TY - JOUR

T1 - Pseudostress-based mixed finite element methods for the Stokes problem in ℝ n with Dirichlet boundary conditions. I

T2 - A priori error analysis

AU - Gatica, Gabriel N.

AU - Márquez, Antonio

AU - Sánchez, Manuel A.

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/7

Y1 - 2012/7

N2 - We consider a non-standard mixed method for the Stokes problem in ℝ n, n ∈ {2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuŝka-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥o for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥o for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

AB - We consider a non-standard mixed method for the Stokes problem in ℝ n, n ∈ {2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuŝka-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥o for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥o for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

KW - Incompressible flow

KW - Mixed finite element

KW - Pseudostress

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U2 - 10.4208/cicp.010311.041011a

DO - 10.4208/cicp.010311.041011a

M3 - Article

AN - SCOPUS:84856614632

VL - 12

SP - 109

EP - 134

JO - Communications in Computational Physics

JF - Communications in Computational Physics

SN - 1815-2406

IS - 1

ER -