Spectral AMGe ($\rho$AMGe)

Abstract

We introduce spectral element-based algebraic multigrid (ρAMGe), 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 ρAMGe 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. ρAMGe 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 ρAMGe along with numerical experiments demonstrating its robustness.

Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients

“…These bounds imply convergence that is independent of the contrast. In the presented preliminary numerical tests, we use geometric agglomerates; however, the algorithm is general and offers some simplifications over the previously proposed spectral agglomerate AMGe methods (cf., [1,2]). …”

“…Introduce a coarser mesh T (1) ⊃ T (0) with parameter h (1) . We assume that each coarse element T c ∈ T (1) is the union of fine elements τ with τ ∈ T (0) .…”

“…We assume that each coarse element T c ∈ T (1) is the union of fine elements τ with τ ∈ T (0) . Define also the subdomains {T } as coarse vertex neighborhoods.…”

“…The resulting multilevel methods are optimal in terms of the contrast. As it turns out, the resulting multilevel methods can be viewed as a version of previously proposed spectral agglomerate algebraic multigrid methods (or agglomerate AMGe), originally in [1] (see also [6]) and its variant that allows ‡ The work of Y.E. is partially supported by NSF and DOE.…”

“…The approach proposed in [3] for the two-level case, which we extend in the present contribution to the multilevel case, needs only identify the "vertices" of the agglomerates; no additional topological relations are required (assuming that we have somehow come up with an agglomeration algorithm or when geometric meshes are simply used as in our present experiments). The methods introduced here simplify the previously proposed similar spectral agglomerate AMGe methods in [1,2]. The simplification occurs due to the fact that the present method uses overlapping subdomains (unions of fine grid agglomerates (elements) that share a common vertex) as domains where the local eigenproblems are posed.…”