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{\large Marco, M Donatelli \\
{\bf An algebraic generalization of the local Fourier analysis}}

Universita' dell'Insubria - sede di Como \\ Dipartimento di Fisica e Matematica \\ via Valleggio \\ 11 - 22100 Como (ITALY)
\\
{\tt marco.donatelli@uninsubria.it}\end{center}

The Local Fourier analysis (LFA) is a classic tool to prove convergence
results for multigrid methods (MGMs). In particular, we
are interested in optimality that is a convergence speed independent of
the size of the involved matrices. For elliptic partial differential
equations (PDEs), a well known optimality result requires that the sum of
the orders of the grid transfer operators is not lower than the order of
the PDE to be solved. Analogously, when dealing with MGMs for Toeplitz
matrices in the literature an optimality condition on the position and on
the order of the zeros of the symbols of the grid transfer operators has
been found. In this work we show that in the case of elliptic PDEs with
constant coefficients, the two different approaches lead to an equivalent
condition. We argue that the analysis for Toeplitz matrices is an
algebraic generalization of the LFA, which allows to deal not only with
differential problems, but also, for instance, with integral problems. We
give also a class of grid
transfer operators related to the B-spline refinement equation and
we study their geometric properties. This analysis suggests further
links between wavelets and multigrid methods. A numerical experimentation
confirms the correctness of the proposed analysis.


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