### An additive Schwarz preconditioner for the spectral element ocean model formulation of the shallow water equations.

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A version of the dynamical systems method (DSM) for solving ill-conditioned linear algebraic systems is studied. An a priori and an a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.

Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${}^{N}$ or on the Segal-Bargmann space over ${\u2102}^{N}$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression ${A}_{n}$ of A to the linear span of the monomials ${z}_{1}^{{k}_{1}}...{z}_{N}^{{k}_{N}}:0\le {k}_{j}\le n$. Unfortunately, in general the spectrum of ${A}_{n}$ does not mimic the spectrum of A as...