Algebraic multigrid based on element interpolation (AMGE)

We introduce AMGe, an algebraic multigrid method for solving the discrete e quations that arise in Ritz-type finite element methods for partial differe ntial equations. Assuming access to the element sti ness matrices, we have that AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measur es are used to determine local representations of algebraically smooth erro r components that provide the basis for constructing effective interpolatio n and, hence, the coarsening process for AMG. Here, we focus on the interpo lation process; choice of the coarse grids based on these measures is the s ubject of current research. We develop a theoretical foundation for AMGe an d present numerical results that demonstrate the efficacy of the method.