Actual source code: ex15.c
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";
13: /*F
14: The $p$-Bratu problem is a combination of the $p$-Laplacian (nonlinear diffusion) and the Brutu solid fuel ignition problem.
15: This problem is modeled by the partial differential equation
17: \begin{equation*}
18: -\nabla\cdot (\eta \nabla u) - \lambda \exp(u) = 0
19: \end{equation*}
21: on $\Omega = (-1,1)^2$ with closure
23: \begin{align*}
24: \eta(\gamma) &= (\epsilon^2 + \gamma)^{(p-2)/2} & \gamma &= \frac 1 2 |\nabla u|^2
25: \end{align*}
27: and boundary conditions $u = 0$ for $(x,y) \in \partial \Omega$
29: A 9-point finite difference stencil is used to discretize
30: the boundary value problem to obtain a nonlinear system of equations.
31: This would be a 5-point stencil if not for the $p$-Laplacian's nonlinearity.
32: F*/
34: /*
35: mpiexec -n 2 ./ex15 -snes_monitor -ksp_monitor log_view
36: */
38: /*
39: Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
40: Include "petscsnes.h" so that we can use SNES solvers. Note that this
41: file automatically includes:
42: petsc.h - base PETSc routines petscvec.h - vectors
43: petscsys.h - system routines petscmat.h - matrices
44: petscis.h - index sets petscksp.h - Krylov subspace methods
45: petscviewer.h - viewers petscpc.h - preconditioners
46: petscksp.h - linear solvers
47: */
49: #include <petscdm.h>
50: #include <petscdmda.h>
51: #include <petscsnes.h>
53: typedef enum {
54: JAC_BRATU,
55: JAC_PICARD,
56: JAC_STAR,
57: JAC_NEWTON
58: } JacType;
59: static const char *const JacTypes[] = {"BRATU", "PICARD", "STAR", "NEWTON", "JacType", "JAC_", 0};
61: /*
62: User-defined application context - contains data needed by the
63: application-provided call-back routines, FormJacobianLocal() and
64: FormFunctionLocal().
65: */
66: typedef struct {
67: PetscReal lambda; /* Bratu parameter */
68: PetscReal p; /* Exponent in p-Laplacian */
69: PetscReal epsilon; /* Regularization */
70: PetscReal source; /* Source term */
71: JacType jtype; /* What type of Jacobian to assemble */
72: PetscBool picard;
73: PetscInt blocks[2];
74: PetscReal kappa;
75: PetscInt initial; /* initial conditions type */
76: } AppCtx;
78: /*
79: User-defined routines
80: */
81: static PetscErrorCode FormRHS(AppCtx *, DM, Vec);
82: static PetscErrorCode FormInitialGuess(AppCtx *, DM, Vec);
83: static PetscErrorCode FormFunctionLocal(DMDALocalInfo *, PetscScalar **, PetscScalar **, AppCtx *);
84: static PetscErrorCode FormFunctionPicardLocal(DMDALocalInfo *, PetscScalar **, PetscScalar **, AppCtx *);
85: static PetscErrorCode FormJacobianLocal(DMDALocalInfo *, PetscScalar **, Mat, Mat, AppCtx *);
86: static PetscErrorCode NonlinearGS(SNES, Vec, Vec, void *);
88: typedef struct _n_PreCheck *PreCheck;
89: struct _n_PreCheck {
90: MPI_Comm comm;
91: PetscReal angle;
92: Vec Ylast;
93: PetscViewer monitor;
94: };
95: PetscErrorCode PreCheckCreate(MPI_Comm, PreCheck *);
96: PetscErrorCode PreCheckDestroy(PreCheck *);
97: PetscErrorCode PreCheckFunction(SNESLineSearch, Vec, Vec, PetscBool *, void *);
98: PetscErrorCode PreCheckSetFromOptions(PreCheck);
100: int main(int argc, char **argv)
101: {
102: SNES snes; /* nonlinear solver */
103: Vec x, r, b; /* solution, residual, rhs vectors */
104: AppCtx user; /* user-defined work context */
105: PetscInt its; /* iterations for convergence */
106: SNESConvergedReason reason; /* Check convergence */
107: PetscBool alloc_star; /* Only allocate for the STAR stencil */
108: PetscBool write_output;
109: char filename[PETSC_MAX_PATH_LEN] = "ex15.vts";
110: PetscReal bratu_lambda_max = 6.81, bratu_lambda_min = 0.;
111: DM da; /* distributed array data structure */
112: PreCheck precheck = NULL; /* precheck context for version in this file */
113: PetscInt use_precheck; /* 0=none, 1=version in this file, 2=SNES-provided version */
114: PetscReal precheck_angle; /* When manually setting the SNES-provided precheck function */
115: SNESLineSearch linesearch;
117: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118: Initialize program
119: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
121: PetscFunctionBeginUser;
122: PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
124: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125: Initialize problem parameters
126: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
127: user.lambda = 0.0;
128: user.p = 2.0;
129: user.epsilon = 1e-5;
130: user.source = 0.1;
131: user.jtype = JAC_NEWTON;
132: user.initial = -1;
133: user.blocks[0] = 1;
134: user.blocks[1] = 1;
135: user.kappa = 1e-3;
136: alloc_star = PETSC_FALSE;
137: use_precheck = 0;
138: precheck_angle = 10.;
139: user.picard = PETSC_FALSE;
140: PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "p-Bratu options", __FILE__);
141: {
142: PetscInt two = 2;
143: PetscCall(PetscOptionsReal("-lambda", "Bratu parameter", "", user.lambda, &user.lambda, NULL));
144: PetscCall(PetscOptionsReal("-p", "Exponent `p' in p-Laplacian", "", user.p, &user.p, NULL));
145: PetscCall(PetscOptionsReal("-epsilon", "Strain-regularization in p-Laplacian", "", user.epsilon, &user.epsilon, NULL));
146: PetscCall(PetscOptionsReal("-source", "Constant source term", "", user.source, &user.source, NULL));
147: PetscCall(PetscOptionsEnum("-jtype", "Jacobian approximation to assemble", "", JacTypes, (PetscEnum)user.jtype, (PetscEnum *)&user.jtype, NULL));
148: PetscCall(PetscOptionsName("-picard", "Solve with defect-correction Picard iteration", "", &user.picard));
