research partners    
Ph.D. students    
consortium members    

Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

Novel massively parallel algorithms for solving nonconforming discretizations of PDEs
Maksymilian Dryja (MIM University of Warsaw) and Zdenek Strakos (Faculty of Mathematics and Physics Charles University in Prague)
The project is carried out by Filip Klawe at University of Warsaw
Mathematical models of physics usually need specific types of discretizations, nonconforming finite element methods, which take into account varying properties of materials, mesh refinement and the design process in CAD environment.

The goal of this project is to design and analyse domain/space decomposition methods ([2], [3], [4]) for solving algebraic systems, which result from nonconforming discretizations of various type on non-matching triangulations. Such types of triangulations are in particular motivated by properties of the underlying PDE problem. Recently, two streams of nonconforming discretizations received a lot of attention: the mortar method [1] and the discontinuous Galerkin method. They provide higher flexibility than the classical finite element or finite volume methods and have already found their way to engineering packages such as NASTRAN. Domain decomposition methods are among the best fitted and the most effective approaches to solving large systems of equations arising from dicretizations of PDEs and have been widely adopted by the engineering community, because modern supercomputers are distributed memory and massively parallel systems.

The aim of the project is exploring these classes of methods for a prescribed mathematical problem and constructing optimal discretizations and iterative solution algorithms that allow for a high level of coarse-grained parallelism. The overall effectiveness and the convergence rate of designed algorithms will be analysed and verified in practice on ICM's and Charles' University parallel computers. The reasearch will be conducted in close cooperation with L. Marcinkowski, P. Krzyzanowski (University of Warsaw) and leading teams in this area, including the BCCS in Bergen, Norway (head: Prof. P. Bjorstad) and the groups of Prof. Zdenek Strakos and Yvon Maday (LJLL in Paris). We plan to incorporate the implementation into one of popular parallel numerical packages: PETSc maintaned in Argonne National Laboratory (Illinois, USA) or the HSL library developed by the Numerical Analysis Group, (Oxford, UK).

[1] C. Bernardi, Y. Maday and A.T. Patera, College de France Seminar, 11 (Paris, 1989-1991).
[2] A. Klawonn, O.B. Widlund, M. Dryja, (2002) SIAM J. Numer. Anal., 40: 159-179.
[3] M. Dryja, J. Galvis and M. Sarkis, (2007) J. Complexity, 23: 715-739.
[4] M. Dryja and O. Widlund, (2007) Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng., 55: 357-364, Springer, Berlin.