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2016 | OriginalPaper | Chapter

Numerical Homogenization Methods for Parabolic Monotone Problems

Author : Assyr Abdulle

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

Publisher: Springer International Publishing

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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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next chapter Virtual Element Implementation for General Elliptic Equations
Footnotes
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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Metadata
Title
Numerical Homogenization Methods for Parabolic Monotone Problems
Author
Assyr Abdulle
Copyright Year
2016
Book
Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Print ISBN: 978-3-319-41638-0

Electronic ISBN: 978-3-319-41640-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-41640-3_1

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