Construction of Discontinuous Enrichment Functions for Enriched FEM's for Interface Elliptic Problems in 1D

TL;DR

This paper develops a new method for constructing discontinuous enrichment functions in enriched finite element methods to effectively solve 1D elliptic interface problems with complex jump conditions.

Contribution

It introduces a novel one-parameter family of enrichment functions based on optimal interpolation, improving the adaptability of enriched FEM for interface problems.

Findings

Effective in solving 1D elliptic interface problems with complex jumps

Enriched elements outperform standard methods in accuracy

Applicable to multi-layer wall models with various interface conditions

Abstract

We introduce an enriched unfitted finite element method to solve 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. We present a novel approach to construct a one-parameter family of discontinuous enrichment functions by finding an optimal order interpolating function to the discontinuous solutions. In the literature, an enrichment function is usually given beforehand, not related to the construction step of an interpolation operator. Furthermore, we recover the well-known continuous enrichment function when the parameter is set to zero. To prove its efficiency, the enriched linear and quadratic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.