Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Discrete exterior calculus
Convex Optimization
Computational Methods for Multiphase Flows in Porous Media (Computational Science and Engineering 2)
Computational Methods for Multiphase Flows in Porous Media (Computational Science and Engineering 2)
The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes
Journal of Computational Physics
Journal of Computational and Applied Mathematics - Special issue on computational and mathematical methods in science and engineering (CMMSE-2004)
Monotonicity of control volume methods
Numerische Mathematik
Journal of Computational Physics
A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.46 |
We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart-Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum-minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.