The Dune framework: Basic concepts and recent developments

This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state Bastian etal. (2008a, 2008b). This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper

Compact and stable discontinuous Galerkin methods for convection-diffusion problems

We present a new scheme, the compact discontinuous Galerkin 2 (CDG2) method, for solving nonlinear convection-diffusion problems together with a detailed comparison to other well-accepted DG methods. The new CDG2 method is similar to the CDG method that was recently introduced in the work of Perraire and Persson for elliptic problems. One main feature of the CDG2 method is the compactness of the stencil which includes only neighboring elements, even for higher order approximation. Theoretical results showing coercivity and stability of CDG2 and CDG for the Poisson and the heat equation are given, providing computable bounds on any free parameters in the scheme. In numerical tests for an elliptic problem, a scalar convection-diffusion equation, and for the compressible Navier–Stokes equations, we demonstrate that the CDG2 method slightly outperforms similar methods in terms of $L^2$-accuracy and CPU time

Compact and Stable Discontinuous Galerkin Methods with Application to Atmospheric Flows

In this work we formulate the Compact Discontinuous Galerkin 2 (CDG2) method introduced in [8] for advection-diffusion problems. We present a proof of stability for the linear heat equation. Numerical results are shown for the compressible Navier-Stokes equation. We compare our new method numerically with two other well-established methods: the Compact Discontinuous Galerkin (CDG) and the Local Discontinuous Galerkin (LDG) method. In contrast to the LDG method, the primal formulation of the CDG2 method only involves the direct neighbors, making it more suitable for execution on parallel computers. The CDG method also has this compactness property, but the performance of the method is not as good as for the CDG2 method in terms of L2-error versus computation time

A generic grid interface for parallel and adaptive scientific computing. Part I : abstract framework

We give a mathematically rigorous definition of a grid for algorithms solving partial differential equations. Unlike previous approaches (Benger 2005, PhD thesis; Berti 2000, PhD thesis), our grids have a hierarchical structure. This makes them suitable for geometric multigrid algorithms and hierarchical local grid refinement. The description is also general enough to include geometrically non-conforming grids. The definitions in this article serve as the basis for an implementation of an abstract grid interface as C++ classes in the framework (Bastian et al. 2008, this issue)

The distributed and unified numerics environment (DUNE

Most finite element or finite volume software is built around a fixed mesh data structure. Therefore, each software package can only be used efficiently for a relatively narrow class of applications. For example, implementations supporting unstructured meshes allow the approximation of complex geometries but are in general much slower and require more memory than implementations using structured meshes. In this paper we show how a generic mesh interface can be defined such that one algorithm, e. g. a finite element discretization scheme, can work efficiently on different mesh implementations. These ideas have also been extended to vectors and sparse matrices where iterative solvers can be written in a generic way using the interface. These components are available within the “Distributed Unified Numerics Environment ” (DUNE).

A generic grid interface for parallel and adaptive scientific computing. Part II : implementation and tests in DUNE

In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations become independent of the underlying grid implementation. Modern C++ template techniques are used to provide an interface implementation without big performance losses. The implementation is realized as part of the software environment DUNE (http://dune-project.org/). Numerical tests demonstrate the flexibility and the efficiency of our approach