AMG TEST IV: Three Dimensional Problem

We consider the linear finite element discretization of Poisson equation in three dimensions with Dirichlet boundary conditions and compare geometric multigrid and algebraic multigrid.

Contents

Conclusion

The algebraic multigrid is robust to the size of the matrix. The iteration steps increases slightly. The preformance is better on structure grids than unstructure grids. The sparsity of the coarse grid is increased.

Uniform Mesh: Geometric Multigrid

clear all
option.solver = 'mg';
cubePoisson;

Time to assemble matrix equation  0.2 s
Direct solver 
#dof:      729,  #nnz:     2493,  iter:   ,  err = 3.6427e-16,  time = 0.46 s
No coarsening of the grid. 
Time to compute the error 0.17 s 
 H1 err  2.6    L2err 0.21 
Time to assemble matrix equation  0.4 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     4913,  #nnz:    23813, iter: 10,   err = 1.9593e-09,   time = 0.84 s
Time to compute the error 0.31 s 
 H1 err  1.4    L2err 0.061 
Time to assemble matrix equation  4.8 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    35937,  #nnz:   208917, iter: 10,   err = 5.1707e-09,   time =  1.1 s
Time to compute the error  2.5 s 
 H1 err 0.69    L2err 0.016 

Uniform Mesh: Algebraic Multigrid

clear all
option.solver = 'amg';
cubePoisson;

Time to assemble matrix equation 0.22 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 6.14,   level:  2,   coarse grid 141,   nnz/Nc 11.71
#dof:     343,    iter:  9,   err = 3.1527e-09,   time = 0.122 s
 
Time to compute the error 0.053 s 
 H1 err  2.6    L2err 0.21 
Time to assemble matrix equation 0.33 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 6.60,   level:  2,   coarse grid 1151,   nnz/Nc 14.37
#dof:    3375,    iter: 11,   err = 2.4172e-09,   time = 0.43 s
 
Time to compute the error 0.19 s 
 H1 err  1.4    L2err 0.061 
Time to assemble matrix equation  3.9 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 6.81,   level:  3,   coarse grid 1752,   nnz/Nc 26.91
#dof:   29791,    iter: 12,   err = 8.7948e-09,   time = 8.33 s
 
Time to compute the error  2.1 s 
 H1 err 0.69    L2err 0.016 

Unstructured Mesh

clear all;
load oilpump;
showboundary3(node,elem);
option.solver = 'amg';
cubePoisson;

Time to assemble matrix equation  1.1 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 7.96,   level:  2,   coarse grid 877,   nnz/Nc 9.81
#dof:    3392,    iter: 12,   err = 2.7210e-09,   time = 0.94 s
 
Time to compute the error 0.35 s 
 H1 err 0.035    L2err 0.00034 
Time to assemble matrix equation  6.2 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 12.35,   level:  3,   coarse grid 1875,   nnz/Nc 19.39
#dof:   40121,    iter: 16,   err = 4.2506e-09,   time =  6.5 s
 
Time to compute the error  3.6 s 
 H1 err 0.019    L2err 0.00014 
Time to assemble matrix equation   93 s
Multigrid W-cycle Preconditioner with Conjugate Gradient Method
  nnz/N: 13.85,   level:  5,   coarse grid 653,   nnz/Nc 22.68
#dof:  373751,    iter: 20,   err = 6.9171e-09,   time = 42.5 s
 
Time to compute the error   47 s 
 H1 err 0.01    L2err 4.6e-05 

Published with MATLAB® 7.14