2: static char help[] = "Solves a variable Poisson problem with KSP.\n\n";
4: /*T
5: Concepts: KSP^basic sequential example
6: Concepts: KSP^Laplacian, 2d
7: Concepts: Laplacian, 2d
8: Processors: 1
9: T*/
11: /*
12: Include "petscksp.h" so that we can use KSP solvers. Note that this file
13: automatically includes:
14: petscsys.h - base PETSc routines petscvec.h - vectors
15: petscmat.h - matrices
16: petscis.h - index sets petscksp.h - Krylov subspace methods
17: petscviewer.h - viewers petscpc.h - preconditioners
18: */
19: #include <petscksp.h> 21: /*
22: User-defined context that contains all the data structures used
23: in the linear solution process.
24: */
25: typedef struct {
26: Vec x,b; /* solution vector, right-hand-side vector */
27: Mat A; /* sparse matrix */
28: KSP ksp; /* linear solver context */
29: PetscInt m,n; /* grid dimensions */
30: PetscScalar hx2,hy2; /* 1/(m+1)*(m+1) and 1/(n+1)*(n+1) */
31: } UserCtx;
33: extern PetscErrorCode UserInitializeLinearSolver(PetscInt,PetscInt,UserCtx*);
34: extern PetscErrorCode UserFinalizeLinearSolver(UserCtx*);
35: extern PetscErrorCode UserDoLinearSolver(PetscScalar*,UserCtx *userctx,PetscScalar *b,PetscScalar *x);
37: int main(int argc,char **args) 38: {
39: UserCtx userctx;
41: PetscInt m = 6,n = 7,t,tmax = 2,i,Ii,j,N;
42: PetscScalar *userx,*rho,*solution,*userb,hx,hy,x,y;
43: PetscReal enorm;
45: /*
46: Initialize the PETSc libraries
47: */
48: PetscInitialize(&argc,&args,(char*)0,help);if (ierr) return ierr;
49: /*
50: The next two lines are for testing only; these allow the user to
51: decide the grid size at runtime.
52: */
53: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
54: PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL);
56: /*
57: Create the empty sparse matrix and linear solver data structures
58: */
59: UserInitializeLinearSolver(m,n,&userctx);
60: N = m*n;
62: /*
63: Allocate arrays to hold the solution to the linear system.
64: This is not normally done in PETSc programs, but in this case,
65: since we are calling these routines from a non-PETSc program, we
66: would like to reuse the data structures from another code. So in
67: the context of a larger application these would be provided by
68: other (non-PETSc) parts of the application code.
69: */
70: PetscMalloc1(N,&userx);
71: PetscMalloc1(N,&userb);
72: PetscMalloc1(N,&solution);
74: /*
75: Allocate an array to hold the coefficients in the elliptic operator
76: */
77: PetscMalloc1(N,&rho);
79: /*
80: Fill up the array rho[] with the function rho(x,y) = x; fill the
81: right-hand-side b[] and the solution with a known problem for testing.
82: */
83: hx = 1.0/(m+1);
84: hy = 1.0/(n+1);
85: y = hy;
86: Ii = 0;
87: for (j=0; j<n; j++) {
88: x = hx;
89: for (i=0; i<m; i++) {
90: rho[Ii] = x;
91: solution[Ii] = PetscSinScalar(2.*PETSC_PI*x)*PetscSinScalar(2.*PETSC_PI*y);
92: userb[Ii] = -2*PETSC_PI*PetscCosScalar(2*PETSC_PI *x)*PetscSinScalar(2*PETSC_PI*y) +
93: 8*PETSC_PI*PETSC_PI*x*PetscSinScalar(2*PETSC_PI *x)*PetscSinScalar(2*PETSC_PI*y);
94: x += hx;
95: Ii++;
96: }
97: y += hy;
98: }
100: /*
101: Loop over a bunch of timesteps, setting up and solver the linear
102: system for each time-step.
104: Note this is somewhat artificial. It is intended to demonstrate how
105: one may reuse the linear solver stuff in each time-step.
106: */
107: for (t=0; t<tmax; t++) {
108: UserDoLinearSolver(rho,&userctx,userb,userx);
110: /*
111: Compute error: Note that this could (and usually should) all be done
112: using the PETSc vector operations. Here we demonstrate using more
113: standard programming practices to show how they may be mixed with
114: PETSc.
115: */
116: enorm = 0.0;
117: for (i=0; i<N; i++) enorm += PetscRealPart(PetscConj(solution[i]-userx[i])*(solution[i]-userx[i]));
118: enorm *= PetscRealPart(hx*hy);
119: PetscPrintf(PETSC_COMM_WORLD,"m %D n %D error norm %g\n",m,n,(double)enorm);
120: }
122: /*
123: We are all finished solving linear systems, so we clean up the
124: data structures.
125: */
126: PetscFree(rho);
127: PetscFree(solution);
128: PetscFree(userx);
129: PetscFree(userb);
130: UserFinalizeLinearSolver(&userctx);
131: PetscFinalize();
132: return ierr;
133: }
135: /* ------------------------------------------------------------------------*/
136: PetscErrorCode UserInitializeLinearSolver(PetscInt m,PetscInt n,UserCtx *userctx)137: {
139: PetscInt N;
141: /*
142: Here we assume use of a grid of size m x n, with all points on the
143: interior of the domain, i.e., we do not include the points corresponding
144: to homogeneous Dirichlet boundary conditions. We assume that the domain
145: is [0,1]x[0,1].
