Andreas Dedner

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.

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).

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

We extend the discontinuous Galerkin framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface in ℝ3. An interior penalty (IP) method is introduced on a discrete surface and we derive a priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretization do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.

Error Control for a Class of Runge-Kutta Discontinuous Galerkin Methods for Nonlinear Conservation Laws

We propose an a posteriori error estimate for the Runge-Kutta discontinuous Galerkin method (RK-DG) of arbitrary order in arbitrary space dimensions. For stabilization of the scheme a general framework of projections is introduced. Finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy. Numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids.

A Fast Parallel Solver for the Forward Problem in Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a noninvasive imaging modality, where imperceptible currents are applied to the skin and the resulting surface voltages are measured. It has the potential to distinguish between ischaemic and haemorrhagic stroke with a portable and inexpensive device. The image reconstruction relies on an accurate forward model of the experimental setup. Because of the relatively small signal in stroke EIT, the finite-element modeling requires meshes of more than 10 million elements. To study the requirements in the forward modeling in EIT and also to reduce the time for experimental image acquisition, it is necessary to reduce the run time of the forward computation. We show the implementation of a parallel forward solver for EIT using the Dune-Fem C++ library and demonstrate its performance on many CPUs of a computer cluster. For a typical EIT application a direct solver was significantly slower and not an alternative to iterative solvers with multigrid preconditioning. With this new solver, we can compute the forward solutions and the Jacobian matrix of a typical EIT application with 30 electrodes on a 15-million element mesh in less than 15 min. This makes it a valuable tool for simulation studies and EIT applications with high precision requirements. It is freely available for download.

A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems

In this paper we present a stabilized Discontinuous Galerkin (DG) method for hyperbolic and convection dominated problems. The presented scheme can be used in several space dimension and with a wide range of grid types. The stabilization method preserves the locality of the DG method and therefore allows to apply the same parallelization techniques used for the underlying DG method. As an example problem we consider the Euler equations of gas dynamics for an ideal gas. We demonstrate the stability and accuracy of our method through the detailed study of several test cases in two space dimension on both unstructured and cartesian grids. We show that our stabilization approach preserves the advantages of the DG method in regions where stabilization is not necessary. Furthermore, we give an outlook to adaptive and parallel calculations in 3d.

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

An efficient implementation of an adaptive and parallel grid in DUNE

L^2

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

-accuracy and CPU time.

Curvature guided level set registration using adaptive finite elements

In this contribution we describe and evaluate an efficient implementation of an adaptive and parallel grid (ALUGrid) within the Distributed and Unified Numerics Environment DUNE. A generalization of the serial grid interface of DUNE, described in [1], to the adaptive and parallel case is discussed and example computations using the grid interface are presented. The computations are compared with computations of the original code, which was optimized for the specific example problem studied here.