Conservative solution of the Fokker–Planck equation for stochastic chemical reactions |
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Authors: | Lars Ferm Per Lötstedt Paul Sjöberg |
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Institution: | 1.Division of Scientific Computing, Department of Information Technology,Uppsala University,Uppsala,Sweden |
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Abstract: | In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block
Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention
to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that
our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as
well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined
efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz
matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block
Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees
of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that
our multigrid solvers lead to optimal order convergence.
AMS subject classification (2000) 65N55, 65F10 |
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Keywords: | multigrid methods iterative methods preconditioning block Toeplitz matrices vector valued problems |
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