146: */
147: userctx->m = m;
148: userctx->n = n;
149: userctx->hx2 = (m+1)*(m+1);
150: userctx->hy2 = (n+1)*(n+1);
151: N = m*n;
153: /*
154: Create the sparse matrix. Preallocate 5 nonzeros per row.
155: */
156: MatCreateSeqAIJ(PETSC_COMM_SELF,N,N,5,0,&userctx->A);
158: /*
159: Create vectors. Here we create vectors with no memory allocated.
160: This way, we can use the data structures already in the program
161: by using VecPlaceArray() subroutine at a later stage.
162: */
163: VecCreateSeqWithArray(PETSC_COMM_SELF,1,N,NULL,&userctx->b);
164: VecDuplicate(userctx->b,&userctx->x);
166: /*
167: Create linear solver context. This will be used repeatedly for all
168: the linear solves needed.
169: */
170: KSPCreate(PETSC_COMM_SELF,&userctx->ksp);
172: return 0;
173: }
175: /*
176: Solves -div (rho grad psi) = F using finite differences.
177: rho is a 2-dimensional array of size m by n, stored in Fortran
178: style by columns. userb is a standard one-dimensional array.
179: */
180: /* ------------------------------------------------------------------------*/
181: PetscErrorCode UserDoLinearSolver(PetscScalar *rho,UserCtx *userctx,PetscScalar *userb,PetscScalar *userx)182: {
184: PetscInt i,j,Ii,J,m = userctx->m,n = userctx->n;
185: Mat A = userctx->A;
186: PC pc;
187: PetscScalar v,hx2 = userctx->hx2,hy2 = userctx->hy2;
189: /*
190: This is not the most efficient way of generating the matrix
191: but let's not worry about it. We should have separate code for
192: the four corners, each edge and then the interior. Then we won't
193: have the slow if-tests inside the loop.
195: Computes the operator
196: -div rho grad
197: on an m by n grid with zero Dirichlet boundary conditions. The rho
198: is assumed to be given on the same grid as the finite difference
199: stencil is applied. For a staggered grid, one would have to change
200: things slightly.
201: */
202: Ii = 0;
203: for (j=0; j<n; j++) {
204: for (i=0; i<m; i++) {
205: if (j>0) {
206: J = Ii - m;
207: v = -.5*(rho[Ii] + rho[J])*hy2;
208: MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
209: }
210: if (j<n-1) {
211: J = Ii + m;
212: v = -.5*(rho[Ii] + rho[J])*hy2;
213: MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
214: }
215: if (i>0) {
216: J = Ii - 1;
217: v = -.5*(rho[Ii] + rho[J])*hx2;
218: MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
219: }
220: if (i<m-1) {
221: J = Ii + 1;
222: v = -.5*(rho[Ii] + rho[J])*hx2;
223: MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);
224: }
225: v = 2.0*rho[Ii]*(hx2+hy2);
226: MatSetValues(A,1,&Ii,1,&Ii,&v,INSERT_VALUES);
227: Ii++;
228: }
229: }
231: /*
232: Assemble matrix
233: */
234: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
235: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
237: /*
238: Set operators. Here the matrix that defines the linear system
239: also serves as the preconditioning matrix. Since all the matrices
240: will have the same nonzero pattern here, we indicate this so the
241: linear solvers can take advantage of this.
242: */
243: KSPSetOperators(userctx->ksp,A,A);
245: /*
246: Set linear solver defaults for this problem (optional).
247: - Here we set it to use direct LU factorization for the solution
248: */
249: KSPGetPC(userctx->ksp,&pc);
250: PCSetType(pc,PCLU);
252: /*
253: Set runtime options, e.g.,
254: -ksp_type <type> -pc_type <type> -ksp_monitor -ksp_rtol <rtol>
255: These options will override those specified above as long as
256: KSPSetFromOptions() is called _after_ any other customization
257: routines.
259: Run the program with the option -help to see all the possible
260: linear solver options.
261: */
262: KSPSetFromOptions(userctx->ksp);
264: /*
265: This allows the PETSc linear solvers to compute the solution
266: directly in the user's array rather than in the PETSc vector.
268: This is essentially a hack and not highly recommend unless you
269: are quite comfortable with using PETSc. In general, users should
270: write their entire application using PETSc vectors rather than
271: arrays.
272: */
273: VecPlaceArray(userctx->x,userx);
274: VecPlaceArray(userctx->b,userb);
276: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
277: Solve the linear system
278: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
280: KSPSolve(userctx->ksp,userctx->b,userctx->x);
282: /*
283: Put back the PETSc array that belongs in the vector xuserctx->x
284: */
285: VecResetArray(userctx->x);
286: VecResetArray(userctx->b);
288: return 0;
289: }
291: /* ------------------------------------------------------------------------*/
292: PetscErrorCode UserFinalizeLinearSolver(UserCtx *userctx)293: {
295: /*
296: We are all done and don't need to solve any more linear systems, so
297: we free the work space. All PETSc objects should be destroyed when
298: they are no longer needed.
299: */
300: KSPDestroy(&userctx->ksp);
301: VecDestroy(&userctx->x);
302: VecDestroy(&userctx->b);
303: MatDestroy(&userctx->A);
304: return 0;
305: }
308: /*TEST
310: test:
311: args: -m 19 -n 20 -ksp_gmres_cgs_refinement_type refine_always
313: TEST*/