149: if (user.picard) {
150: user.jtype = JAC_PICARD;
151: PetscCheck(user.p == 3, PETSC_COMM_WORLD, PETSC_ERR_SUP, "Picard iteration is only supported for p == 3");
152: /* the Picard linearization only requires a 5 point stencil, while the Newton linearization requires a 9 point stencil */
153: /* hence allocating the 5 point stencil gives the same convergence as the 9 point stencil since the extra stencil points are not used */
154: PetscCall(PetscOptionsBool("-alloc_star", "Allocate for STAR stencil (5-point)", "", alloc_star, &alloc_star, NULL));
155: }
156: PetscCall(PetscOptionsInt("-precheck", "Use a pre-check correction intended for use with Picard iteration 1=this version, 2=library", "", use_precheck, &use_precheck, NULL));
157: PetscCall(PetscOptionsInt("-initial", "Initial conditions type (-1: default, 0: zero-valued, 1: peaked guess)", "", user.initial, &user.initial, NULL));
158: if (use_precheck == 2) { /* Using library version, get the angle */
159: PetscCall(PetscOptionsReal("-precheck_angle", "Angle in degrees between successive search directions necessary to activate step correction", "", precheck_angle, &precheck_angle, NULL));
160: }
161: PetscCall(PetscOptionsIntArray("-blocks", "number of coefficient interfaces in x and y direction", "", user.blocks, &two, NULL));
162: PetscCall(PetscOptionsReal("-kappa", "diffusivity in odd regions", "", user.kappa, &user.kappa, NULL));
163: PetscCall(PetscOptionsString("-o", "Output solution in vts format", "", filename, filename, sizeof(filename), &write_output));
164: }
165: PetscOptionsEnd();
166: if (user.lambda > bratu_lambda_max || user.lambda < bratu_lambda_min) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "WARNING: lambda %g out of range for p=2\n", (double)user.lambda));
168: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
169: Create nonlinear solver context
170: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
171: PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
173: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174: Create distributed array (DMDA) to manage parallel grid and vectors
175: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
176: PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, alloc_star ? DMDA_STENCIL_STAR : DMDA_STENCIL_BOX, 4, 4, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da));
177: PetscCall(DMSetFromOptions(da));
178: PetscCall(DMSetUp(da));
180: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
181: Extract global vectors from DM; then duplicate for remaining
182: vectors that are the same types
183: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
184: PetscCall(DMCreateGlobalVector(da, &x));
185: PetscCall(VecDuplicate(x, &r));
186: PetscCall(VecDuplicate(x, &b));
188: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
189: User can override with:
190: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
191: (unless user explicitly sets preconditioner)
192: -snes_mf_operator : form preconditioning matrix as set by the user,
193: but use matrix-free approx for Jacobian-vector
194: products within Newton-Krylov method
196: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
198: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
199: Set local function evaluation routine
200: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201: PetscCall(DMSetApplicationContext(da, &user));
202: PetscCall(SNESSetDM(snes, da));
203: if (user.picard) {
204: /*
205: This is not really right requiring the user to call SNESSetFunction/Jacobian but the DMDASNESSetPicardLocal() cannot access
206: the SNES to set it
207: */
208: PetscCall(DMDASNESSetPicardLocal(da, INSERT_VALUES, (PetscErrorCode(*)(DMDALocalInfo *, void *, void *, void *))FormFunctionPicardLocal, (PetscErrorCode(*)(DMDALocalInfo *, void *, Mat, Mat, void *))FormJacobianLocal, &user));
209: PetscCall(SNESSetFunction(snes, NULL, SNESPicardComputeFunction, &user));
210: PetscCall(SNESSetJacobian(snes, NULL, NULL, SNESPicardComputeJacobian, &user));
211: } else {
212: PetscCall(DMDASNESSetFunctionLocal(da, INSERT_VALUES, (PetscErrorCode(*)(DMDALocalInfo *, void *, void *, void *))FormFunctionLocal, &user));
213: PetscCall(DMDASNESSetJacobianLocal(da, (PetscErrorCode(*)(DMDALocalInfo *, void *, Mat, Mat, void *))FormJacobianLocal, &user));
214: }
216: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
217: Customize nonlinear solver; set runtime options
218: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
219: PetscCall(SNESSetFromOptions(snes));
220: PetscCall(SNESSetNGS(snes, NonlinearGS, &user));
221: PetscCall(SNESGetLineSearch(snes, &linesearch));
222: /* Set up the precheck context if requested */
223: if (use_precheck == 1) { /* Use the precheck routines in this file */
224: PetscCall(PreCheckCreate(PETSC_COMM_WORLD, &precheck));
225: PetscCall(PreCheckSetFromOptions(precheck));
226: PetscCall(SNESLineSearchSetPreCheck(linesearch, PreCheckFunction, precheck));
227: } else if (use_precheck == 2) { /* Use the version provided by the library */
228: PetscCall(SNESLineSearchSetPreCheck(linesearch, SNESLineSearchPreCheckPicard, &precheck_angle));
229: }
231: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
232: Evaluate initial guess
233: Note: The user should initialize the vector, x, with the initial guess
234: for the nonlinear solver prior to calling SNESSolve(). In particular,
235: to employ an initial guess of zero, the user should explicitly set
236: this vector to zero by calling VecSet().
