Implementation of $hp$-adaptive discontinuous finite element methods in Dune-Fem

TL;DR

This paper presents algorithms and data structures for implementing $hp$-adaptive discontinuous finite element methods in the Dune-Fem library, including data transfer during adaptation and a reference implementation for $hp$-adaptive spaces.

Contribution

It introduces a generalized approach for $p$- and $hp$-adaptation in Dune-Fem, with an extensible implementation and a practical $hp$-adaptive interior penalty method.

Findings

Successful implementation of $hp$-adaptive interior penalty method

Enhanced data transfer during adaptive refinement

Extensible framework for $hp$-discontinuous finite element spaces

Abstract

In this paper we describe generic algorithms and data structures for the implementation of $hp$-adaptive discontinuous finite element methods in the Dune-Fem library. Special attention is given to the often tedious and error-prone task of transferring user data during adaptation. Simultaneously, we generalize the approach to the restriction and prolongation of data currently implemented in Dune-Fem to the case of $p$- and $hp$-adaptation. The dune-fem-hpdg module described in this paper provides an extensible reference implementation of $hp$-adaptive discontinuous discrete function spaces. We give details on its implementation and the extended adaptive interface. As proof of concept we present the practical realization of an $hp$-adaptive interior penalty method for elliptic problems.