Scalable Recovery-based Adaptation on Quadtree Meshes for Advection-Diffusion-Reaction Problems

TL;DR

This paper introduces a scalable mesh adaptation method for quadtree meshes, utilizing a recovery-based a posteriori error estimator to improve the discretization of advection-diffusion-reaction problems, especially with discontinuities and internal layers.

Contribution

It presents a novel mesh adaptation algorithm driven by a recovery-based error estimator, demonstrating high accuracy and scalability for complex scalar PDEs on Cartesian quadtree meshes.

Findings

Performs well with discontinuities and internal layers

Outperforms standard estimate-mark-refine strategies

Exhibits excellent parallel scalability

Abstract

We propose a mesh adaptation procedure for Cartesian quadtree meshes, to discretize scalar advection-diffusion-reaction problems. The adaptation process is driven by a recovery-based a posteriori estimator for the $L^2(\Omega)$-norm of the discretization error, based on suitable higher order approximations of both the solution and the associated gradient. In particular, a metric-based approach exploits the information furnished by the estimator to iteratively predict the new adapted mesh. The new mesh adaptation algorithm is successfully assessed on different configurations, and turns out to perform well also when dealing with discontinuities in the data as well as in the presence of internal layers not aligned with the Cartesian directions. A cross-comparison with a standard estimate--mark--refine approach and with other adaptive strategies available in the literature shows the remarkable accuracy and parallel scalability of the proposed approach.