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

Abstract

These figures show the fate of the product in a transient transport-controlled bimolecular reaction under vortex-stirred mixing. The left figure is obtained using a popular numerical formulation, which violates the non-negative constraint. The right figure is based on the proposed computational framework. These figures clearly illustrate the main contribution of this paper: The proposed computational framework produces physically meaningful results for advective–diffusive–reactive systems, which is not the case with many popular formulations. 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 Péclet 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.