Piotr Krzyżanowski

On Block Preconditioners for Nonsymmetric Saddle Point Problems

We discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations, in particular the Navier--Stokes equation. We prove that block diagonal and block triangular preconditioners based on symmetric, positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h.

WELL-POSEDNESS AND CONVERGENCE TO THE STEADY STATE FOR A MODEL OF MORPHOGEN TRANSPORT

Well-posedness and large time convergence to the unique steady state are shown for a model which describes the spreading of morphogens by a nonlinear transport mechanism (transcytosis) and couples a quasilinear parabolic partial differential equation with an ordinary differential equation. A simpler model which assumes linear transport is also investigated for comparison. The analysis of both models requires the construction of specific Liapunov functionals. The study is supplemented by numerical simulations of the sensitivity of the models to the variation of their parameters.

On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright

Domain Decomposition Solvers for Large Scale Industrial Finite Element Problems

The European research project PARASOL aimed to design and develop a public domain library of scalable sparse matrix solvers for distributed memory computers. Parallab was a partner in the project and developed a domain decomposition code for solving large scale finite element problems in a robust, yet efficient way. Although the PARASOL project finished in June 1999, Parallab has continued the development of the solver. In this paper, we report on the present status of the solver and show its performance on some challenging industrial problems.

A Flexible 2-Level Neumann-Neumann Method for Structural Analysis Problems

We discuss a new approach for the construction of the second-level Neumann-Neumann coarse space. Our method is based on an inexpensive and parallel analysis of the lower part spectrum of each subdomain stiffness matrix. We show that the method is flexible enough to converge fast on nonstandard decompositions and various types of finite elements used in structural analysis packages.

On the Heat Convection Equations with Dissipation Term in Regions with Moving Boundaries

We consider an initial value problem for a system of equations describing the motion and the heat convection in a viscous and incompressible fluid which occupies a smooth region Ω t ⊂ R 3 depending on time. In the equation for the distribution of temperature in the fluid we take into account not only the convective term but also the term responsible for the dissipation of energy. We prove local in time existence and uniqueness of solutions of the considered problem, and global in time existence for sufficiently small data.

Additive Schwarz Method for DG Discretization of Anisotropic Elliptic Problems

Second order elliptic problem with discontinuous anisotropic coefficients in 2-D is considered. The problem is discretized by a Discontinuous Galerkin (DG) finite element method with triangular elements and piecewise-linear functions. The resulting discrete problem is solved by a special two-level additive Schwarz method. It is proved that the rate of convergence of the method is independent of the jumps of coefficients if they appear inside of substructures on which the original region is partitioned. Numerical experiments are reported which confirm theoretical results.

A massively parallel nonoverlapping additive Schwarz method for discontinuous Galerkin discretization of elliptic problems

A second order elliptic problem with discontinuous coefficient in 2-D or 3-D is considered. The problem is discretized by a symmetric weighted interior penalty discontinuous Galerkin finite element method with nonmatching simplicial elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element. In this way the method has a potential for a very high level of fine grained parallelism. Condition number estimate depending on the relative sizes of the underlying grids is provided. The rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers’ subdomain boundaries. Numerical experiments are reported which confirm theoretical results.

Aggregation Algorithms for Perturbed Markov Chains with Applications to Networks Modeling

A powerly perturbed Markov chain (MC) is a family of finite MCs given by transition probability matrices

Additive Schwarz Methods for DG Discretization of Elliptic Problems with Discontinuous Coefficient

P(\varepsilon)=(p_{ij}(\varepsilon))