Non-Overlapping Domain Decomposition via BURA Preconditioning of the Schur Complement

A new class of high-performance preconditioned iterative solution methods for large-scale finite element method (FEM) elliptic systems is proposed and analyzed. The non-overlapping domain decomposition (DD) naturally introduces coupling operator at the interface γ . In general, γ is a manifold of lower dimensions. At the operator level, a key property is that the energy norm associated with the Steklov-Poincaré operator is spectrally equivalent to the Sobolev norm of index 1/2. We define the new multiplicative non-overlapping DD preconditioner by approximating the Schur complement using the best uniform rational approximation (BURA) of L γ 1 / 2 . Here, L γ 1 / 2 denotes the discrete Laplacian over the interface γ . The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the BURA-based non-overlapping DD preconditioner has optimal computational complexity O ( n ) , where n is the number of unknowns (degrees of freedom) of the FEM linear system. All theoretical estimates are robust, with respect to the geometry of the interface γ . Results of systematic numerical experiments are given at the end to illustrate the convergence properties of the new method, as well as the choice of the involved parameters.