237: */
239: PetscCall(FormInitialGuess(&user, da, x));
240: PetscCall(FormRHS(&user, da, b));
242: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
243: Solve nonlinear system
244: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
245: PetscCall(SNESSolve(snes, b, x));
246: PetscCall(SNESGetIterationNumber(snes, &its));
247: PetscCall(SNESGetConvergedReason(snes, &reason));
249: PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%s Number of nonlinear iterations = %" PetscInt_FMT "\n", SNESConvergedReasons[reason], its));
251: if (write_output) {
252: PetscViewer viewer;
253: PetscCall(PetscViewerVTKOpen(PETSC_COMM_WORLD, filename, FILE_MODE_WRITE, &viewer));
254: PetscCall(VecView(x, viewer));
255: PetscCall(PetscViewerDestroy(&viewer));
256: }
258: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
259: Free work space. All PETSc objects should be destroyed when they
260: are no longer needed.
261: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
263: PetscCall(VecDestroy(&x));
264: PetscCall(VecDestroy(&r));
265: PetscCall(VecDestroy(&b));
266: PetscCall(SNESDestroy(&snes));
267: PetscCall(DMDestroy(&da));
268: PetscCall(PreCheckDestroy(&precheck));
269: PetscCall(PetscFinalize());
270: return 0;
271: }
273: /* ------------------------------------------------------------------- */
274: /*
275: FormInitialGuess - Forms initial approximation.
277: Input Parameters:
278: user - user-defined application context
279: X - vector
281: Output Parameter:
282: X - vector
283: */
284: static PetscErrorCode FormInitialGuess(AppCtx *user, DM da, Vec X)
285: {
286: PetscInt i, j, Mx, My, xs, ys, xm, ym;
287: PetscReal temp1, temp, hx, hy;
288: PetscScalar **x;
290: PetscFunctionBeginUser;
291: PetscCall(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));
293: hx = 1.0 / (PetscReal)(Mx - 1);
294: hy = 1.0 / (PetscReal)(My - 1);
295: temp1 = user->lambda / (user->lambda + 1.);
297: /*
298: Get a pointer to vector data.
299: - For default PETSc vectors, VecGetArray() returns a pointer to
300: the data array. Otherwise, the routine is implementation dependent.
301: - You MUST call VecRestoreArray() when you no longer need access to
302: the array.
303: */
304: PetscCall(DMDAVecGetArray(da, X, &x));
306: /*
307: Get local grid boundaries (for 2-dimensional DA):
308: xs, ys - starting grid indices (no ghost points)
309: xm, ym - widths of local grid (no ghost points)
311: */
312: PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));
314: /*
315: Compute initial guess over the locally owned part of the grid
316: */
317: for (j = ys; j < ys + ym; j++) {
318: temp = (PetscReal)(PetscMin(j, My - j - 1)) * hy;
319: for (i = xs; i < xs + xm; i++) {
320: if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
321: /* boundary conditions are all zero Dirichlet */
322: x[j][i] = 0.0;
323: } else {
324: if (user->initial == -1) {
325: if (user->lambda != 0) {
326: x[j][i] = temp1 * PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i, Mx - i - 1)) * hx, temp));
327: } else {
328: /* The solution above is an exact solution for lambda=0, this avoids "accidentally" starting
329: * with an exact solution. */
330: const PetscReal xx = 2 * (PetscReal)i / (Mx - 1) - 1, yy = 2 * (PetscReal)j / (My - 1) - 1;
331: x[j][i] = (1 - xx * xx) * (1 - yy * yy) * xx * yy;
332: }
333: } else if (user->initial == 0) {
334: x[j][i] = 0.;
335: } else if (user->initial == 1) {
336: const PetscReal xx = 2 * (PetscReal)i / (Mx - 1) - 1, yy = 2 * (PetscReal)j / (My - 1) - 1;
337: x[j][i] = (1 - xx * xx) * (1 - yy * yy) * xx * yy;
338: } else {
339: if (user->lambda != 0) {
340: x[j][i] = temp1 * PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i, Mx - i - 1)) * hx, temp));
341: } else {
342: x[j][i] = 0.5 * PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i, Mx - i - 1)) * hx, temp));
343: }
344: }
345: }
346: }
347: }
348: /*
349: Restore vector
350: */
351: PetscCall(DMDAVecRestoreArray(da, X, &x));
352: PetscFunctionReturn(PETSC_SUCCESS);
353: }
355: /*
356: FormRHS - Forms constant RHS for the problem.
