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Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1291–1327 | Cite as

High dimensional finite elements for time-space multiscale parabolic equations

  • Wee Chin Tan
  • Viet Ha HoangEmail author
Article

Abstract

The paper develops the essentially optimal sparse tensor product finite element method for a parabolic equation in a domain in \(\mathbb {R}^{d}\) which depends on a microscopic scale in space and a microscopic scale in time. We consider the critical self similar case which has the most interesting homogenization limit. We solve the high dimensional time-space multiscale homogenized equation, which provides the solution to the homogenized equation which describes the multiscale equation macroscopically, and the corrector which encodes the microscopic information. For obtaining an approximation within a prescribed accuracy, the method requires an essentially optimal number of degrees of freedom that is essentially equal to that for solving a macroscopic parabolic equation in a domain in \(\mathbb {R}^{d}\). A numerical corrector is deduced from the finite element solution. Numerical examples for one and two dimensional problems confirm the theoretical results. Although the theory is developed for problems with one spatial microscopic scale, we show numerically that the method is capable of solving problems with more than one spatial microscopic scale.

Keywords

High dimensional finite elements Time-space multiscale parabolic equations Optimal complexity Numerical corrector 

Mathematics Subject Classification (2010)

35B27 65M12 65M60 

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Notes

Acknowledgments

The research topic originates from a discussion with Professor Christoph Schwab, ETH, Zurich. The authors gratefully acknowledge a postgraduate scholarship of A*Star, Singapore, the AcRF Tier 1 grant 2016-T1-001-202 RG30/16, the Singapore A*Star SERC grant 122-PSF-0007, and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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