- A generalized spectral problem ^ = &^ is considered for two Poincare"-Steklov operators [3,4] corresponding to two adjacent domains in a plane. A sufficient condition for the problem's spectrum to be discrete is suggested. A continuous spectrum is known to impair the convergence of domain-decomposition iterative methods. A Poincaro-Steklov operator is understood to be the operator that transforms the boundary values of a harmonic function to the values of the normal derivative of the function. It is shown that under certain conditions imposed on the domains the spectrum of this problem is discrete, it converges to = 1, and all the eigenvalues other than = 1 have finite multiplicities. This implies that the eigenvalues, except for a finite number of them, are contained in a small neighbourhood of unity. Moreover, it is shown that the spectrum of every problem of this kind is related to the spectrum of an integral operator with an analytic kernel, which can be sometimes calculated explicitly. It is also emphasized that the spectrum is invariant under conformal mapping of the two domains. The problem is investigated by the methods of the theory of complex-variable functions and the theory of one-dimensional singular equations.
Russian Journal of Numerical Analysis and Mathematical Modelling – de Gruyter
Published: Jan 1, 1993
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