Nonconforming Immersed Finite Element Spaces For Elliptic Interface Problems

TL;DR

This paper develops a unified framework for nonconforming immersed finite element spaces with integral degrees of freedom, proving their existence, uniqueness, and optimal approximation capabilities for elliptic interface problems.

Contribution

It introduces a unified approach to construct and analyze nonconforming IFE spaces with integral degrees of freedom, ensuring their well-posedness and optimal approximation properties.

Findings

Proved existence and uniqueness of shape functions on interface elements.

Established optimal approximation capabilities of the IFE spaces.

Developed a multi-edge expansion and identities for nonconforming IFE functions.

Abstract

In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise polynomials defined on sub-elements separated either by the actual interface or its line approximation. In this unified framework, we use the invertibility of the well known Sherman-Morison systems to prove the existence and uniqueness of shape functions on each interface element in either rectangular or triangular mesh. Furthermore, we develop a multi-edge expansion for piecewise functions and a group of identities for nonconforming IFE functions which enable us to show that these IFE spaces have the optimal approximation capability.