Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Approximations

TL;DR

This paper develops robust, locally optimal a priori error estimates for mixed finite element methods applied to elliptic interface problems with low regularity, guiding improved adaptive computational techniques.

Contribution

It introduces new robust and local optimal a priori error estimates for mixed finite element approximations in low-regularity interface problems, enhancing accuracy and robustness.

Findings

Error estimates are robust to diffusion coefficient variations.

Estimates are optimal relative to local solution regularity.

Guidance for constructing robust a posteriori error estimates.

Abstract

For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the Raviart-Thomas and Brezzi-Douglas-Marini mixed finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to the local regularity of the solution. Several versions of the robust best approximations of the flux and the potential approximations are obtained. These robust and local optimal a priori estimates provide guidance for constructing robust a posteriori error estimates and adaptive methods for the mixed approximations.