358: Input Parameters:
359: user - user-defined application context
360: B - RHS vector
362: Output Parameter:
363: B - vector
364: */
365: static PetscErrorCode FormRHS(AppCtx *user, DM da, Vec B)
366: {
367: PetscInt i, j, Mx, My, xs, ys, xm, ym;
368: PetscReal hx, hy;
369: PetscScalar **b;
371: PetscFunctionBeginUser;
372: PetscCall(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));
374: hx = 1.0 / (PetscReal)(Mx - 1);
375: hy = 1.0 / (PetscReal)(My - 1);
376: PetscCall(DMDAVecGetArray(da, B, &b));
377: PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));
378: for (j = ys; j < ys + ym; j++) {
379: for (i = xs; i < xs + xm; i++) {
380: if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
381: b[j][i] = 0.0;
382: } else {
383: b[j][i] = hx * hy * user->source;
384: }
385: }
386: }
387: PetscCall(DMDAVecRestoreArray(da, B, &b));
388: PetscFunctionReturn(PETSC_SUCCESS);
389: }
391: static inline PetscReal kappa(const AppCtx *ctx, PetscReal x, PetscReal y)
392: {
393: return (((PetscInt)(x * ctx->blocks[0])) + ((PetscInt)(y * ctx->blocks[1]))) % 2 ? ctx->kappa : 1.0;
394: }
395: /* p-Laplacian diffusivity */
396: static inline PetscScalar eta(const AppCtx *ctx, PetscReal x, PetscReal y, PetscScalar ux, PetscScalar uy)
397: {
398: return kappa(ctx, x, y) * PetscPowScalar(PetscSqr(ctx->epsilon) + 0.5 * (ux * ux + uy * uy), 0.5 * (ctx->p - 2.));
399: }
400: static inline PetscScalar deta(const AppCtx *ctx, PetscReal x, PetscReal y, PetscScalar ux, PetscScalar uy)
401: {
402: return (ctx->p == 2) ? 0 : kappa(ctx, x, y) * PetscPowScalar(PetscSqr(ctx->epsilon) + 0.5 * (ux * ux + uy * uy), 0.5 * (ctx->p - 4)) * 0.5 * (ctx->p - 2.);
403: }
405: /* ------------------------------------------------------------------- */
406: /*
407: FormFunctionLocal - Evaluates nonlinear function, F(x).
408: */
409: static PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **x, PetscScalar **f, AppCtx *user)
410: {
411: PetscReal hx, hy, dhx, dhy, sc;
412: PetscInt i, j;
413: PetscScalar eu;
415: PetscFunctionBeginUser;
416: hx = 1.0 / (PetscReal)(info->mx - 1);
417: hy = 1.0 / (PetscReal)(info->my - 1);
418: sc = hx * hy * user->lambda;
419: dhx = 1 / hx;
420: dhy = 1 / hy;
421: /*
422: Compute function over the locally owned part of the grid
423: */
424: for (j = info->ys; j < info->ys + info->ym; j++) {
425: for (i = info->xs; i < info->xs + info->xm; i++) {
426: PetscReal xx = i * hx, yy = j * hy;
427: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
428: f[j][i] = x[j][i];
429: } else {
430: const PetscScalar u = x[j][i], ux_E = dhx * (x[j][i + 1] - x[j][i]), uy_E = 0.25 * dhy * (x[j + 1][i] + x[j + 1][i + 1] - x[j - 1][i] - x[j - 1][i + 1]), ux_W = dhx * (x[j][i] - x[j][i - 1]), uy_W = 0.25 * dhy * (x[j + 1][i - 1] + x[j + 1][i] - x[j - 1][i - 1] - x[j - 1][i]), ux_N = 0.25 * dhx * (x[j][i + 1] + x[j + 1][i + 1] - x[j][i - 1] - x[j + 1][i - 1]), uy_N = dhy * (x[j + 1][i] - x[j][i]), ux_S = 0.25 * dhx * (x[j - 1][i + 1] + x[j][i + 1] - x[j - 1][i - 1] - x[j][i - 1]), uy_S = dhy * (x[j][i] - x[j - 1][i]), e_E = eta(user, xx, yy, ux_E, uy_E), e_W = eta(user, xx, yy, ux_W, uy_W), e_N = eta(user, xx, yy, ux_N, uy_N), e_S = eta(user, xx, yy, ux_S, uy_S), uxx = -hy * (e_E * ux_E - e_W * ux_W), uyy = -hx * (e_N * uy_N - e_S * uy_S);
431: if (sc) eu = PetscExpScalar(u);
432: else eu = 0.;
433: /* For p=2, these terms decay to:
434: uxx = (2.0*u - x[j][i-1] - x[j][i+1])*hydhx
435: uyy = (2.0*u - x[j-1][i] - x[j+1][i])*hxdhy
436: */
437: f[j][i] = uxx + uyy - sc * eu;
438: }
439: }
440: }
441: PetscCall(PetscLogFlops(info->xm * info->ym * (72.0)));
442: PetscFunctionReturn(PETSC_SUCCESS);
443: }
445: /*
446: This is the opposite sign of the part of FormFunctionLocal that excludes the A(x) x part of the operation,
447: 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)
449: */
450: static PetscErrorCode FormFunctionPicardLocal(DMDALocalInfo *info, PetscScalar **x, PetscScalar **f, AppCtx *user)
451: {
452: PetscReal hx, hy, sc;
453: PetscInt i, j;
455: PetscFunctionBeginUser;
456: hx = 1.0 / (PetscReal)(info->mx - 1);
457: hy = 1.0 / (PetscReal)(info->my - 1);
458: sc = hx * hy * user->lambda;
459: /*
460: Compute function over the locally owned part of the grid
461: */
462: for (j = info->ys; j < info->ys + info->ym; j++) {
463: for (i = info->xs; i < info->xs + info->xm; i++) {
464: if (!(i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1)) {
465: const PetscScalar u = x[j][i];
466: f[j][i] = sc * PetscExpScalar(u);
467: } else {
468: f[j][i] = 0.0; /* this is zero because the A(x) x term forces the x to be zero on the boundary */
469: }
470: }
471: }
472: PetscCall(PetscLogFlops(info->xm * info->ym * 2.0));
473: PetscFunctionReturn(PETSC_SUCCESS);
474: }
476: /*
477: FormJacobianLocal - Evaluates Jacobian matrix.
