A Stabilized Dual Mixed Hybrid Finite Element Method with Lagrange Multipliers for Three-Dimensional Elliptic Problems with Internal Interfaces
A Stabilized Dual Mixed Hybrid Finite Element Method with Lagrange Multipliers for Three-Dimensional Elliptic Problems with Internal Interfaces
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DOI：
10.1007/s10915-020-01163-7
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发表时间：
2020-02
影响因子：
2.5
通讯作者：
R. Sacco;A. Mauri;G. Guidoboni
中科院分区：
文献类型：
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作者：
R. Sacco;A. Mauri;G. Guidoboni
本文研究了由选择性界面分隔的两种介质组成的三维区域中具有扩散项、平流项和反应项的椭圆型边值问题。对于该问题的数值逼近，我们首次提出了一种新的方法，它结合了：(1)基于最低阶Raviart-Thomas空间（RT0）的双混合混合（DMH）有限元法（FEM）；(2)三场公式；(3)流线迎风/ Petrov-Galerkin （SUPG）稳定化方法。在证明了该方法的弱形式及其数值对应形式都是唯一可解的，并且有限元格式对离散化参数具有最优收敛性之后，我们提出了一种基于静态凝聚的有效实现，将该方法简化为三维简单体网格上的非一致性有限元方法。大量的计算试验表明：(1)理论收敛性得到了验证；(2) DMH-RT0有限元方法在溶液及其法向通量存在明显的界面跳变不连续的情况下仍然准确稳定；(3)在强平流项占主导地位的情况下，SUPG稳定性可以准确地解析陡峭的边界和/或内层，而不会引入虚假的非物理振荡或溶液前缘的过度涂抹。
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.