Abstract: For a vast number of models for real-life problems, including various partial differential equations, the numerical solution is ultimately reduced to the solution of linear algebraic systems. The efficiency of this last step often depends on the proper choice of a preconditioning matrix. A class of efficient preconditioners for discretized elliptic problems can be obtained via equivalent operator preconditioning. This means that the preconditioner is chosen as the discretization of a suitable auxiliary operator that is equivalent to the original one, Under proper conditions one can thus achieve mesh independent convergence rates. Hence, if the discretized auxiliary problems possess efficient optimal order solvers (e.g. of multigrid type) regarding the number of arithmetic operations, then the overall iteration also yields an optimal order solution, i.e. the cost O(N) is proportional to the degrees of freedom.

In this talk first some theoretical background is summarized, including both linear and superlinear mesh independent convergence, then various applications are shown. The results can be applied, among other things, for parallel preconditioning of transport type systems,
for streamline diffusion preconditioning of convection-diffusion problems, and to achieve superlinear convergence under shifted Laplace preconditioners for Helmholtz equations.

(The talk will be in Hungarian.)