Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements

TL;DR

This paper develops a reliable residual-based a posteriori error estimator for nonconforming linear finite element methods applied to interface problems, effectively handling discontinuous coefficients without restrictive assumptions.

Contribution

It introduces a new direct approach to analyze estimator reliability, independent of Helmholtz decomposition and quasi-monotonicity assumptions.

Findings

Estimator is reliable with a constant independent of coefficient jumps

Numerical tests confirm effectiveness on problems with intersecting interfaces

No need for Helmholtz decomposition or quasi-monotone coefficients

Abstract

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented.