An arbitrarily high order unfitted finite element method for elliptic interface problems with automatic mesh generation, Part II. Piecewise-smooth interfaces

TL;DR

This work considers the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities and extends the previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces.

Abstract

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An $hp$ a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

Figures (17)

• Figure 2.1: Illustration of the domain $\Omega$ and the Cartesian mesh $\mathcal{T}_0$ (left) and $\mathcal{T}$ (right).
• Figure 2.2: Different types of interface elements. From left to right, a type $\mathcal{T}_1,\mathcal{T}_2,\mathcal{T}_3$ element.
• Figure 2.3: Illustration of $K_{ij}^h$ of the regular interface large elements (top) or singular interface large elements. $\Gamma_{iK}^h$ are denoted by the dashed line, $i=1,2,j=1,\cdots,J_i^K$.
• Figure 3.1: Examples of the singular pattern.
• Figure 3.2: The figures used in the proof of Lemma \ref{['lem:3.1']}.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.1
• proof
• Theorem 3.1