478: */
479: static PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **x, Mat J, Mat B, AppCtx *user)
480: {
481: PetscInt i, j;
482: MatStencil col[9], row;
483: PetscScalar v[9];
484: PetscReal hx, hy, hxdhy, hydhx, dhx, dhy, sc;
486: PetscFunctionBeginUser;
487: PetscCall(MatZeroEntries(B));
488: hx = 1.0 / (PetscReal)(info->mx - 1);
489: hy = 1.0 / (PetscReal)(info->my - 1);
490: sc = hx * hy * user->lambda;
491: dhx = 1 / hx;
492: dhy = 1 / hy;
493: hxdhy = hx / hy;
494: hydhx = hy / hx;
496: /*
497: Compute entries for the locally owned part of the Jacobian.
498: - PETSc parallel matrix formats are partitioned by
499: contiguous chunks of rows across the processors.
500: - Each processor needs to insert only elements that it owns
501: locally (but any non-local elements will be sent to the
502: appropriate processor during matrix assembly).
503: - Here, we set all entries for a particular row at once.
504: */
505: for (j = info->ys; j < info->ys + info->ym; j++) {
506: for (i = info->xs; i < info->xs + info->xm; i++) {
507: PetscReal xx = i * hx, yy = j * hy;
508: row.j = j;
509: row.i = i;
510: /* boundary points */
511: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
512: v[0] = 1.0;
513: PetscCall(MatSetValuesStencil(B, 1, &row, 1, &row, v, INSERT_VALUES));
514: } else {
515: /* interior grid points */
516: const PetscScalar ux_E = dhx * (x[j][i + 1] - x[j][i]), uy_E = 0.25 * dhy * (x[j + 1][i] + x[j + 1][i + 1] - x[j - 1][i] - x[j - 1][i + 1]), ux_W = dhx * (x[j][i] - x[j][i - 1]), uy_W = 0.25 * dhy * (x[j + 1][i - 1] + x[j + 1][i] - x[j - 1][i - 1] - x[j - 1][i]), ux_N = 0.25 * dhx * (x[j][i + 1] + x[j + 1][i + 1] - x[j][i - 1] - x[j + 1][i - 1]), uy_N = dhy * (x[j + 1][i] - x[j][i]), ux_S = 0.25 * dhx * (x[j - 1][i + 1] + x[j][i + 1] - x[j - 1][i - 1] - x[j][i - 1]), uy_S = dhy * (x[j][i] - x[j - 1][i]), u = x[j][i], e_E = eta(user, xx, yy, ux_E, uy_E), e_W = eta(user, xx, yy, ux_W, uy_W), e_N = eta(user, xx, yy, ux_N, uy_N), e_S = eta(user, xx, yy, ux_S, uy_S), de_E = deta(user, xx, yy, ux_E, uy_E), de_W = deta(user, xx, yy, ux_W, uy_W), de_N = deta(user, xx, yy, ux_N, uy_N), de_S = deta(user, xx, yy, ux_S, uy_S), skew_E = de_E * ux_E * uy_E, skew_W = de_W * ux_W * uy_W, skew_N = de_N * ux_N * uy_N, skew_S = de_S * ux_S * uy_S, cross_EW = 0.25 * (skew_E - skew_W), cross_NS = 0.25 * (skew_N - skew_S), newt_E = e_E + de_E * PetscSqr(ux_E), newt_W = e_W + de_W * PetscSqr(ux_W), newt_N = e_N + de_N * PetscSqr(uy_N), newt_S = e_S + de_S * PetscSqr(uy_S);
517: /* interior grid points */
518: switch (user->jtype) {
519: case JAC_BRATU:
520: /* Jacobian from p=2 */
521: v[0] = -hxdhy;
522: col[0].j = j - 1;
523: col[0].i = i;
524: v[1] = -hydhx;
525: col[1].j = j;
526: col[1].i = i - 1;
527: v[2] = 2.0 * (hydhx + hxdhy) - sc * PetscExpScalar(u);
528: col[2].j = row.j;
529: col[2].i = row.i;
530: v[3] = -hydhx;
531: col[3].j = j;
532: col[3].i = i + 1;
533: v[4] = -hxdhy;
534: col[4].j = j + 1;
535: col[4].i = i;
536: PetscCall(MatSetValuesStencil(B, 1, &row, 5, col, v, INSERT_VALUES));
537: break;
538: case JAC_PICARD:
539: /* Jacobian arising from Picard linearization */
540: v[0] = -hxdhy * e_S;
541: col[0].j = j - 1;
542: col[0].i = i;
543: v[1] = -hydhx * e_W;
544: col[1].j = j;
545: col[1].i = i - 1;
546: v[2] = (e_W + e_E) * hydhx + (e_S + e_N) * hxdhy;
547: col[2].j = row.j;
548: col[2].i = row.i;
549: v[3] = -hydhx * e_E;
550: col[3].j = j;
551: col[3].i = i + 1;
552: v[4] = -hxdhy * e_N;
553: col[4].j = j + 1;
554: col[4].i = i;
555: PetscCall(MatSetValuesStencil(B, 1, &row, 5, col, v, INSERT_VALUES));
556: break;
557: case JAC_STAR:
558: /* Full Jacobian, but only a star stencil */
559: col[0].j = j - 1;
560: col[0].i = i;
561: col[1].j = j;
562: col[1].i = i - 1;
563: col[2].j = j;
564: col[2].i = i;
565: col[3].j = j;
566: col[3].i = i + 1;
567: col[4].j = j + 1;
568: col[4].i = i;
569: v[0] = -hxdhy * newt_S + cross_EW;
570: v[1] = -hydhx * newt_W + cross_NS;
