The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

Research output: Contribution to journalArticle

117 Citations (Scopus)

Abstract

An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal muitigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.

Original languageEnglish (US)
Pages (from-to)215-235
Number of pages21
JournalComputing (Vienna/New York)
Volume56
Issue number3
DOIs
StatePublished - Jan 1 1996

Fingerprint

Preconditioning Techniques
Unstructured Grid
Multigrid Method
Preconditioner
Grid
Relaxation Scheme
Biharmonic Equation
Partial Differential Operators
Precondition
Programming
Finite Element
Numerical Results

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Computer Science Applications
  • Computational Theory and Mathematics
  • Computational Mathematics

Cite this

@article{5537f55c08a14e37b8b351f5c0a1f5e4,
title = "The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids",
abstract = "An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal muitigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.",
author = "Jinchao Xu",
year = "1996",
month = "1",
day = "1",
doi = "10.1007/BF02238513",
language = "English (US)",
volume = "56",
pages = "215--235",
journal = "Computing (Vienna/New York)",
issn = "0010-485X",
publisher = "Springer Wien",
number = "3",

}

The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. / Xu, Jinchao.

In: Computing (Vienna/New York), Vol. 56, No. 3, 01.01.1996, p. 215-235.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

AU - Xu, Jinchao

PY - 1996/1/1

Y1 - 1996/1/1

N2 - An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal muitigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.

AB - An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal muitigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.

UR - http://www.scopus.com/inward/record.url?scp=0030398920&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030398920&partnerID=8YFLogxK

U2 - 10.1007/BF02238513

DO - 10.1007/BF02238513

M3 - Article

AN - SCOPUS:0030398920

VL - 56

SP - 215

EP - 235

JO - Computing (Vienna/New York)

JF - Computing (Vienna/New York)

SN - 0010-485X

IS - 3

ER -