BIT Numerical Mathematics

, Volume 48, Issue 2, pp 215–229

Rational approximation to trigonometric operators


DOI: 10.1007/s10543-008-0185-9

Cite this article as:
Grimm, V. & Hochbruck, M. Bit Numer Math (2008) 48: 215. doi:10.1007/s10543-008-0185-9


We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior 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 propose and analyze 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. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

Key words

rational Krylov subspace methods trigonometric operator function Hilbert space wave equations trigonometric integrators highly oscillatory problems finite element discretization 

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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