Non-negative mixed finite element formulations for a tensorial diffusion equation

TL;DR

This paper introduces two non-negative mixed finite element formulations for tensorial diffusion equations, ensuring solutions adhere to the maximum-minimum principle on arbitrary meshes using constrained optimization techniques.

Contribution

It presents novel non-negative mixed finite element formulations based on quadratic programming, applicable to low-order elements and arbitrary meshes, with analysis of their properties and solution strategies.

Findings

Producing non-negative solutions on arbitrary meshes

Effect of non-negative constraints on local mass balance

Performance demonstrated on canonical test problems

Abstract

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. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). 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 is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. 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.