Christoph Gersbacher | Searcharxiv

arXiv2017

Constrained Reconstruction in MUSCL-type Finite Volume Schemes

In this paper we are concerned with the stabilization of MUSCL-type finite volume schemes in arbitrary space dimensions. We consider a number of limited reconstruction techniques which are defined in terms inequality-constrained linear or quadratic programming problems on individual grid elements. No restrictions to the conformity of the grid or the shape of its elements are made. In the special case of Cartesian meshes a novel QP reconstruction is shown to coincide with the widely used Minmod reconstruction. The accuracy and overall efficiency of the stabilized second-order finite volume schemes is supported by numerical experiments.

math.NA↗

arXiv2016

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

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.

cs.MS↗