An Anisotropic $hp$-Adaptation Framework for Ultraweak Discontinuous Petrov-Galerkin Formulations

TL;DR

This paper introduces an anisotropic $hp$-refinement framework for ultraweak DPG methods in 3D, utilizing residual-based error estimation to adaptively refine meshes for improved accuracy.

Contribution

It develops a novel anisotropic $hp$-refinement strategy for ultraweak DPG formulations, combining residual-based error estimation with adaptive mesh refinement in 3D.

Findings

Effective anisotropic $hp$-refinement demonstrated on hexahedral meshes.

Achieves prescribed error tolerance efficiently through adaptive cycles.

Improves solution accuracy with fewer degrees of freedom.

Abstract

In this article, we present a three-dimensional anisotropic $hp$-mesh refinement strategy for ultraweak discontinuous Petrov--Galerkin (DPG) formulations with optimal test functions. The refinement strategy utilizes the built-in residual-based error estimator accompanying the DPG discretization. The refinement strategy is a two-step process: (a) use the built-in error estimator to mark and isotropically $hp$-refine elements of the (coarse) mesh to generate a finer mesh; (b) use the reference solution on the finer mesh to compute optimal $h$- and $p$-refinements of the selected elements in the coarse mesh. The process is repeated with coarse and fine mesh being generated in every adaptation cycle, until a prescribed error tolerance is achieved. We demonstrate the performance of the proposed refinement strategy using several numerical examples on hexahedral meshes.