On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

Abstract

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain . One subdivides into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h. Mathematics Subject Classification (2000)65N55 · 65N22 · 65F10

A Maximum Principle for L 2-Trace Norms with an Application to Optimized Schwarz Methods

“…This result is more general than the one we presented in [9], where it is assumed that v is exactly (a, c)-harmonic. The structure of this article is as follows.…”

“…We also mention that, while we proved Theorem 1 in the plane, it also holds in higher dimensions and on manifolds, under suitable hypotheses; see [9]. This is important because one cannot rely, e.g., on conformal maps in dimensions higher than 2 to prove results for general domains.…”

“…We prove this theorem in [9], but without the benefit of the ǫ > 0 parameter. Although we have not proved it, we hope that the robustness of the ǫ-harmonicity condition can be used to show that the Optimized Schwarz Method can also use inexact local solvers.…”

“…In a recent paper [9], we have shown that trace norms can be used in a similar way to show the convergence of Optimized Schwarz Methods, which use Robin boundary conditions on the interfaces between the subdomains.…”

Scharz Preconditioners for Krylov Methods: Theory and Practice

“…Scientists from five DOE labs presented work at the Twentieth International Conference on Domain Decomposition Methods in San Diego in 2011, and the meeting was in fact co-sponsored by Sandia and Livermore. We have made several contributions to the algebraic understanding of these preconditioners, and in some instances proved convergence results, or proposed new alternatives, which could not have been done without this algebraic approach [21], [22], [4], [23], [20], [34], [5], [32], [31], [6], [39], [35], [40], [1], [27], [29], [28], [30], [12], [24]. As one example, let us mention that the default Schwarz preconditioner in PETSc (from Argonne) is Restricted Additive Schwarz [7], and its convergence properties are understood thanks to our work [23].…”