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An adaptive homotopy approach for non-selfadjoint eigenvalue problems

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Abstract

This paper presents adaptive algorithms for eigenvalue problems associated with non-selfadjoint partial differential operators. The basis for the developed algorithms is a homotopy method which departs from a well-understood selfadjoint problem. Apart from the adaptive grid refinement, the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources. The first algorithm balances the homotopy, discretization and approximation errors with respect to a fixed stepsize τ in the homotopy. The second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance ε. The outcome of the analysis leads to the third algorithm which allows the complete adaptivity in space, homotopy stepsize as well as the iterative algebraic eigenvalue solver. All three algorithms are compared in numerical examples.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    C. Carstensen & J. Gedicke

  2. Department of Computational Science and Engineering, Yonsei University, 120–749, Seoul, Korea

    C. Carstensen

  3. Institut für Mathematik, MA 4-5, TU Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    V. Mehrmann & A. Miedlar

Authors
  1. C. Carstensen
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  2. J. Gedicke
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  3. V. Mehrmann
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  4. A. Miedlar
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Correspondence to V. Mehrmann.

Additional information

V. Mehrmann: Supported by the DFG Research Center Matheon “Mathematics for Key Technologies” in Berlin. J. Gedicke and A. Miedlar: Supported by the DFG graduate school BMS “Berlin Mathematical School” in Berlin. C. Carstensen: World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0.

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Carstensen, C., Gedicke, J., Mehrmann, V. et al. An adaptive homotopy approach for non-selfadjoint eigenvalue problems. Numer. Math. 119, 557–583 (2011). https://doi.org/10.1007/s00211-011-0388-x

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