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

kornev@rpi.edu

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^{-2}p^{3})
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.