Multigrid for Locally Refined Meshes

Yair Shapira

Los Alamos National Laboratory
Mail Stop B-256
Los Alamos, NM 87545


A multi-level method for the solution of finite element schemes on locally refined meshes is introduced. The upper bound on the condition number (in the two-level SPD case) is independent of the mesh-size (as is verified numerically for a diffusion problem with discontinuous coefficients) and of the jump in the diffusion coefficient. The curves of discontinuity need not be aligned with the coarse mesh. Indeed, numerical applications with ten levels of local refinement yield a rapid convergence of the corresponding ten-level multigrid V-cycle and other more parallelizable multigrid cycles, even when the discontinuities are invisible on most of the coarse meshes.