:    test:
906:       suffix: 4
907:       args: -snes_monitor_short -snes_type newtonls -npc_snes_type ngs -snes_npc_side left -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -ksp_monitor_short -pc_type none
908:       requires: !single

910:    test:
911:       suffix: lag_jac
912:       nsize: 4
913:       args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 6.0 -lambda 0 -jtype NEWTON -snes_type ngmres -npc_snes_type newtonls -npc_snes_lag_jacobian 5 -npc_pc_type asm -npc_ksp_converged_reason -npc_snes_lag_jacobian_persists
914:       requires: !single

916:    test:
917:       suffix: lag_pc
918:       nsize: 4
919:       args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 6.0 -lambda 0 -jtype NEWTON -snes_type ngmres -npc_snes_type newtonls -npc_snes_lag_preconditioner 5 -npc_pc_type asm -npc_ksp_converged_reason -npc_snes_lag_preconditioner_persists
920:       requires: !single

922:    test:
923:       suffix: nleqerr
924:       args: -snes_monitor_short -snes_type newtonls -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -snes_linesearch_monitor -pc_type lu -snes_linesearch_type nleqerr
925:       requires: !single

927: TEST*/