Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Least Squares Ranking on Graphs

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.Comment: Added missing references, comparison of linear solvers overhauled, conclusion section added, some new figures adde

Low-order finite element preconditioner for spectral element pressure solver in Navier-Stokes equations

High-order spectral element methods (SEM) while very accurate for computational fluid dynamics (CFD) simulations can be prohibitively expensive for meshes with difficult geometries. Controlling the number of iterations in the pressure solver can significantly reduce the computing time of CFD applications. A low-order finite element (FEM) operator collocated on the Gauss-Lobatto-Legendre (GLL) points in the SEM discretization is proposed as preconditioner. Three different versions of the preconditioner based on combinations of the low-order stiffness and mass matrices are tested for 2D and 3D geometries. When building the preconditioning operators a new meshing approach that allows elements to overlap and need not fill out the volume of the mesh are explored and proven to be better than traditional schemes. With these preconditioners a bound on the number of iterations is attained regardless of mesh geometry or polynomial degree used. This novel meshing strategy achieves a reduction up to 30% in the number of iterations compared to the best current schemes without increasing the computational cost of the preconditioners, and it also overcomes the shortcommings of other well known preconditioners such as the hybrid Schwarz preconditioner