Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

TL;DR

This paper investigates the failure of classical finite element methods to respect maximum principles in anisotropic diffusion with decay and introduces an optimization-based approach to enforce these principles on general grids.

Contribution

The paper identifies limitations of classical Galerkin formulations and proposes a novel optimization-based method to enforce maximum principles in anisotropic diffusion with decay.

Findings

Classical Galerkin method violates maximum principles, especially with high decay coefficients.

Mesh refinement reduces violations in isotropic cases but not in anisotropic cases.

The proposed optimization approach effectively enforces maximum principles on general grids.

Abstract

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation.