Weak approximation properties of elliptic projections with functional constraints

Robert Scheichl, Panayot S. Vassilevski, Ludmil T. Zikatanov

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble-Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted L 2 norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided.

Original languageEnglish (US)
Pages (from-to)1677-1699
Number of pages23
JournalMultiscale Modeling and Simulation
Volume9
Issue number4
DOIs
StatePublished - Dec 1 2011

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Weak Approximation
Approximation Property
projection
Projection
pulse detonation engines
coefficients
approximation
Coefficient
Upscaling
Varying Coefficients
Elliptic PDE
Saddle Point Problems
upscaling
Interpolants
eigenvalue
saddle points
norms
Elliptic Problems
Hilbert
Eigenvalue Problem

All Science Journal Classification (ASJC) codes

  • Chemistry(all)
  • Modeling and Simulation
  • Ecological Modeling
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

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Weak approximation properties of elliptic projections with functional constraints. / Scheichl, Robert; Vassilevski, Panayot S.; Zikatanov, Ludmil T.

In: Multiscale Modeling and Simulation, Vol. 9, No. 4, 01.12.2011, p. 1677-1699.

Research output: Contribution to journalArticle

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