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

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.

Thermal-Mechanical Coupling Model Based on the Hybrid Finite Element Method for Solving Bipolar the Plate Deformation of Hydrogen Fuel Cells

“…At the same time, some research has considered mixed finite element methods for fracture Darcy flow based on interface-unfitted meshes [7]. A few papers have used the mixed finite element method to solve the elliptic interface problems based on interface-fitted meshes [8]. In [9], an extended mixed finite element method was proposed for elliptic interface problems using the lowest-order Brezzi-Douglas-Marini finite element pair (BDMP 1 , P 0 ) to construct our extended mixed finite element space.…”

“…Here, n is the unit normal vector of the boundary ∂Ω, and then the differential Equation ( 7) is converted to an integral expression, Equation (8):…”

“…The geometric appearance is mainly a rectangular plate, with four small circular holes and a central elliptical hole inside. The length of the rectangular plate is L = 12 m, and the width is H = 12 m. The center coordinates of the four small circles are (10,8) in turn. The radius of the four small circles is r = 0.8 m. The center elliptical hole long axis a = 2 m, and the short axis b = 1 m. The two-dimensional linear elastic mechanics equation solved by this example is:…”

Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model

“…This paper adopts the Rackwitz-Fiessler probability transformation and merges this transformation with the Lagrange multiplier method. At the same time, this paper proposes a primary reliability method based on Rackwitz-Fiessler probability transformation and the Lagrange multiplier method [2]. In this paper, the rational polynomial method is introduced into the derivation of the inverse function expression of the distribution function.…”

An extended mixed finite element method for elliptic interface problems

“…Whereas, to the best of our knowledge, there are a few papers using the mixed finite element method to handle elliptic interface problems. Several works related to mixed finite element methods for the elliptic interface problems are based on interface-fitted meshes, please see [32], [36], [37], [38], [39]. There are also some works ( [2], [30]) concerned with mixed finite element methods for the fracture Darcy flow based on interface-unfitted meshes.…”