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High-dimensional finite elements for elliptic problems with multiple scales

by V. H. Hoang and Ch. Schwab

(Report number 2003-14)

Abstract
Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. They are discretized by a sparse tensor product finite element method (FEM) which resolves all scales of the solution throughout the physical domain. We prove that this FEM has accuracy, work and memory requirement comparable of FEM for single scale problems in the physical domain $\Omega$ and performs independently of the scale parameters. Numerical examples for problems with two and three scales confirm our results.

Keywords:

@Techreport{HS03_326,
  author = {V. H. Hoang and Ch. Schwab},
  title = {High-dimensional finite elements for elliptic problems with multiple scales},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2003-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2003/2003-14.pdf },
  year = {2003}
}

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