/*T Concepts: KSP^solving a system of linear equations Concepts: KSP^Laplacian, 2d Processors: n T*/ /* Laplacian in 2D. Modeled by the partial differential equation div grad u = f, 0 < x,y < 1, with forcing function f = e^{-(1 - x)^2/\nu} e^{-(1 - y)^2/\nu} with pure Neumann boundary conditions The functions are cell-centered This uses multigrid to solve the linear system Contributed by Andrei Draganescu Note the nice multigrid convergence despite the fact it is only using peicewise constant interpolation/restriction. This is because cell-centered multigrid does not need the same rule: polynomial degree(interpolation) + polynomial degree(restriction) + 2 > degree of PDE that vertex based multigrid needs. */ static char help[] = "Solves 2D inhomogeneous Laplacian using multigrid.\n\n"; #include #include #include #include extern PetscErrorCode ComputeMatrix(DMMG,Mat,Mat); extern PetscErrorCode ComputeRHS(DMMG,Vec); typedef enum {DIRICHLET, NEUMANN} BCType; typedef struct { PetscScalar nu; BCType bcType; } UserContext; #undef __FUNCT__ #define __FUNCT__ "main" int main(int argc,char **argv) { DMMG *dmmg; DM da; UserContext user; PetscReal norm; const char *bcTypes[2] = {"dirichlet","neumann"}; PetscErrorCode ierr; PetscInt l,bc; PetscInitialize(&argc,&argv,(char *)0,help); ierr = DMMGCreate(PETSC_COMM_WORLD,3,PETSC_NULL,&dmmg);CHKERRQ(ierr); ierr = DMDACreate2d(PETSC_COMM_WORLD, DMDA_BOUNDARY_NONE, DMDA_BOUNDARY_NONE,DMDA_STENCIL_STAR,3,3,PETSC_DECIDE,PETSC_DECIDE,1,1,0,0,&da);CHKERRQ(ierr); ierr = DMDASetInterpolationType(da, DMDA_Q0);CHKERRQ(ierr); ierr = DMMGSetDM(dmmg,(DM)da);CHKERRQ(ierr); ierr = DMDestroy(&da);CHKERRQ(ierr); for (l = 0; l < DMMGGetLevels(dmmg); l++) { ierr = DMMGSetUser(dmmg,l,&user);CHKERRQ(ierr); } ierr = PetscOptionsBegin(PETSC_COMM_WORLD, "", "Options for the inhomogeneous Poisson equation", "DMMG"); user.nu = 0.1; ierr = PetscOptionsScalar("-nu", "The width of the Gaussian source", "ex29.c", 0.1, &user.nu, PETSC_NULL);CHKERRQ(ierr); bc = (PetscInt)NEUMANN; ierr = PetscOptionsEList("-bc_type","Type of boundary condition","ex29.c",bcTypes,2,bcTypes[0],&bc,PETSC_NULL);CHKERRQ(ierr); user.bcType = (BCType)bc; ierr = PetscOptionsEnd(); ierr = DMMGSetKSP(dmmg,ComputeRHS,ComputeMatrix);CHKERRQ(ierr); if (user.bcType == NEUMANN) { ierr = DMMGSetNullSpace(dmmg,PETSC_TRUE,0,PETSC_NULL);CHKERRQ(ierr); } ierr = DMMGSolve(dmmg);CHKERRQ(ierr); ierr = MatMult(DMMGGetJ(dmmg),DMMGGetx(dmmg),DMMGGetr(dmmg));CHKERRQ(ierr); ierr = VecAXPY(DMMGGetRHS(dmmg),-1.0,DMMGGetr(dmmg));CHKERRQ(ierr); ierr = VecNorm(DMMGGetr(dmmg),NORM_2,&norm);CHKERRQ(ierr); /* ierr = PetscPrintf(PETSC_COMM_WORLD,"Residual norm %g\n",norm);CHKERRQ(ierr); */ ierr = DMMGDestroy(dmmg);CHKERRQ(ierr); ierr = PetscFinalize(); return 0; } #undef __FUNCT__ #define __FUNCT__ "ComputeRHS" PetscErrorCode ComputeRHS(DMMG dmmg, Vec b) { DM da = dmmg->dm; UserContext *user = (UserContext *) dmmg->user; PetscErrorCode ierr; PetscInt i,j,mx,my,xm,ym,xs,ys; PetscScalar Hx,Hy; PetscScalar **array; PetscFunctionBegin; ierr = DMDAGetInfo(da, 0, &mx, &my, 0,0,0,0,0,0,0,0,0,0);CHKERRQ(ierr); Hx = 1.0 / (PetscReal)(mx); Hy = 1.0 / (PetscReal)(my); ierr = DMDAGetCorners(da,&xs,&ys,0,&xm,&ym,0);CHKERRQ(ierr); ierr = DMDAVecGetArray(da, b, &array);CHKERRQ(ierr); for (j=ys; jnu)*PetscExpScalar(-(((PetscReal)j+0.5)*Hy)*(((PetscReal)j+0.5)*Hy)/user->nu)*Hx*Hy; } } ierr = DMDAVecRestoreArray(da, b, &array);CHKERRQ(ierr); ierr = VecAssemblyBegin(b);CHKERRQ(ierr); ierr = VecAssemblyEnd(b);CHKERRQ(ierr); /* force right hand side to be consistent for singular matrix */ /* note this is really a hack, normally the model would provide you with a consistent right handside */ if (user->bcType == NEUMANN) { MatNullSpace nullspace; ierr = KSPGetNullSpace(dmmg->ksp,&nullspace);CHKERRQ(ierr); ierr = MatNullSpaceRemove(nullspace,b,PETSC_NULL);CHKERRQ(ierr); } PetscFunctionReturn(0); } #undef __FUNCT__ #define __FUNCT__ "ComputeMatrix" PetscErrorCode ComputeMatrix(DMMG dmmg, Mat J,Mat jac) { DM da = dmmg->dm; UserContext *user = (UserContext *) dmmg->user; PetscErrorCode ierr; PetscInt i,j,mx,my,xm,ym,xs,ys,num, numi, numj; PetscScalar v[5],Hx,Hy,HydHx,HxdHy; MatStencil row, col[5]; PetscFunctionBegin; ierr = DMDAGetInfo(da,0,&mx,&my,0,0,0,0,0,0,0,0,0,0);CHKERRQ(ierr); Hx = 1.0 / (PetscReal)(mx); Hy = 1.0 / (PetscReal)(my); HxdHy = Hx/Hy; HydHx = Hy/Hx; ierr = DMDAGetCorners(da,&xs,&ys,0,&xm,&ym,0);CHKERRQ(ierr); for (j=ys; jbcType == DIRICHLET) { v[0] = 2.0*(HxdHy + HydHx); ierr = MatSetValuesStencil(jac,1,&row,1,&row,v,INSERT_VALUES);CHKERRQ(ierr); SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"Dirichlet boundary conditions not supported !\n"); } else if (user->bcType == NEUMANN) { num = 0; numi=0; numj=0; if (j!=0) { v[num] = -HxdHy; col[num].i = i; col[num].j = j-1; num++; numj++; } if (i!=0) { v[num] = -HydHx; col[num].i = i-1; col[num].j = j; num++; numi++; } if (i!=mx-1) { v[num] = -HydHx; col[num].i = i+1; col[num].j = j; num++; numi++; } if (j!=my-1) { v[num] = -HxdHy; col[num].i = i; col[num].j = j+1; num++; numj++; } v[num] = (PetscReal)(numj)*HxdHy + (PetscReal)(numi)*HydHx; col[num].i = i; col[num].j = j; num++; ierr = MatSetValuesStencil(jac,1,&row,num,col,v,INSERT_VALUES);CHKERRQ(ierr); } } else { v[0] = -HxdHy; col[0].i = i; col[0].j = j-1; v[1] = -HydHx; col[1].i = i-1; col[1].j = j; v[2] = 2.0*(HxdHy + HydHx); col[2].i = i; col[2].j = j; v[3] = -HydHx; col[3].i = i+1; col[3].j = j; v[4] = -HxdHy; col[4].i = i; col[4].j = j+1; ierr = MatSetValuesStencil(jac,1,&row,5,col,v,INSERT_VALUES);CHKERRQ(ierr); } } } ierr = MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); ierr = MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); PetscFunctionReturn(0); }