A posteriori error analysis of the virtual element method for second-order quasilinear elliptic PDEs

TL;DR

This paper develops a $C^0$-conforming virtual element method for second-order quasilinear elliptic PDEs, providing a residual-based a posteriori error estimator and demonstrating its effectiveness through adaptive refinement and numerical tests.

Contribution

It introduces a new a posteriori error analysis and residual-based estimator for the virtual element method applied to quasilinear elliptic PDEs, enabling adaptive mesh refinement.

Findings

Estimator is fully computable and explicit in local mesh size.

Adaptive algorithm effectively refines meshes based on the estimator.

Numerical tests confirm the estimator's reliability and efficiency.

Abstract

In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.