A posteriori virtual element method for the acoustic vibration problem

TL;DR

This paper develops and analyzes an a posteriori error estimator for the acoustic spectral problem using the virtual element method, demonstrating reliability, efficiency, and superconvergence in two dimensions.

Contribution

It introduces a novel a posteriori error estimator for the virtual element method applied to acoustic problems, with proven reliability and efficiency, and confirms performance through numerical tests.

Findings

The error estimator is reliable in the L2-norm.

The estimator is globally efficient.

Numerical results confirm the estimator's effectiveness.

Abstract

In two dimensions, we propose and analyze an a posteriori error estimator for the acoustic spectral problem based on the virtual element method in $\H(\div;\Omega)$. Introducing an auxiliary unknown, we use the fact that the primal formulation of the acoustic problem is equivalent to a mixed formulation, in order to prove a superconvergence result, necessary to despise high order terms. Under the virtual element approach, we prove that our local indicator is reliable and globally efficient in the $\L^2$-norm. We provide numerical results to assess the performance of the proposed error estimator.