Flux-preserving enforcement of inhomogeneous Dirichlet boundary conditions for strongly divergence-free mixed finite element methods for flow problems

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Download Full-text

Pseudostress-Based Mixed Finite Element Methods for the Stokes Problem in ℝn with Dirichlet Boundary Conditions. I: A Priori Error Analysis

Communications in Computational Physics ◽

10.4208/cicp.010311.041011a ◽

2012 ◽

Vol 12 (1) ◽

pp. 109-134 ◽

Cited By ~ 7

Author(s):

Gabriel N. Gatica ◽

Antonio Márquez ◽

Manuel A. Sánchez

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Stokes Problem ◽

Mixed Finite Element ◽

A Priori ◽

Dirichlet Boundary Conditions ◽

Equilibrium Equations ◽

Galerkin Scheme ◽

Piecewise Polynomials

AbstractWe consider a non-standard mixed method for the Stokes problem in ℝn, n Є {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 Babuška-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≥0 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≥0 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.

Download Full-text

Mixed Finite Element Methods for Problems with Robin Boundary Conditions

SIAM Journal on Numerical Analysis ◽

10.1137/09077970x ◽

2011 ◽

Vol 49 (1) ◽

pp. 285-308

Author(s):

Juho Könnö ◽

Dominik Schötzau ◽

Rolf Stenberg

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

Finite Element Methods ◽

Mixed Finite Element ◽

Mixed Finite Element Methods ◽

Robin Boundary Conditions ◽

Robin Boundary

Download Full-text