Algebraic Multigrid Based on Element Interpolation (AMGe)

• Published in: SIAM journal on scientific computing Vol. 22; no. 5; pp. 1570 - 1592
• Main Authors: Brezina, M., Cleary, A. J., Falgout, R. D., Henson, V. E., Jones, J. E., Manteuffel, T. A., McCormick, S. F., Ruge, J. W.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2001
• Subjects: Algebra; Exact sciences and technology; Explicit knowledge; Laboratories; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Sciences and techniques of general use; Choice; Grid pattern; Computer simulation; Differential equation; Numerical method; Representation; Algorithm; Partial differential equation; Algebraic multigrid; Finite element method; Interpolation; Multigrid; Linear algebra; Algebraic method; Stiffness; Ritz method; Component; Differential method; Current; Finite element
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/S1064827598344303

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, 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 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.