Algebraic multigrid methods for elastic structures with highly discontinuous coefficients

Abstract

In this paper, we propose two types of algebraic multigrid (AMG) methods to find the numerical solution of elastic structures with highly discontinuous coefficients. One is the interface preserving coarsening AMG method. The idea of this method is to capture the discontinuous behavior of the gradient of the displacement functions along the interfaces. It selects coarse grid points so that all the coarse grids are aligned with the interface for regular interface problems on structured grids and for irregular interface problems on unstructured grids in a purely algebraic way. As a result, AMG with simple linear interpolation and point block Gauss–Seidel smoothing is sufficient to obtain the usual rapid multigrid convergence. The process of coarse grid selection given in this paper can be performed in parallel. Another method introduced in this paper is an AMG method by applying a special block Gauss–Seidel smoother with blocks corresponding to the constant coefficient regions and their interfaces. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the resulting AMG methods are more robust and efficient than the commonly used AMG method in both CPU times and numbers of iteration for elastic structures with highly discontinuous coefficients.