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Space-Time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients

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Advanced Finite Element Methods with Applications (FEM 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 128))

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Abstract

We introduce a completely unstructured, conforming space-time finite element method for the numerical solution of parabolic initial-boundary value problems with variable in space and time, possibly discontinuous diffusion coefficients. Discontinuous diffusion coefficients allow the treatment of moving interfaces. We show stability of the method and an a priori error estimate , including the case of local stabilizations which are important for adaptivity . To study the method in practice, we consider several typical model problems in one, two, and three spatial dimensions. The implementation of our space-time finite element method is fully parallelized with MPI. Extensive numerical tests were performed to study the convergence behavior of the stabilized space-time finite element discretization method and the scaling properties of the parallel AMG-preconditioned GMRES solver that we use to solve the huge system of space-time finite element equations.

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  1. 1.

    https://www.llnl.gov/casc/hypre/.

  2. 2.

    https://www.ricam.oeaw.ac.at/hpc/.

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Acknowledgements

This work was supported by the Austrian Science Fund (FWF) under the grant W1214, project DK4. This support is gratefully acknowledged. Furthermore, the authors would like to express their thanks to the anonymous referees for their helpful hints and valuable suggestions.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Ulrich Langer & Martin Neumüller

  2. Doctoral Program “Computational Mathematics”, Johannes Kepler University Linz, Linz, Austria

    Andreas Schafelner

Authors
  1. Ulrich Langer
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  2. Martin Neumüller
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  3. Andreas Schafelner
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Corresponding author

Correspondence to Ulrich Langer .

Editor information

Editors and Affiliations

  1. Institut für Mathematik & Computergestützte Simulation, Universität der Bundeswehr München, Neubiberg, Germany

    Thomas Apel

  2. Institute for Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Ulrich Langer

  3. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Arnd Meyer

  4. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria

    Olaf Steinbach

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Langer, U., Neumüller, M., Schafelner, A. (2019). Space-Time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_13

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