**Ivan Dimov Lirkov**

Chapter 1:

The use of circulant preconditioners for the solution of elliptic
problems is analysed. Finite element method is applied and spline
elements are used in the discretization. It is proven that circulant
preconditioners can be constructed such that the relative condition
number of the preconditioned system can be reduced from
to
where *N* is size of the discrete problem.

Chapter 2:

New circulant block-factorization (CBF) preconditioners are introduced
and studied. The general approach is first formulated for the case of
block tridiagonal sparse matrices. Then estimates of the relative
condition number for a model Dirichlet boundary value problem are
derived in the form
.
In the case of *y*-periodic
problems the CBF preconditioner is shown to give an optimal
convergence rate. Finally, using a proper embedding of the original
Dirichlet boundary value problem to a *y*-periodic one, a
preconditioner of optimal convergence rate for the general case is
obtained. An application of the CBF preconditioners for a Dirichlet
boundary value problem in *L*-shaped domain is studied. Various
numerical tests that demonstrate the features of the CBF
preconditioners are presented.

Chapter 3:

An estimate of the relative condition number for a model anisotropic Dirichlet boundary value problem is derived. This is one of the important benchmark problems for the robustness of the iterative methods. Here we consider problems with a fixed direction of dominating anisotropy and derive an exact relative condition number estimate . Various numerical tests demonstrating the behavior of the CBF preconditioners for anisotropic problems are presented.

Chapter 4:

This chapter contains results for parallel properties of the proposed in the previous chapters algorithms and their implementation on different parallel architectures. The parallel numerical solution of large scale elliptic boundary value problems is discussed. We analyze the parallel complexity of the circulant preconditioners when solving elliptic problems by preconditioned conjugate gradient method. Estimations for speed-up and parallel efficiency for different models of computer architecture are obtained. The efficiency of considered algorithm on various parallel architecture models is compared. Numerical experiments for a large number test problems are described and their results analyzed.