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

Abstract

This work studies an elliptic boundary value problem with diffusive, advective and reactive terms, in a three-dimensional domain composed of two media separated by a selective interface. For the numerical approximation of the problem we propose a novel approach that combines, 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 formulation; and (3) a Streamline Upwind/Petrov–Galerkin (SUPG) stabilization method. After proving that the weak formulation of the proposed method and its numerical counterpart are both uniquely solvable and that the finite element scheme enjoys optimal convergence properties with respect to the discretization parameter, we present an efficient implementation based on static condensation, which reduces the method to a nonconforming finite element approach on a grid made by three-dimensional simplices. Extensive computational tests indicate that: (1) the theoretical convergence properties are verified; (2) the DMH-RT0 FEM is accurate and stable even in the presence of marked interface jump discontinuities in the solution and its associated normal flux; and (3) in the case of strongly dominating advective terms, the SUPG stabilization resolves accurately steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.