571: v[2] = hxdhy * (newt_N + newt_S) + hydhx * (newt_E + newt_W) - sc * PetscExpScalar(u);
572: v[3] = -hydhx * newt_E - cross_NS;
573: v[4] = -hxdhy * newt_N - cross_EW;
574: PetscCall(MatSetValuesStencil(B, 1, &row, 5, col, v, INSERT_VALUES));
575: break;
576: case JAC_NEWTON:
577: /* The Jacobian is
579: -div [ eta (grad u) + deta (grad u0 . grad u) grad u0 ] - (eE u0) u
581: */
582: col[0].j = j - 1;
583: col[0].i = i - 1;
584: col[1].j = j - 1;
585: col[1].i = i;
586: col[2].j = j - 1;
587: col[2].i = i + 1;
588: col[3].j = j;
589: col[3].i = i - 1;
590: col[4].j = j;
591: col[4].i = i;
592: col[5].j = j;
593: col[5].i = i + 1;
594: col[6].j = j + 1;
595: col[6].i = i - 1;
596: col[7].j = j + 1;
597: col[7].i = i;
598: col[8].j = j + 1;
599: col[8].i = i + 1;
600: v[0] = -0.25 * (skew_S + skew_W);
601: v[1] = -hxdhy * newt_S + cross_EW;
602: v[2] = 0.25 * (skew_S + skew_E);
603: v[3] = -hydhx * newt_W + cross_NS;
604: v[4] = hxdhy * (newt_N + newt_S) + hydhx * (newt_E + newt_W) - sc * PetscExpScalar(u);
605: v[5] = -hydhx * newt_E - cross_NS;
606: v[6] = 0.25 * (skew_N + skew_W);
607: v[7] = -hxdhy * newt_N - cross_EW;
608: v[8] = -0.25 * (skew_N + skew_E);
609: PetscCall(MatSetValuesStencil(B, 1, &row, 9, col, v, INSERT_VALUES));
610: break;
611: default:
612: SETERRQ(PetscObjectComm((PetscObject)info->da), PETSC_ERR_SUP, "Jacobian type %d not implemented", user->jtype);
613: }
614: }
615: }
616: }
618: /*
619: Assemble matrix, using the 2-step process:
620: MatAssemblyBegin(), MatAssemblyEnd().
621: */
622: PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
623: PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
625: if (J != B) {
626: PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
627: PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
628: }
630: /*
631: Tell the matrix we will never add a new nonzero location to the
632: matrix. If we do, it will generate an error.
633: */
634: if (user->jtype == JAC_NEWTON) PetscCall(PetscLogFlops(info->xm * info->ym * (131.0)));
635: PetscFunctionReturn(PETSC_SUCCESS);
636: }
638: /***********************************************************
639: * PreCheck implementation
640: ***********************************************************/
641: PetscErrorCode PreCheckSetFromOptions(PreCheck precheck)
642: {
643: PetscBool flg;
645: PetscFunctionBeginUser;
646: PetscOptionsBegin(precheck->comm, NULL, "PreCheck Options", "none");
647: PetscCall(PetscOptionsReal("-precheck_angle", "Angle in degrees between successive search directions necessary to activate step correction", "", precheck->angle, &precheck->angle, NULL));
648: flg = PETSC_FALSE;
649: PetscCall(PetscOptionsBool("-precheck_monitor", "Monitor choices made by precheck routine", "", flg, &flg, NULL));
650: if (flg) PetscCall(PetscViewerASCIIOpen(precheck->comm, "stdout", &precheck->monitor));
651: PetscOptionsEnd();
652: PetscFunctionReturn(PETSC_SUCCESS);
653: }
655: /*
656: Compare the direction of the current and previous step, modify the current step accordingly
657: */
658: PetscErrorCode PreCheckFunction(SNESLineSearch linesearch, Vec X, Vec Y, PetscBool *changed, void *ctx)
659: {
660: PreCheck precheck;
661: Vec Ylast;
662: PetscScalar dot;
663: PetscInt iter;
664: PetscReal ynorm, ylastnorm, theta, angle_radians;
665: SNES snes;
667: PetscFunctionBeginUser;
668: PetscCall(SNESLineSearchGetSNES(linesearch, &snes));
669: precheck = (PreCheck)ctx;
670: if (!precheck->Ylast) PetscCall(VecDuplicate(Y, &precheck->Ylast));
671: Ylast = precheck->Ylast;
672: PetscCall(SNESGetIterationNumber(snes, &iter));
673: if (iter < 1) {
674: PetscCall(VecCopy(Y, Ylast));
675: *changed = PETSC_FALSE;
676: PetscFunctionReturn(PETSC_SUCCESS);
677: }
679: PetscCall(VecDot(Y, Ylast, &dot));
680: PetscCall(VecNorm(Y, NORM_2, &ynorm));
681: PetscCall(VecNorm(Ylast, NORM_2, &ylastnorm));
682: /* Compute the angle between the vectors Y and Ylast, clip to keep inside the domain of acos() */
683: theta = PetscAcosReal((PetscReal)PetscClipInterval(PetscAbsScalar(dot) / (ynorm * ylastnorm), -1.0, 1.0));
