A Continuous $hp-$Mesh Model for Discontinuous Petrov-Galerkin Finite Element Schemes with Optimal Test Functions

TL;DR

This paper introduces an anisotropic $hp$-mesh adaptation strategy for DPG finite element schemes, leveraging residual-based error estimation and local problem solving to optimize mesh and polynomial distribution for improved accuracy.

Contribution

It extends previous $h$-adaptation work by developing a continuous mesh model for $hp$-adaptation in DPG schemes, enabling anisotropic mesh refinement based on error estimators.

Findings

Effective anisotropic $hp$-mesh adaptation demonstrated on triangular grids.

Parallelizable local problem approach enhances computational efficiency.

Improved accuracy in DPG solutions shown through numerical examples.

Abstract

We present an anisotropic $hp-$mesh adaptation strategy using a continuous mesh model for discontinuous Petrov-Galerkin (DPG) finite element schemes with optimal test functions, extending our previous work on $h-$adaptation. The proposed strategy utilizes the inbuilt residual-based error estimator of the DPG discretization to compute both the polynomial distribution and the anisotropy of the mesh elements. In order to predict the optimal order of approximation, we solve local problems on element patches, thus making these computations highly parallelizable. The continuous mesh model is formulated either with respect to the error in the solution, measured in a suitable norm, or with respect to certain admissible target functionals. We demonstrate the performance of the proposed strategy using several numerical examples on triangular grids. Keywords: Discontinuous Petrov-Galerkin, Continuous mesh models, $hp-$ adaptations, Anisotropy