Study of a non-overlapping domain decomposition method: Poisson and Stokes problems

Abstract

The purpose of this work is to perform a theoretical analysis of the domain decomposition method introduced by J.L. Lions in: [Parallel stabilization of hyperbolic and Petrowsky systems, in: S. Idelsohn, E. Oñate, E. Dvorkin (Eds.), Computational Mechanics: New Trends and Applications, CIMNE, Barcelona, 1998]; and in: [C. R. Acad. Sci. Paris Ser. I 330 (2000) 937]. We motivate and introduce an improvement of this method and carry out the analysis when it is applied to solving two model problems such as the Poisson problem and the incompressible Stokes equations. The improvement that we propose turns out to be a non-overlapping domain decomposition method based on a penalty term on the interface of the subdomains that enforces the appropriated transmission conditions. Under the usual regularity assumptions on the true solution, we obtain error estimates in the natural norms that are optimal in terms of the space discretization in the sense that we obtain an O(h k) error estimate when we use a space approximation of order k. The solution is computed via an iterative process that turns out to be rather slow. This convergence speed is greatly improved when a standard acceleration technique is applied. Some numerical tests are also presented.