684: angle_radians = precheck->angle * PETSC_PI / 180.;
685: if (PetscAbsReal(theta) < angle_radians || PetscAbsReal(theta - PETSC_PI) < angle_radians) {
686: /* Modify the step Y */
687: PetscReal alpha, ydiffnorm;
688: PetscCall(VecAXPY(Ylast, -1.0, Y));
689: PetscCall(VecNorm(Ylast, NORM_2, &ydiffnorm));
690: alpha = ylastnorm / ydiffnorm;
691: PetscCall(VecCopy(Y, Ylast));
692: PetscCall(VecScale(Y, alpha));
693: if (precheck->monitor) PetscCall(PetscViewerASCIIPrintf(precheck->monitor, "Angle %g degrees less than threshold %g, corrected step by alpha=%g\n", (double)(theta * 180. / PETSC_PI), (double)precheck->angle, (double)alpha));
694: } else {
695: PetscCall(VecCopy(Y, Ylast));
696: *changed = PETSC_FALSE;
697: if (precheck->monitor) PetscCall(PetscViewerASCIIPrintf(precheck->monitor, "Angle %g degrees exceeds threshold %g, no correction applied\n", (double)(theta * 180. / PETSC_PI), (double)precheck->angle));
698: }
699: PetscFunctionReturn(PETSC_SUCCESS);
700: }
702: PetscErrorCode PreCheckDestroy(PreCheck *precheck)
703: {
704: PetscFunctionBeginUser;
705: if (!*precheck) PetscFunctionReturn(PETSC_SUCCESS);
706: PetscCall(VecDestroy(&(*precheck)->Ylast));
707: PetscCall(PetscViewerDestroy(&(*precheck)->monitor));
708: PetscCall(PetscFree(*precheck));
709: PetscFunctionReturn(PETSC_SUCCESS);
710: }
712: PetscErrorCode PreCheckCreate(MPI_Comm comm, PreCheck *precheck)
713: {
714: PetscFunctionBeginUser;
715: PetscCall(PetscNew(precheck));
717: (*precheck)->comm = comm;
718: (*precheck)->angle = 10.; /* only active if angle is less than 10 degrees */
719: PetscFunctionReturn(PETSC_SUCCESS);
720: }
722: /*
723: Applies some sweeps on nonlinear Gauss-Seidel on each process
725: */
726: PetscErrorCode NonlinearGS(SNES snes, Vec X, Vec B, void *ctx)
727: {
728: PetscInt i, j, k, xs, ys, xm, ym, its, tot_its, sweeps, l, m;
729: PetscReal hx, hy, hxdhy, hydhx, dhx, dhy, sc;
730: PetscScalar **x, **b, bij, F, F0 = 0, J, y, u, eu;
731: PetscReal atol, rtol, stol;
732: DM da;
733: AppCtx *user = (AppCtx *)ctx;
734: Vec localX, localB;
735: DMDALocalInfo info;
737: PetscFunctionBeginUser;
738: PetscCall(SNESGetDM(snes, &da));
739: PetscCall(DMDAGetLocalInfo(da, &info));
741: hx = 1.0 / (PetscReal)(info.mx - 1);
742: hy = 1.0 / (PetscReal)(info.my - 1);
743: sc = hx * hy * user->lambda;
744: dhx = 1 / hx;
745: dhy = 1 / hy;
746: hxdhy = hx / hy;
747: hydhx = hy / hx;
749: tot_its = 0;
750: PetscCall(SNESNGSGetSweeps(snes, &sweeps));
751: PetscCall(SNESNGSGetTolerances(snes, &atol, &rtol, &stol, &its));
752: PetscCall(DMGetLocalVector(da, &localX));
753: if (B) PetscCall(DMGetLocalVector(da, &localB));
754: if (B) {
755: PetscCall(DMGlobalToLocalBegin(da, B, INSERT_VALUES, localB));
756: PetscCall(DMGlobalToLocalEnd(da, B, INSERT_VALUES, localB));
757: }
758: if (B) PetscCall(DMDAVecGetArrayRead(da, localB, &b));
759: PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
760: PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
761: PetscCall(DMDAVecGetArray(da, localX, &x));
762: for (l = 0; l < sweeps; l++) {
763: /*
764: Get local grid boundaries (for 2-dimensional DMDA):
765: xs, ys - starting grid indices (no ghost points)
766: xm, ym - widths of local grid (no ghost points)
767: */
768: PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));
769: for (m = 0; m < 2; m++) {
770: for (j = ys; j < ys + ym; j++) {
771: for (i = xs + (m + j) % 2; i < xs + xm; i += 2) {
772: PetscReal xx = i * hx, yy = j * hy;
773: if (B) bij = b[j][i];
774: else bij = 0.;
776: if (i == 0 || j == 0 || i == info.mx - 1 || j == info.my - 1) {
777: /* boundary conditions are all zero Dirichlet */
778: x[j][i] = 0.0 + bij;
779: } else {
780: const PetscScalar u_E = x[j][i + 1], u_W = x[j][i - 1], u_N = x[j + 1][i], u_S = x[j - 1][i];
