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Title: High dimensional finite elements for multiscale wave equations
Authors: Xia, Bingxing
Hoang, Viet Ha
Keywords: DRNTU::Science::Mathematics::Applied mathematics::Simulation and modeling
Issue Date: 2014
Source: Xia, B., & Hoang, V. H. (2014). High dimensional finite elements for multiscale wave equations. Multiscale modeling & simulation, 12(4), 1622-1666.
Series/Report no.: Multiscale modeling & simulation
Abstract: For locally periodic multiscale wave equations in $\mathbb{R}^d$ that depend on a macroscopic scale and n microscopic separated scales, we solve the high dimensional limiting multiscale homogenized problem that is posed in $(n+1)d$ dimensions and is obtained by multiscale convergence. We consider the full and sparse tensor product finite element methods, and analyze both the spatial semidiscrete and the fully (both temporal and spatial) discrete approximating problems. With sufficient regularity, the sparse tensor product approximation achieves a convergence rate essentially equal to that for the full tensor product approximation, but requires only an essentially equal number of degrees of freedom as for solving an equation in $\mathbb{R}^d$ for the same level of accuracy. For the initial condition $u(0,x)=0$, we construct a numerical corrector from the finite element solution. In the case of two scales, we derive an explicit homogenization error which, together with the finite element error, produces an explicit rate of convergence for the numerical corrector. Numerical examples for two- and three-scale problems in one or two dimensions confirm our analysis.
ISSN: 1540-3459
DOI: 10.1137/120902409
Rights: © 2014 Society for Industrial and Applied Mathematics. This paper was published in Multiscale Modeling & Simulation and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [].  One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.
Fulltext Permission: open
Fulltext Availability: With Fulltext
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