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

Abstract

Abstract. We consider the tensorial diffusion equation, and address the discrete maximumminimum 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 maximumminimum 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.

Mixed finite elements for solving 2-D diffusion-type equations

“…The basic idea of the method is to rewrite the MFE formulation as a problem of optimization under linear constraints explicitly introduced to satisfy positivity of the solution. This technique may, however, induce violation of the local mass balance in a part of the domain [Nakshatrala and Valocchi, 2009].…”

“…[8] In the field of water resources, applicability and worth of MFEs were shown for a wide range of problems, including steady state [Arbogast et al, 1997;Chavent and Roberts, 1991;Durlofsky, 1994;James and Graham, 1999] and transient [Chavent and Roberts, 1991;Ackerer et al, 1999;Younes et al, 1999a] single-phase flow, flow in unsaturated media [Farthing et al, 2003], multiphase flow [Chen and Ewing, 1997;Dawson et al, 1998;Bergamaschi and Putti, 1999;Huber and Helmig, 1999;Nayagum et al, 2004;Hoteit and Firoozabadi, 2008], flow with heat transfer [Chounet et al, 1999;Holstad, 2001], multiphase flow in fractured media Firoozabadi, 2005, 2008], and numerical upscaling [Durlofsky, 1998;Ma et al, 2006]. The MFE method was also successfully used to obtain a locally mass conservative multiscale approach.…”

On the performance of high‐order finite elements with respect to maximum principles and the nonnegative constraint for diffusion‐type equations

“…Anisotropic diffusion in a square domain with a hole. This problem has been used in References [21,19,23,22] with respect to the enforcement of the non-negative constraint but in the context of low-order approximation. The computational domain is a bi-unit square with a square hole of dimension [4/9, 5/9] × [4/9, 5/9].…”

“…In References [23,22], optimization-based techniques have been employed to meet these constraints under both mixed and single-field finite element formulations but have restricted their studies to low-order finite elements.…”

Do Current Lattice Boltzmann Methods for Diffusion and Advection-Diffusion Equations Respect Maximum Principle and the Non-Negative Constraint?

“…Several factors such as the physical properties of the medium, topology of the domain, and the spatial and temporal discretization determine the performance of a numerical solution in preserving the discrete versions of the mentioned mathematical properties. A discussion on the influence of these factors in the context of the finite element method can be found in [Nakshatrala and Valocchi, 2009]. Violations of these mathematical properties can make a numerical solution inappropriate for scientific and engineering applications.…”