Martin Nolte

Performance Pitfalls in the Dune Grid Interface

We discuss performance issues and common pitfalls one can encounter when dealing with the Dune-Grid interface. This discussion includes the implementation of Cartesian grids in Dune as well as the implementation of meta grids. Furthermore, for the use of local grid adaptivity we present several approaches to data restriction and prolongation and discuss their advantages and disadvantages in terms of performance. Finally, we also compare a general Dune implementation of a Finite Volume scheme with a special purpose implementation.

Construction of Local Finite Element Spaces Using the Generic Reference Elements

Based on the recursive definition of the generic reference elements in Dune-Grid, we present an effective framework for the implementation of finite element shape functions. Such a shape function set is described by a set of functionals defining the degrees of freedom and an arbitrary basis of the finite element space on the reference element. To illustrate the power of this approach we show how Lagrange shape functions, Raviart-Thomas shape functions, and L 2-orthonormal shape functions fit into this framework.

Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids

Improved and enhanced oil recovery methods require sophisticated simulation tools to predict the injected flow pass together with the chemical reactions inside it. One approach is application of higher-order numerical schemes to avoid excessive numerical diffusion that is very typical for transport processes. In this work, we provide a first step towards higher-order schemes applicable on general polyhedral and corner-point grids typically used in reservoir simulation. We compare three possible approaches of linear reconstruction and slope limiting techniques on a variety of different meshes in two and three spatial dimensions and discuss advantages and disadvantages.

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.

Comparison of Linear Reconstructions for Second Order Finite Volume Schemes on Polyhedral Grids

Improved and enhanced oil recovery methods require sophisticated simulation tools to predict the injected flow pass together with the chemical reactions inside it. One approach is the application of higher order numerical schemes to avoid excessive numerical diffusion that is very typical for transport processes. In this work we provide a first step towards higher order schemes applicable on general polyhedral and corner-point grids typically used in reservoir simulation. We compare two possible approaches of linear reconstruction and slope limiting techniques on a variety of different meshes in two and three space dimensions and discuss advantages and disadvantages.

Dune-Fem: A General Purpose Discretization Toolbox for Parallel and Adaptive Scientific Computing

Dune-Fem is a free discretization toolbox for parallel and adaptive scientific computing based on Dune. The implementation of discretization schemes such as finite elements, finite volumes or discontinuous Galerkin schemes is based on abstractions that are very close to the mathematical description of the underlying methods. In this contribution we will give a compact overview on the design and abstraction principles of Dune-Fem and demonstrate its wide range of applicability in numerical experiments ranging from the solution of flow processes on surfaces to parallel and adaptive fluid flow in three space dimensions. A more detailed presentation of the abstraction principles is given in [Dedner et al. A generic interface for parallel and adaptive discretization scheme: abstraction principles and the Dune-Fem module. Computing 90 (2010), no. 3-4, 165-196]. In the whole design of Dune-Fem efficiency was a main concern. In this paper we will give some indication to what degree the generic programming principals used in Dune-Fem can lead to the generation of efficient code.