A Unified Equilibrated Flux Recovery Framework with Robust A Posteriori Error Estimation

TL;DR

This paper presents a unified flux recovery framework, EARM, for elliptic interface problems applicable to various finite element methods, providing robust a posteriori error estimation in multiple dimensions.

Contribution

The paper introduces EARM, a novel equilibrated flux recovery method applicable to multiple discretizations and dimensions, with a new ON-EARM variant ensuring flux uniqueness and robustness.

Findings

EARM provides efficient, explicit flux recovery matching state-of-the-art methods.

The method yields a robust a posteriori error estimator unaffected by coefficient jumps.

Numerical tests confirm the theoretical robustness and effectiveness of EARM.

Abstract

We introduce the Equilibrated Averaging Residual Method (EARM), a unified equilibrated flux-recovery framework for elliptic interface problems that applies to a broad class of finite element discretizations. The method is applicable in both two and three dimensions and for arbitrary polynomial orders, and it enables the construction of computationally efficient recovered fluxes. We develop EARM for both discontinuous Galerkin (DG) and conforming finite element discretizations. For DG methods, EARM can be applied directly and yields an explicit recovered flux that coincides with state-of-the-art conservative flux reconstructions. For conforming discretizations, we further propose the Orthogonal Null-space--Eliminated EARM (ON-EARM), which ensures uniqueness by restricting the correction flux to the orthogonal complement of the divergence-free null space. We prove local conservation and establish a robust a~posteriori error estimator for the recovered flux in two dimensions, with robustness measured with respect to jumps in the diffusion coefficient. Numerical results in two and three dimensions confirm the theoretical findings.