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Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales

Author: Assyr Abdulle
Journal: Math. Comp. 81 (2012), 687-713
MSC (2010): Primary 65N30, 65M60; Secondary 74Q05, 35J15
Published electronically: July 26, 2011
MathSciNet review: 2869033
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Abstract: An analysis of a multiscale symmetric interior penalty discontinuous Galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed. This new method, first described in [A. Abdulle, C.R. Acad. Sci. Paris, Ser. I 346 (2008)] is based on numerical homogenization. It allows to significantly reduce the computational cost of a fine scale discontinuous Galerkin method by probing the fine scale data on sampling domains within a macroscopic partition of the computational domain. Macroscopic numerical fluxes, an essential ingredient of discontinuous Galerkin finite elements, can be recovered from the computation on the sampling domains with negligible computation overhead. Fully discrete a priori error bounds are derived in the $ L^2$ and $ H^1$ norms.

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Additional Information

Assyr Abdulle
Affiliation: Section of Mathematics, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland

Keywords: Heterogeneous multiscale method, discontinuous Galerkin methods, a priori error analysis, fully discrete error, elliptic homogenization
Received by editor(s): October 5, 2009
Received by editor(s) in revised form: January 29, 2011
Published electronically: July 26, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.