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.

N1 - Funding Information:
Over the last decade, preconditioners, in particular those based on domain decomposition, have attracted increasing interest among numerical analysts; seven annual, international symposia tThis work was supported by the U.S. Department of Energy under Contract DE-FG-05-92ER25142 and by the State of Texas under Contract 1059. tThis work was supported in part by the National Science Foundation under Grant NSF-CCR-9204255 and, in part, by the U.S. Department of Energy under Contracts DE-FG02-92ER25127 and DF_,-FG02-88ER25053.

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

U2 - 10.1016/s0898-1221(96)00230-1

DO - 10.1016/s0898-1221(96)00230-1

M3 - Article

AN - SCOPUS:0030826037

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 -