A simple virtual element-based flux recovery on quadtree

TL;DR

This paper presents a straightforward virtual element-based flux recovery method for quadtree meshes that handles irregular hanging nodes without special adjustments, enabling effective a posteriori error estimation.

Contribution

It introduces a simple, robust flux recovery technique on quadtree meshes using virtual elements, accommodating irregular hanging nodes without ad hoc modifications.

Findings

Reliable a posteriori error estimator constructed

Method verified through numerical experiments

Handles irregular hanging nodes seamlessly

Abstract

In this paper, we introduce a simple local flux recovery for $\mathcal{Q}_k$ finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on $l$-irregular ($l\geq 2$) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient a posteriori error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics.