TY - JOUR

T1 - On Fast Domain Decomposition Solving Procedures for hp-Discretizations of 3-D Elliptic Problems

AU - Korneev, V.

AU - Langer, U.

AU - Xanthis, L. S.

PY - 2003

Y1 - 2003

N2 - A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by K-interpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently. The relative condition number, provided by the DD preconditioner under consideration, is O((1 + log p)3.5) and its total arithmetic cost is O((1 + log p)1.75[(1 + log p)(1 + log(1 + log p))p3R + pR2]), where R is the number of finite elements. The term pR2 is due to the solver for the wire basket subsystem. We outline, how the cost of this component may be reduced to O(pR). The presented DD algorithms are highly parallelizable.

AB - A DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by K-interpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently. The relative condition number, provided by the DD preconditioner under consideration, is O((1 + log p)3.5) and its total arithmetic cost is O((1 + log p)1.75[(1 + log p)(1 + log(1 + log p))p3R + pR2]), where R is the number of finite elements. The term pR2 is due to the solver for the wire basket subsystem. We outline, how the cost of this component may be reduced to O(pR). The presented DD algorithms are highly parallelizable.

KW - domain decomposition

KW - fast solvers

KW - hp finite element discretizations

KW - preconditioning

UR - http://www.scopus.com/inward/record.url?scp=85013123146&partnerID=8YFLogxK

U2 - 10.2478/cmam-2003-0034

DO - 10.2478/cmam-2003-0034

M3 - Article

AN - SCOPUS:85013123146

VL - 3

SP - 536

EP - 559

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 4

ER -