Recent Developments in Schwarz Algorithms for the h-p-version of the Finite Element Method

Vadim Korneev
State University Sankt Petersburg
Institute for Mathematics and Mechanics
Bibliotechnaya square 2, Russia

Current address:
Rensselaer Poytechnic Institute
Department of Computer Science
Troy, NY, 12180, USA


We consider the Dirichlet-Dirichlet Domain Decomposition (DD) algorithms for systems of algebraic equations of h-p-version of the finite element method for second order elliptic equations. We split the problem of the DD preconditioning into the subproblems of preconditioning of the internal problems on the subdomains of decomposition, of Schur complement and of the discrete harmonic prolongations of polynomials from the interface boundary inside subdomains of decomposition. In the case of the square reference element equipped with the tensor product polynomial space, we are able to suggest efficient and cheap preconditioners for every mentioned component. As a result, we come to the DD preconditioner which is spectrally equivalent to the global stiffness matrix and requires O(h-2p3) operations for solving the system with such preconditioner for the matrix. The obtained finite-difference-like preconditioner for internal problems makes it possible further improvement of algorithm on the basis of known efficient solution techniques. Most of the results except for preconditioning of internal problems are expandable to the case of triangular elements. The algorithms are highly parallelizable. Some results of the report have been obtained jointly with S. Jensen and with J. Fish and J. Flaherty.