Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems

• Published in: SIAM journal on numerical analysis Vol. 39; no. 4; pp. 1343 - 1365
• Main Authors: Feng, Xiaobing, Karakashian, Ohannes A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2002
• Subjects: Approximation; Decomposition; Decomposition methods; Estimates; Estimation methods; Exact sciences and technology; Finite element method; Function discontinuity; Galerkin methods; Inequality; Linear systems; Mathematical discontinuity; Mathematical functions; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Polynomials; Sciences and techniques of general use; Second order equation; Parameter estimation; Grid pattern; Decomposition method; Two level method; Partial differential equation; Discretization method; Equation system; Elliptic equation; Condition number; Numerical simulation; Galerkin method; Domain decomposition; Preconditioning; Schwarz additive method
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142900378480

We present some two-level nonoverlapping and overlapping additive Schwarz methods for a discontinuous Galerkin method for solving second order elliptic problems. It is shown that the condition numbers of the preconditioned systems are of the order $O({H \over {h}})$ for the nonoverlapping Schwarz methods and of the order $O({H \over {\delta}})$ for the overlapping Schwarz methods, where h and H stand for the fine-mesh size and the coarse-mesh size, respectively, and δ denotes the size of the overlaps between subdomains. Numerical experiments are provided to gauge the efficiency of the methods and to validate the theory.