In this paper, we present 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 discretizations.Firstly, we construct the unique weak solution for each problem by using the method of regularization combined with monotone theory operators and compactness properties.Secondly, we study the convergence of the finite element approximation by estimating the a priori error.Thirdly, for the computation of the finite element solution, we formulate three algorithms namely; projection like algorithm couple with Uzawa iteration, the alternative direction method of multiplier and a active set strategy. Finally some numerical experiments are performed to confirm the theoretical findings and the efficiency of the schemes formulated.

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This paper discusses a novel three field formulation for the Darcy-Forchheimer flow with a nonlinear viscosity depending on the temperature coupled with the heat equation. We show unique solvability of the variational problem by using; Galerkin method, Brouwer's fixed point and some compactness properties. We propose and study in detail a finite element approximation. A priori error estimate is then derived and convergence is obtained. A solution technique is formulated to solve the nonlinear problem and numerical experiments that validate the theoretical findings are presented.