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Aspects of Guaranteed Error Control in CPDEs

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Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 45))

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Abstract

Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretization error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bound (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects: first, the variational crimes from a nonconforming finite element discretization and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB; second, the reliable approximation of the discretization error on curved boundaries; and finally, the reliable bounds of the error with respect to some goal functional, namely, the error in the approximation of the directional derivative at a given point.

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Acknowledgements

This work was written while the first author enjoyed the kind hospitality of the Oxford PDE Centre.

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Authors and Affiliations

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

    C. Carstensen & C. Merdon

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

    C. Carstensen & C. Merdon

  3. Weierstraß-Institut, Mohrenstr. 39, 10117, Berlin, Germany

    J. Neumann

Authors
  1. C. Carstensen
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  2. C. Merdon
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  3. J. Neumann
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Corresponding author

Correspondence to C. Carstensen .

Editor information

Editors and Affiliations

  1. , Dept. of Flow and Material Simulation, Fraunhofer Institute for Industrial Math, Fraunhofer-Platz 1, Kaiserslautern, 67663, Germany

    Oleg P. Iliev

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. 25A, Sofia, 1113, Bulgaria

    Svetozar D. Margenov

  3. , Dept. of Mathematical and Statistical Sc, University of Alberta, 677 Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Peter D Minev

  4. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, East Avenue 7000, Livermore, 94550, California, USA

    Panayot S. Vassilevski

  5. , Department of Mathematics, Pennsylvania State University, 310 McAllister Building, University Park, 16802, Pennsylvania, USA

    Ludmil T Zikatanov

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Carstensen, C., Merdon, C., Neumann, J. (2013). Aspects of Guaranteed Error Control in CPDEs. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_6

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