A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem^*,**

¹ Département de Mathématiques, Université de Jijel, BP 98 Aouled Aissa, 18000 Jijel, Algeria.
² Université de Toulouse, Institut Mathématique de Toulouse, Département de Génie Mathématique, INSA de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. h.benhassine@gmail.com; abendali@insa-toulouse.fr

Received: 20 August 2009
Revised: 20 June 2010

Abstract

This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at the cross points, that is, points shared by more than two subdomains, one can interpret them as iterative solvers of the actual discrete problem directly issued from the finite element scheme. In this way, the good stability properties of continuous-pressure mixed finite element approximations of the Stokes system are preserved. Estimates ensuring that each iteration can be performed in a stable way as well as a proof of the convergence of the iterative process provide a theoretical background for the application of the related solving procedure. Finally some numerical experiments are given to demonstrate the effectiveness of the approach, and particularly to compare its efficiency with an adaptation to this framework of a standard FETI-DP method.