Abstract: Space-fractional diffusion processes were observed in the last decades in many real-life situations. To simulate such phenomena, we first need an appropriate model. It turned out that a classical topic, the so-called fractional-order calculus can be applied here. We also discuss some approaches for the numerical solution of the corresponding problems, both finite difference and finite element methods. The matrix transformation method is investigated in details. We also mention some open questions of the topic.
Abstract: The method of waveform relaxation (WR) is an iterative method which can be applied for a large class of problems. It was first used to solve a system of ordinary differential equations describing large scale circuits. Since then many works have been devoted to investigate the convergence of the method for different types of problems, for example reaction--diffusion equations, delay differential equations. All of them consider time dependent problems that are either a system of ODEs originally or obtained from partial differential equations by spatial discretization. The key to prove convergence in every case is the Lipschitz property of the function acting on the right hand side.
In PDEs describing diffusion or advection processes the spatial differentiation is not a Lipschitz-continuous operation. Consequently the usual formulation of WR fails when the method is applied directly on reaction-diffusion or reaction-advection problems. Furthermore the convergence rates of the spatially discretized problem depend on the discretization parameter, thus the results can not automatically be transferred to the original continuous model.
In this talk, I propose to apply the WR method directly on the continuous problem.
Using the concept of strongly continuous one-parameter semigroups a large class of continuous problems can be discussed, including systems of reaction-diffusion and reaction-advection equations in multiple spatial dimensions. Better error estimations can be given, that are also explicit.
After that the subproblems can be solved in each iteration numerically, with spatial discretization. This scenario allows to investigate the effect of the numerical treatment as well, an overall error estimation can be formulated which includes the iteration error and the cumulative numerical error.
Faster convergence can be achieved by dividing the time interval into subintervals and applying WR on these time windows one after another. This procedure is called windowing. Convergence of windowing is proven for a large class of PDEs.
(The talk will be in Hungarian.)
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.)