On enforcing maximum principles and achieving element-wise species balance for advection-diffusion-reaction equations under the finite element method

TL;DR

This paper introduces a finite element framework that enforces maximum principles, non-negativity, and species balance in advection-diffusion-reaction systems, handling complex media and high Péclet numbers effectively.

Contribution

It integrates advection into a non-negative finite element framework using convex quadratic programming with explicit constraints, a novel approach for such systems.

Findings

Robustness in advection-dominated regimes

Convergence of the proposed method

Effective handling of complex media and anisotropy

Abstract

We present a robust computational framework for advective-diffusive-reactive systems that satisfies maximum principles, the non-negative constraint, and element-wise species balance property. The proposed methodology is valid on general computational grids, can handle heterogeneous anisotropic media, and provides accurate numerical solutions even for very high P\'eclet numbers. The significant contribution of this paper is to incorporate advection (which makes the spatial part of the differential operator non-self-adjoint) into the non-negative computational framework, and overcome numerical challenges associated with advection. We employ low-order mixed finite element formulations based on least-squares formalism, and enforce explicit constraints on the discrete problem to meet the desired properties. The resulting constrained discrete problem belongs to convex quadratic programming for which a unique solution exists. Maximum principles and the non-negative constraint give rise to bound constraints while element-wise species balance gives rise to equality constraints. The resulting convex quadratic programming problems are solved using an interior-point algorithm. Several numerical results pertaining to advection-dominated problems are presented to illustrate the robustness, convergence, and the overall performance of the proposed computational framework.