Solving Three-Dimensional Interface Problems with Immersed Finite Elements: A-Priori Error Analysis
Solving Three-Dimensional Interface Problems with Immersed Finite Elements: A-Priori Error Analysis
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DOI：
10.1016/j.jcp.2021.110445
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发表时间：
2020-04
期刊：
影响因子：
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通讯作者：
Ruchi Guo;Xu Zhang
中科院分区：
文献类型：
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作者：
Ruchi Guo;Xu Zhang
浸没有限元方法是用来解决界面不匹配网格上的界面问题的。然而，大多数研究，尤其是分析，主要局限于二维案例。本文给出了求解长方体笛卡尔网格上三维椭圆界面问题的三线性浸没有限元方法的先验分析。我们建立了具有任意界面切割构型的界面单元的三线性IFE函数的迹和逆不等式。严格地证明了能量范数和L范数下的最优先验误差估计，并且明确规定了误差界中的常数与界面位置无关及其与系数对比度的依赖关系。数值算例不仅验证了我们的理论结果，而且证明了这种IFE方法在处理一些真实世界的3D界面模型中的适用性。
Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish the trace and inverse inequalities for trilinear IFE functions for interface elements with arbitrary interface-cutting configuration. Optimal a priori error estimates are rigorously proved in both energy and L 2 norms, with the constant in the error bound independent of the interface location and its dependence on coefficient contrast explicitly specified. Numerical examples are provided not only to verify our theoretical results but also to demonstrate the applicability of this IFE method in tackling some real-world 3D interface models.