Two-grid $hp$-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

TL;DR

This paper extends two-grid $hp$-discontinuous Galerkin methods for quasilinear elliptic PDEs to agglomerated meshes, providing error analysis, an adaptive algorithm, and demonstrating efficiency through numerical experiments.

Contribution

It introduces an extension of two-grid $hp$-discontinuous Galerkin methods to agglomerated meshes with error analysis and an automatic adaptive algorithm.

Findings

Effective error estimates for agglomerated meshes.

Successful implementation of an $hp$-adaptive two-grid method.

Numerical results show improved computational performance.

Abstract

This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid $hp$-version discontinuous Galerkin finite element method from 10.1007/s10915-012-9644-1 for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an $hp$-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed $hp$-adaptive two-grid method.