Markus Blatt

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).

Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems

SUMMARY We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin (DG) discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections, and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system associated with this space is constructed algebraically using a Galerkin approach with the natural embedding as the prolongation operator. This embedding operator needs to be provided, which means that the approach is not fully algebraic. For the construction of the linear systems on the subsequent coarser levels, non-smoothed aggregation AMG techniques are used. In a series of numerical experiments, we establish for the first time the efficiency and robustness of an AMG method for various symmetric and non-symmetric interior penalty DG methods (including the higher-order cases) on problems with complicated, high contrast jumps in the coefficients. The solver is robust with respect to an increase in the polynomial degree of the DG approximation space (at least up to degree 6), computationally efficient, and affected only mildly by the coefficient jumps and by the mesh size h (i.e., number of iterations = O(log h−1)). Copyright

C++ components describing parallel domain decomposition and communication

Large-scale parallel codes require the data to be decomposed between the set of processes active in the computation. This data decomposition implies recurring communication schemes. The paper introduces generic template classes in C++ for describing the data decomposition. The aim is to store the data in arbitrary existent efficient sequential data structures. Each entry in the sequential data structure corresponds to an entry in the virtual global view of the container. Once the decomposition is setup the needed communication schemes can be created automatically and can be used to communicate values from containers of various types. Even containers with a varying number of values associated with an entry are possible. The framework abstracts the decomposition information and the communication in the client code from the eventual parallel paradigm choice. A prototype based on Message Passing Interface standard is presented. It relieves the user from specifying information that is already known at compile time.