GLS methods for Stokes equations under boundary condition of friction type: formulation-analysis-numerical schemes and simulations

Abstract

In this paper, we present in two and three dimensional space Galerkin least squares (GLS) methods allowing the use of equal order approximation for both the velocity and pressure modeling the Stokes equations under Tresca’s boundary condition. We propose and analyse two finite element formulations in bounded domains. Firstly, we construct the unique weak solution for each problem by using the method of regularization combined with the monotone operators theory and compactness properties. Secondly, we study the convergence of the finite element approximation by deriving a priori error estimate. Thirdly, we formulate three numerical algorithms namely; projection-like algorithm couple with Uzawa’s iteration, the alternative direction method of multiplier and an active set strategy. Finally some numerical experiments are performed to confirm the theoretical findings and the efficiency of the schemes formulated.