ALGEBRAIC MULTIGRID METHOD
We consider solving an SPD matrix equation Ax = b, where A could be obtained by the finite element discretization on a unstructured grids. A coarsening of the graph of A is needed and restriction and prolongation can be constructued based on the coarsening. We focus on the classic algebraic multigrid based on the strong connectness. A breif introduction can be found at Falgout, RD. An Introduction to Algebraic Multigrid
Contents
Coarsening
See coarsenAMGdoc
Prolongation
The submatrix A_{cf} is used to construct the interpolation of values on fine nodes from that of coarse nodes. The weight is normalized to preserve the constant.
Test: Different meshes
We consider linear finite element discretization of the Poisson equation with homongenous Dirichlet boundary condition on different meshes. We summarize the results in AMG Test I which shows our AMG solver is of complexity O(Nlog(logN)).
Test: Different boundary conditions
We consider the linear finite element discretization of Poisson equation on the unstructured mesh with Dirichlet or Neumann boundary conditions. We summarize the results in AMG Test II which shows our AMG solver is robust to different boundary conditions.
Test: Different time stepsize
We consider the linear finite element discretization of heat equation on the unstructured mesh with Neumann boundary conditions. We test the implicit time discretization with various time stepsizes dt = 1/h^4, 1/h^2, 1/h, 1. We summarize the results in AMG Test III which shows our AMG solver is robust to different time steps.
Test: Three dimensional problems
We consider the linear finite element discretization of Poisson equation in three dimensions with Dirichlet boundary conditions and compare geometric multigrid and algebraic multigrid in AMG Test IV.
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