A Stabilized Dual Mixed Hybrid Finite Element Method with Lagrange multipliers for Three-Dimensional Problems with Internal Interfaces

TL;DR

This paper introduces a novel stabilized dual mixed hybrid finite element method for 3D heterogeneous elliptic problems with internal interfaces, ensuring stability, optimal convergence, and accurate resolution of sharp solution features.

Contribution

It combines a dual mixed hybrid FEM, a three-field formulation, and SUPG stabilization for the first time in this context, with an efficient implementation and proven stability.

Findings

Method achieves optimal convergence and stability.

Accurately resolves steep layers without oscillations.

Effective for problems with strong interface discontinuities.

Abstract

This work focuses on a class of elliptic boundary value problems with diffusive, advective and reactive terms, motivated by the study of three-dimensional heterogeneous physical systems composed of two or more media separated by a selective interface. We propose a novel approach for the numerical approximation of such heterogeneous systems combining, for the first time: (1) a dual mixed hybrid (DMH) finite element method (FEM) based on the lowest order Raviart-Thomas space (RT0); (2) a Three-Field (3F) formulation; and (3) a Streamline Upwind/Petrov-Galerkin (SUPG) stabilization method. Using the abstract theory for generalized saddle-point problems and their approximation, we show that the weak formulation of the proposed method and its numerical counterpart are both uniquely solvable and that the resulting finite element scheme enjoys optimal convergence properties with respect to the discretization parameter. In addition, an efficient implementation of the proposed formulation is presented. The implementation is based on a systematic use of static condensation which reduces the method to a nonconforming finite element approach on a grid made by three-dimensional simplices. Extensive computational tests demonstrate the theoretical conclusions and indicate that the proposed DMH-RT0 FEM scheme is accurate and stable even in the presence of marked interface jump discontinuities in the solution and its associated normal flux. Results also show that in the case of strongly dominating advective terms, the proposed method with the SUPG stabilization is capable of resolving accurately steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.