Resolvent Krylov subspace approximation to operator functions
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We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of φ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments.
KeywordsOperator/matrix functions Rational Krylov subspace method φ-functions Evolution equations Exponential integrators Hyperbolic problems Highly oscillatory equations Finite element discretisation
Mathematics Subject Classification (2010)65F60 65F10 65M60 65N22
This work is supported by the Deutsche Forschungsgemeinschaft (DFG) via research grant GR 3787/1-1.
The author thanks Maria Carmela De Bonis and Tanja Göckler for a careful reading of the manuscript, Maria Carmela De Bonis for providing a copy of her joint work with G. Mastroianni and M. Viggiano , and two anonymous referees for pointing to the work of Beckermann and Reichel .
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