TY - JOUR

T1 - Domain decomposition preconditioning in the hierarchical p-version of the finite element method

AU - Korneev, Vadim G.

AU - Jensen, Søren

N1 - Funding Information: Recently p-version finite element methods for solving second order elliptic equations have been intensively studied analytically \[2,3,5-7,13,22-28,33,37-39,43-47,49\] and numerically \[4,5,15-17,19, Research supported in part by grants from the International Science Foundation, by the US National Research Council under the CAST program, and by a grant from Office of Naval Research N00014-90-J-1238. * Corresponding author. E-mall: komev@rpi.edu. I Deceased.

PY - 1999/4

Y1 - 1999/4

N2 - The p-version finite element method for solving linear second order elliptic equations in an arbitrary sufficiently smooth domain is studied in the framework of the Domain Decomposition (DD) method. Curvilinear elements associated with the square reference elements and satisfying conditions of generalized quasiuniformity are used to approximate the boundary and boundary conditions. Two types of square reference elements are primarily considered with the products of integrated Legendre polynomials for coordinate functions. Condition number estimates are given, and preconditioning of the problems arising on subdomains and of the Schur complement together with derivation of the global DD preconditioner are all considered. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). The paper consists of seven sections. We give some preliminary results for the ID case, condition number estimates and some inequalities for the 2D reference element. The preconditioning of the Schur complement is detailed for the 2D reference element, the p-version with curvilinear finite elements is considered next, and the DD preconditioning of the entire stiffness matrix is introduced and analyzed. Some related reference elements using Lobatto-Chebyshev nodal bases on the boundary are introduced and studied with respect to the effect on the preconditioner - the (log p)3 factor may thus be removed. We discuss also some specific features of the algorithms stemming from the suggested preconditioners, which provide a low computational cost and a high degree of parallelization.

AB - The p-version finite element method for solving linear second order elliptic equations in an arbitrary sufficiently smooth domain is studied in the framework of the Domain Decomposition (DD) method. Curvilinear elements associated with the square reference elements and satisfying conditions of generalized quasiuniformity are used to approximate the boundary and boundary conditions. Two types of square reference elements are primarily considered with the products of integrated Legendre polynomials for coordinate functions. Condition number estimates are given, and preconditioning of the problems arising on subdomains and of the Schur complement together with derivation of the global DD preconditioner are all considered. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). The paper consists of seven sections. We give some preliminary results for the ID case, condition number estimates and some inequalities for the 2D reference element. The preconditioning of the Schur complement is detailed for the 2D reference element, the p-version with curvilinear finite elements is considered next, and the DD preconditioning of the entire stiffness matrix is introduced and analyzed. Some related reference elements using Lobatto-Chebyshev nodal bases on the boundary are introduced and studied with respect to the effect on the preconditioner - the (log p)3 factor may thus be removed. We discuss also some specific features of the algorithms stemming from the suggested preconditioners, which provide a low computational cost and a high degree of parallelization.

KW - Domain decomposition

KW - H-p version of the finite element method

UR - http://www.scopus.com/inward/record.url?scp=0033117093&partnerID=8YFLogxK

U2 - 10.1016/S0168-9274(98)00077-4

DO - 10.1016/S0168-9274(98)00077-4

M3 - Article

AN - SCOPUS:0033117093

VL - 29

SP - 479

EP - 518

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 4

ER -