Scaling by the volume element causes the rediscretized coarse grid problem to be scaled like a Galerkin coarse operator. This is done automatically when you use finite element methods.
Jason Lefley <[email protected]> writes: >> On Jun 26, 2017, at 7:52 PM, Matthew Knepley <[email protected]> wrote: >> >> On Mon, Jun 26, 2017 at 8:37 PM, Jason Lefley <[email protected] >> <mailto:[email protected]>> wrote: >>> Okay, when you say a Poisson problem, I assumed you meant >>> >>> div grad phi = f >>> >>> However, now it appears that you have >>> >>> div D grad phi = f >>> >>> Is this true? It would explain your results. Your coarse operator is >>> inaccurate. AMG makes the coarse operator directly >>> from the matrix, so it incorporates coefficient variation. Galerkin >>> projection makes the coarse operator using R A P >>> from your original operator A, and this is accurate enough to get good >>> convergence. So your coefficient representation >>> on the coarse levels is really bad. If you want to use GMG, you need to >>> figure out how to represent the coefficient on >>> coarser levels, which is sometimes called "renormalization". >>> >>> Matt >> >> I believe we are solving the first one. The discretized form we are using is >> equation 13 in this document: >> https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/Projects/prjpoisson.pdf >> >> <https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/Projects/prjpoisson.pdf> >> Would you clarify why you think we are solving the second equation? >> >> Something is wrong. The writeup is just showing the FD Laplacian. Can you >> take a look at SNES ex5, and let >> me know how your problem differs from that one? There were use GMG and can >> converge is a few (5-6) iterates, >> and if you use FMG you converge in 1 iterate. In fact, that is in my class >> notes on the CAAM 519 website. Its possible >> that you have badly scaled boundary values, which can cause convergence to >> deteriorate. >> >> Thanks, >> >> Matt >> > > I went through ex5 and some of the other Poisson/multigrid examples again and > noticed that they arrange the coefficients in a particular way. > > Our original attempt (solver_test.c) and some related codes that solve > similar problems use an arrangement like this: > > > u(i-1,j,k) - 2u(i,j,k) + u(i+1,j,k) u(i,j-1,k) - 2u(i,j,k) + > u(i,j+1,k) u(i,j,k-1) - 2u(i,j,k) + u(i,j,k+1) > ---------------------------------------- + > ---------------------------------------- + > ---------------------------------------- = f > dx^2 dy^2 > dz^2 > > That results in the coefficient matrix containing -2 * (1/dx^2 + 1/dy^2 + > 1/dz^2) on the diagonal and 1/dx^2, 1/dy^2 and 1/dz^2 on the off-diagonals. > I’ve also looked at some codes that assume h = dx = dy = dz, multiply f by > h^2 and then use -6 and 1 for the coefficients in the matrix. > > It looks like snes ex5, ksp ex32, and ksp ex34 rearrange the terms like this: > > > dy dz (u(i-1,j,k) - 2u(i,j,k) + u(i+1,j,k)) dx dz (u(i,j-1,k) > - 2u(i,j,k) + u(i,j+1,k)) dx dy (u(i,j,k-1) - 2u(i,j,k) + > u(i,j,k+1)) > --------------------------------------------------- + > --------------------------------------------------- + > --------------------------------------------------- = f dx dy dz > dx > dy dz > > > I changed our code to use this approach and we observe much better > convergence with the mg pre-conditioner. Is this renormalization? Can anyone > explain why this change has such an impact on convergence with geometric > multigrid as a pre-conditioner? It does not appear that the arrangement of > coefficients affects convergence when using conjugate gradient without a > pre-conditioner. Here’s output from some runs with the coefficients and right > hand side modified as described above: > > > $ mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 > -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 5 > -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale > -mg_levels_ksp_max_it 5 > right hand side 2 norm: 0.000244141 > right hand side infinity norm: 4.76406e-07 > 0 KSP preconditioned resid norm 3.578255383614e+00 true resid norm > 2.441406250000e-04 ||r(i)||/||b|| 1.000000000000e+00 > 1 KSP preconditioned resid norm 1.385321366208e-01 true resid norm > 4.207234652404e-05 ||r(i)||/||b|| 1.723283313625e-01 > 2 KSP preconditioned resid norm 4.459925861922e-03 true resid norm > 1.480495515589e-06 ||r(i)||/||b|| 6.064109631854e-03 > 3 KSP preconditioned resid norm 4.311025848794e-04 true resid norm > 1.021041953365e-07 ||r(i)||/||b|| 4.182187840984e-04 > 4 KSP preconditioned resid norm 1.619865162873e-05 true resid norm > 5.438265013849e-09 ||r(i)||/||b|| 2.227513349673e-05 > Linear solve converged due to CONVERGED_RTOL iterations 4 > KSP final norm of residual 5.43827e-09 > Residual 2 norm 5.43827e-09 > Residual infinity norm 6.25328e-11 > > > $ mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 > -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 5 > -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale > -mg_levels_ksp_max_it 5 -pc_mg_type full > 0 KSP preconditioned resid norm 3.459879233358e+00 true resid norm > 2.441406250000e-04 ||r(i)||/||b|| 1.000000000000e+00 > 1 KSP preconditioned resid norm 1.169574216505e-02 true resid norm > 4.856676267753e-06 ||r(i)||/||b|| 1.989294599272e-02 > 2 KSP preconditioned resid norm 1.158728408668e-04 true resid norm > 1.603345697667e-08 ||r(i)||/||b|| 6.567303977645e-05 > 3 KSP preconditioned resid norm 6.035498575583e-07 true resid norm > 1.613378731540e-10 ||r(i)||/||b|| 6.608399284389e-07 > Linear solve converged due to CONVERGED_RTOL iterations 3 > KSP final norm of residual 1.61338e-10 > Residual 2 norm 1.61338e-10 > Residual infinity norm 1.95499e-12 > > > $ mpirun -n 64 ./solver_test -da_grid_x 512 -da_grid_y 512 -da_grid_z 512 > -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 8 > -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale > -mg_levels_ksp_max_it 5 -pc_mg_type full -bc_type neumann > right hand side 2 norm: 3.05176e-05 > right hand side infinity norm: 7.45016e-09 > 0 KSP preconditioned resid norm 5.330711358065e+01 true resid norm > 3.051757812500e-05 ||r(i)||/||b|| 1.000000000000e+00 > 1 KSP preconditioned resid norm 4.687628546610e-04 true resid norm > 2.452752396888e-08 ||r(i)||/||b|| 8.037179054124e-04 > Linear solve converged due to CONVERGED_RTOL iterations 1 > KSP final norm of residual 2.45275e-08 > Residual 2 norm 2.45275e-08 > Residual infinity norm 8.41572e-10
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