Finite element methods for 3D interface problems on local anisotropic hybrid meshes

We propose a new finite element method designed to address three-dimensional interface problems. This method employs a quasi-uniform, unfitted mesh as the foundation for constructing the grid, which incorporates anisotropic tetrahedral, pyramidal, and prism elements near the interface. We conduct a rigorous analysis of the optimal approximation capabilities of anisotropic elements, with a specific focus on their linear convergence rates in the H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1$$\end{document}-norm, excluding a logarithmic factor related to the intersection of the interface and element edges. Additionally, we thoroughly investigate errors arising from transitioning between the continuous and discretized interfaces. After applying suitable approximations to the discretized interface, this logarithmic factor is expressed as |lnh|1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ln h|<^>{1/2}$$\end{document}. The convergence rate in the H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1$$\end{document}-norm is quantified as O(|logh|1/2h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|\log h|<^>{1/2} h)$$\end{document}. Numerical experiments are presented to corroborate these theoretical results.