Interface Reduction for Elliptic Interface Problems with Conservative Flux Reconstruction

TL;DR

This paper introduces a low-dimensional interface reduction method for elliptic interface problems that ensures locally conservative fluxes and demonstrates that the solution accuracy depends mainly on interface data approximation.

Contribution

The paper presents a novel interface reduction technique combining finite element discretization with flux recovery, achieving machine-precision interface condition satisfaction.

Findings

Reduced solution error is controlled by interface data approximation error.

Numerical experiments confirm high accuracy when interface data is well-represented.

The method concentrates problem complexity on the interface, simplifying analysis.

Abstract

We propose a low-dimensional interface reduction method for elliptic interface problems based on conservative flux reconstruction. The approach combines a fitted $P_1$ finite element discretization with a flux recovery procedure following \cite{ChouTang2000}, yielding locally conservative fluxes that satisfy interface conditions to machine precision. A central result shows that the error of the reduced solution is controlled entirely by the approximation error of the interface data. Numerical experiments for both continuous and discontinuous interface conditions confirm that once the interface data is accurately represented, the full solution is recovered to roundoff accuracy. These results indicate that the essential complexity of elliptic interface problems is concentrated on the interface.