A posteriori error estimators for fourth order elliptic problems with concentrated loads

TL;DR

This paper develops and analyzes two residual-based a posteriori error estimators for the biharmonic equation with concentrated loads, demonstrating their efficiency, reliability, and robustness through theoretical proofs and numerical validation.

Contribution

It introduces two novel a posteriori error estimators for fourth-order elliptic problems, including one based on Dirac delta projection and extends them to general boundary conditions.

Findings

Both estimators are proven to be efficient and reliable.

Numerical experiments confirm the estimators' robustness and accuracy.

The methods improve adaptive finite element solutions for biharmonic problems.

Abstract

In this paper, we study two residual-based a posteriori error estimators for the $C^0$ interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples.