### Abstract

Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for p-version finite element methods based on continuous, piecewise Q_{p} functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using Gauss-Lobatto-Legendre quadrature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p)^{2}. These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory.

Original language | English (US) |
---|---|

Pages (from-to) | 193-209 |

Number of pages | 17 |

Journal | Computers and Mathematics with Applications |

Volume | 33 |

Issue number | 1-2 |

State | Published - Jan 1997 |

### Fingerprint

### Keywords

- Domain decomposition
- Gauss-Lobatto-Legendre quadrature
- Iterative substructuring
- Preconditioned conjugate gradient methods
- Spectral finite element approximation

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Computers and Mathematics with Applications*,

*33*(1-2), 193-209.

**Iterative substructuring methods for spectral elements : Problems in three dimensions based on numerical quadrature.** / Pavarino, L. F.; Widlund, O. B.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 33, no. 1-2, pp. 193-209.

}

TY - JOUR

T1 - Iterative substructuring methods for spectral elements

T2 - Problems in three dimensions based on numerical quadrature

AU - Pavarino, L. F.

AU - Widlund, O. B.

PY - 1997/1

Y1 - 1997/1

N2 - Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for p-version finite element methods based on continuous, piecewise Qp functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using Gauss-Lobatto-Legendre quadrature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p)2. These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory.

AB - Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for p-version finite element methods based on continuous, piecewise Qp functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using Gauss-Lobatto-Legendre quadrature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p)2. These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory.

KW - Domain decomposition

KW - Gauss-Lobatto-Legendre quadrature

KW - Iterative substructuring

KW - Preconditioned conjugate gradient methods

KW - Spectral finite element approximation

UR - http://www.scopus.com/inward/record.url?scp=0030826037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030826037&partnerID=8YFLogxK

M3 - Article

VL - 33

SP - 193

EP - 209

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 1-2

ER -