A High Order Geometry Conforming Immersed Finite Element for Elliptic Interface Problems

TL;DR

This paper introduces a high-order immersed finite element method that exactly satisfies interface conditions for elliptic problems, using a nonlinear transformation to efficiently construct conforming IFE functions applicable in standard DG schemes.

Contribution

The paper develops a novel high-order IFE space based on a nonlinear transformation, enabling cost-effective, robust, and conforming solutions for elliptic interface problems with optimal approximation properties.

Findings

Achieves optimal convergence under mesh and polynomial degree refinements.

Constructs interface-conforming IFE functions using a nonlinear Frenet-Serret frame transformation.

Demonstrates robustness and efficiency of the method through numerical examples.

Abstract

We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions exactly and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet-Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Numerical examples are provided to showcase the convergence properties of the method under $h$ and $p$ refinements.