Non-Overlapping Domain Decomposition via BURA Preconditioning of the Schur Complement - Nikola Kosturski; Svetozar Margenov; Yavor Vutov (Mathematics, 2022)
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 <semantics>γ</semantics>. In general, <semantics>γ</semantics> 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 <semantics><msubsup>Lγ1/2</msubsup></semantics>. Here, <semantics><msubsup>Lγ1/2</msubsup></semantics> denotes the discrete Laplacian over the interface <semantics>γ</semantics>. 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 <semantics>O(n)</semantics>, 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 <semantics>γ</semantics>. 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.