Spectral AMGe ($\rho$AMGe)

We introduce spectral element-based algebraic multigrid ($\rho$AMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically "smooth" vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of "strong" connections (relatively large coefficients in the operator matrix). One aim of $\rho$AMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. $\rho$AMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for $\rho$AMGe along with numerical experiments demonstrating its robustness.