Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes
This paper considers a number of variants of the additive Schwarz method introduced by \textit{M. Dryja} and \textit{O. B. Widlund} in 1987. The focus here is on extensions of this algorithm to unstructured meshes. The preconditioners defined by domain decomposition methods of this family use two triangulations, a coarse with characteristic mesh size \(H\), and a fine on which the finite element problem is defined. A global component of the preconditioner is constructed from a finite element problem on the coarse mesh and, in addition, potentially many local components of the preconditioner are given in terms of the fine mesh and an overlapping decomposition of the region. The theory was first developed in detail for the case when the coarse space is a subspace of the finite element space on the fine mesh, and the two meshes are quasi uniform. The generalization considered here addresses situations where the coarse mesh is selected independently of the fine mesh and where the elements are shape regular but the meshes not necessarily quasi uniform. The estimate of the condition number of a domain decomposition method of this class always involves a decomposition of the elements of the entire space into components of the subspaces and the study of the stability of this decomposition. Often, the element of the coarse mesh subspace is chosen as the \(L_ 2\) projection onto the coarse space, or in the general case, the standard finite element interpolant of that projection. This is where the requirement of quasi uniformity enters the analysis. That condition can be relaxed by using an idea, employed in approximation theory for many years, namely averaging and interpolation. This is in fact how the first good proof of the optimality of this Schwarz method was constructed; cf. \textit{M. Dryja} and \textit{O. B. Widlund}, in ``Iterative methods for large linear systems'', Academic Press (1989; Zbl 0703.68010). That proof extends immediately to any mesh with shape regular elements. The other main ingredient of the proof of good bounds for the condition number in the general case is a proof of the stability of the standard finite element interpolant when restricted to the coarse space. Such a result, which is not trivial nor very difficult, appears first to have been given by \textit{Xiao-Chuan Cai} [The use of pointwise interpolation in domain decomposition methods with nonnested meshes, SIAM J. Sci. Comput., 16, No. 1, 250-256 (1995)]. In this paper, a number of variants of the basic Schwarz method are explored. Special attention is given to estimates of the dependence of the condition number on the relative overlap of the subregions.
scientific article; zbMATH DE number 720557