### 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 language | English (US) |
---|---|

Pages (from-to) | 1677-1699 |

Number of pages | 23 |

Journal | Multiscale Modeling and Simulation |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2011 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Multiscale Modeling and Simulation*,

*9*(4), 1677-1699. https://doi.org/10.1137/110821639

}

*Multiscale Modeling and Simulation*, vol. 9, no. 4, pp. 1677-1699. https://doi.org/10.1137/110821639

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Weak approximation properties of elliptic projections with functional constraints

AU - Scheichl, Robert

AU - Vassilevski, Panayot S.

AU - Zikatanov, Ludmil T.

PY - 2011/12/1

Y1 - 2011/12/1

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.

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

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

U2 - 10.1137/110821639

DO - 10.1137/110821639

M3 - Article

AN - SCOPUS:84856540404

VL - 9

SP - 1677

EP - 1699

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 4

ER -