A Group of Immersed Finite Element Spaces For Elliptic Interface Problems

TL;DR

This paper introduces a unified framework for constructing immersed finite element spaces for elliptic interface problems, enabling the use of various polynomial types and ensuring optimal approximation properties on interface-independent meshes.

Contribution

The paper develops a new unified framework for immersed finite element spaces with different polynomial types, ensuring unisolvence and optimal approximation capabilities for elliptic interface problems.

Findings

Constructed IFE spaces with linear, bilinear, and rotated-Q1 polynomials.

Proved unisolvence via Sherman-Morrison matrix invertibility.

Established optimal approximation properties using a unified Taylor expansion approach.

Abstract

We present a unified framework for developing and analysing immersed finite element (IFE) spaces for solving typical elliptic interface problems with interface independent meshes. This framework allows us to construct a group of new IFE spaces with either linear, or bilinear, or the rotated-Q1 polynomials. Functions in these IFE spaces are locally piecewise polynomials defined according to the sub-elements formed by the interface itself instead of its line approximation. We show that the unisolvence for these IFE spaces follows from the invertibility of the Sherman-Morrison matrix. A group of estimates and identities are established for the interface geometry and shape functions that are applicable to all of these IFE spaces. Most importantly, these fundamental preparations enable us to develop a unified multipoint Taylor expansion procedure for proving that these IFE spaces have the expected optimal approximation capability according to the involved polynomials.