Weak approximation properties of elliptic projections with functional constraints

Robert Scheichl, P S Vassilevski, L 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
Pages (from-to)1677-1699
Number of pages23
JournalMultiscale Modeling and Simulation
Volume9
Issue number4
DOIs
Publication statusPublished - 2011

Fingerprint

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

Keywords

  • algebraic multigrid
  • overlapping Schwarz method
  • elliptic problems with large coefficient variation
  • coarse spaces
  • abstract Bramble-Hilbert-type lemma

Cite this

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

In: Multiscale Modeling and Simulation, Vol. 9, No. 4, 2011, p. 1677-1699.

Research output: Contribution to journalArticle

@article{56f89996855f4eba90345289dc0649a9,
title = "Weak approximation properties of elliptic projections with functional constraints",
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.",
keywords = "algebraic multigrid, overlapping Schwarz method, elliptic problems with large coefficient variation, coarse spaces, abstract Bramble-Hilbert-type lemma",
author = "Robert Scheichl and Vassilevski, {P S} and Zikatanov, {L T}",
year = "2011",
doi = "10.1137/110821639",
language = "English",
volume = "9",
pages = "1677--1699",
journal = "Multiscale Modeling and Simulation",
issn = "1540-3459",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "4",

}

TY - JOUR

T1 - Weak approximation properties of elliptic projections with functional constraints

AU - Scheichl, Robert

AU - Vassilevski, P S

AU - Zikatanov, L T

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - algebraic multigrid

KW - overlapping Schwarz method

KW - elliptic problems with large coefficient variation

KW - coarse spaces

KW - abstract Bramble-Hilbert-type lemma

UR - http://www.scopus.com/inward/record.url?scp=84856540404&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1137/110821639

U2 - 10.1137/110821639

DO - 10.1137/110821639

M3 - Article

VL - 9

SP - 1677

EP - 1699

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 4

ER -