Finite Element Methods for Saddlepoint Problems with Application to Darcy and Stokes Flow

The goal of this short course is to review the analysis and the systematic discretization of saddlepoint systems that arise in the weak formulation of certain fluid flow problems by Galekrin methods. To demonstrate the applicability of the main arguments in various situations, we consider the slow flow of water through a channel surrounded by a porous aquiver. While the free flow of the fluid may be described by the Stokes system, the flow of the water through the confining porous medium is governed by the Darcy equations. The coupling of these two rather different fluid flow models is accomplished by the interface conditions of Beavers, Joseph, and Saffmann. We establish the wellposedness of this interface problem in the framework of mixed variational problems by verifying the assumptions of Brezzi’s splitting theorem. Based on the proper variational setting, we then discuss the Galerkin discretization for the Darcy and Stokes problem. The discretizations are analyzed with the same arguments as the continuous problem, and we will highlight the importance of discrete stability conditions for the construction of reliable schemes. In particular, we will illustrate by explicit examples, that a straight forward discretization will in general not lead to stable numerical schemes. Based on appropriate discretization strategies for the Darcy and Stokes problems, we then propose a combined finite element method which yields quasi-optimal approximations for the coupled Darcy-Stokes flow problem.