Two-Level Schwarz Methods for the Biharmonic Problem Discretized Conforming $C^1 $ Elements

Abstract

Additive Schwarz methods are overlapping domain decomposition methods proposed by Dryja and Widlund for second-order elliptic problems. In this paper, we consider two-level additive Schwarz methods for the biharmonic Dirichlet problem discretized by conforming $C^1 $ finite elements. Most of these elements are nonnested in the sense that the finite element space defined on the coarse mesh is not a subspace of the space defined on the finer mesh. We construct certain intergrid transfer operators and establish that the algorithms have optimal convergence properties. Our algorithms include the cases when the two-level triangulation are nonnested and the subspaces on the coarse and the fine grids are defined by different finite elements. Our analysis is based on the theory of Dryja and Widlund and our estimates use the stable approximation properties of the finite elements and the intergrid transfer operators.