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Numerical Homogenization Methods for Parabolic Monotone Problems

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Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

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Abstract

In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A-stable or explicit stabilized methods). We discuss the construction and the analysis of such methods for a range of problems, from linear parabolic problems to nonlinear monotone parabolic problems in the very general L p(W 1, p) setting. We also show that under appropriate assumptions, a computationally attractive linearized method can be constructed for nonlinear problems.

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Notes

  1. 1.

    We concentrate on simplicial elements for simplicity but note that many results presented in this paper can be extended to rectangular elements (see for example [9]).

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Acknowledgement

This research is partially supported by the Swiss National Foundation under Grant 200021_150019. 

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Authors and Affiliations

  1. ANMC, Mathematics Section, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Assyr Abdulle

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  1. Assyr Abdulle
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Correspondence to Assyr Abdulle .

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Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

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Abdulle, A. (2016). Numerical Homogenization Methods for Parabolic Monotone Problems. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_1

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