Algebraic Multigrid Based on Element Interpolation (AMGe)

We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically “smooth” error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse “grids” based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.