An Enriched Immersed Finite Element Method for 3D Interface Problems

TL;DR

This paper presents an enriched immersed finite element method for 3D interface problems with non-homogeneous jump conditions, combining homogenization and enrichment functions for improved stability and efficiency.

Contribution

It introduces a novel enriched IFE approach that incorporates jump data without extra degrees of freedom, ensuring stable conditioning and enabling fast multigrid solvers.

Findings

Optimal $\\mathcal{O}(h^2)$ conditioning independent of interface location

Iteration numbers are unaffected by mesh size or interface position

Numerical experiments confirm theoretical stability and efficiency

Abstract

We introduce an enriched immersed finite element method for addressing interface problems characterized by general non-homogeneous jump conditions. Unlike many existing unfitted mesh methods, our approach incorporates a homogenization concept. The IFE trial function set is composed of two components: the standard homogeneous IFE space and additional enrichment IFE functions. These enrichment functions are directly determined by the jump data, without adding extra degrees of freedom to the system. Meanwhile, the homogeneous IFE space is isomorphic to the standard finite element space on the same mesh. This isomorphism remains stable regardless of interface location relative to the mesh, ensuring optimal $\mathcal{O}(h^2)$ conditioning that is independent of the interface location and facilitates an immediate development of a multigrid fast solver; namely the iteration numbers are independent of not only the mesh size but also the relative interface location. Theoretical analysis and extensive numerical experiments are carried out in the efforts to demonstrate these features.