781: const PetscScalar uy_E = 0.25 * dhy * (x[j + 1][i] + x[j + 1][i + 1] - x[j - 1][i] - x[j - 1][i + 1]), uy_W = 0.25 * dhy * (x[j + 1][i - 1] + x[j + 1][i] - x[j - 1][i - 1] - x[j - 1][i]), ux_N = 0.25 * dhx * (x[j][i + 1] + x[j + 1][i + 1] - x[j][i - 1] - x[j + 1][i - 1]), ux_S = 0.25 * dhx * (x[j - 1][i + 1] + x[j][i + 1] - x[j - 1][i - 1] - x[j][i - 1]);
782: u = x[j][i];
783: for (k = 0; k < its; k++) {
784: const PetscScalar ux_E = dhx * (u_E - u), ux_W = dhx * (u - u_W), uy_N = dhy * (u_N - u), uy_S = dhy * (u - u_S), e_E = eta(user, xx, yy, ux_E, uy_E), e_W = eta(user, xx, yy, ux_W, uy_W), e_N = eta(user, xx, yy, ux_N, uy_N), e_S = eta(user, xx, yy, ux_S, uy_S), de_E = deta(user, xx, yy, ux_E, uy_E), de_W = deta(user, xx, yy, ux_W, uy_W), de_N = deta(user, xx, yy, ux_N, uy_N), de_S = deta(user, xx, yy, ux_S, uy_S), newt_E = e_E + de_E * PetscSqr(ux_E), newt_W = e_W + de_W * PetscSqr(ux_W), newt_N = e_N + de_N * PetscSqr(uy_N), newt_S = e_S + de_S * PetscSqr(uy_S), uxx = -hy * (e_E * ux_E - e_W * ux_W), uyy = -hx * (e_N * uy_N - e_S * uy_S);
786: if (sc) eu = PetscExpScalar(u);
787: else eu = 0;
789: F = uxx + uyy - sc * eu - bij;
790: if (k == 0) F0 = F;
791: J = hxdhy * (newt_N + newt_S) + hydhx * (newt_E + newt_W) - sc * eu;
792: y = F / J;
793: u -= y;
794: tot_its++;
795: if (atol > PetscAbsReal(PetscRealPart(F)) || rtol * PetscAbsReal(PetscRealPart(F0)) > PetscAbsReal(PetscRealPart(F)) || stol * PetscAbsReal(PetscRealPart(u)) > PetscAbsReal(PetscRealPart(y))) break;
796: }
797: x[j][i] = u;
798: }
799: }
800: }
801: }
802: /*
803: x Restore vector
804: */
805: }
806: PetscCall(DMDAVecRestoreArray(da, localX, &x));
807: PetscCall(DMLocalToGlobalBegin(da, localX, INSERT_VALUES, X));
808: PetscCall(DMLocalToGlobalEnd(da, localX, INSERT_VALUES, X));
809: PetscCall(PetscLogFlops(tot_its * (118.0)));
810: PetscCall(DMRestoreLocalVector(da, &localX));
811: if (B) {
812: PetscCall(DMDAVecRestoreArrayRead(da, localB, &b));
813: PetscCall(DMRestoreLocalVector(da, &localB));
814: }
815: PetscFunctionReturn(PETSC_SUCCESS);
816: }
818: /*TEST
820: test:
821: nsize: 2
822: args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype NEWTON
823: requires: !single
825: test:
826: suffix: 2
827: nsize: 2
828: args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype PICARD -precheck 1
829: requires: !single
831: test:
832: suffix: 3
833: nsize: 2
834: args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype PICARD -picard -precheck 1 -p 3
835: requires: !single
837: test:
838: suffix: 3_star
839: nsize: 2
840: args: -snes_monitor_short -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -jtype PICARD -picard -precheck 1 -p 3 -alloc_star
841: output_file: output/ex15_3.out
842: requires: !single
844: test:
845: suffix: 4
846: 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
847: requires: !single
849: test:
850: suffix: 5
851: args: -snes_monitor_short -snes_type newtontr -npc_snes_type ngs -snes_npc_side right -da_grid_x 20 -da_grid_y 20 -p 1.3 -lambda 1 -ksp_converged_reason -pc_type ilu
852: requires: !single
854: test:
855: suffix: lag_jac
856: nsize: 4
857: 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
858: requires: !single
860: test:
861: suffix: lag_pc
862: nsize: 4
863: 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
864: requires: !single
866: test:
867: suffix: nleqerr
868: 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
869: requires: !single
871: test:
872: suffix: mf
873: args: -snes_monitor_short -pc_type lu -da_refine 4 -p 3 -ksp_rtol 1.e-12 -snes_mf_operator
874: requires: !single
876: test:
877: suffix: mf_picard
878: args: -snes_monitor_short -pc_type lu -da_refine 4 -p 3 -ksp_rtol 1.e-12 -snes_mf_operator -picard
879: requires: !single
880: output_file: output/ex15_mf.out
882: test:
883: suffix: fd_picard
884: args: -snes_monitor_short -pc_type lu -da_refine 2 -p 3 -ksp_rtol 1.e-12 -snes_fd -picard
885: requires: !single
887: test:
888: suffix: fd_color_picard
889: args: -snes_monitor_short -pc_type lu -da_refine 4 -p 3 -ksp_rtol 1.e-12 -snes_fd_color -picard
890: requires: !single
891: output_file: output/ex15_mf.out
893: TEST*/