A posteriori error estimator for elliptic interface problems in the fictitious formulation

TL;DR

This paper develops and validates a residual-based a posteriori error estimator for elliptic interface problems in the fictitious domain formulation, ensuring reliable adaptive refinement across complex geometries and coefficient jumps.

Contribution

It introduces a new a posteriori error estimator for elliptic interface problems with discontinuous coefficients, demonstrating its reliability and efficiency both theoretically and numerically.

Findings

Estimator is reliable and efficient for constant and smooth coefficient jumps.

Numerical experiments confirm optimal convergence and robustness.

Adaptive refinement effectively handles geometric singularities.

Abstract

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.