TY - JOUR

T1 - Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes

AU - Korneev, Vadim

AU - Flaherty, J. E.

AU - Oden, J. T.

AU - Fish, J.

N1 - Funding Information: ✩ Research supported in part by grants from the Office of Naval Research N000014-97-1-0687, from the Department of Energy B341495 and B347883, from the Army Research Office NDAAG55-98-1-0200 and DAAH04-96-1-002, and from the Air Force Office of Scientific Research NF49620-95-1-0407. * Corresponding author. E-mail addresses: vadim.korneev@pobox.spbu.ru (V. Korneev), flahej@rpi.edu (J.E. Flaherty), oden@ticam.utexas.edu (J.T. Oden), fishj@rpi.edu (J. Fish).

PY - 2002/12

Y1 - 2002/12

N2 - Highly parallelizable domain decomposition Dirichlet-Dirichlet solvers for hp-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. The edge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes, or hierarchical polynomials of several types. These coordinate functions are defined within elements as being arbitrary or discrete quasi-harmonic coordinate functions. The latter are obtained from explicit and inexpensive prolongation operators. In all situations, we are able to suggest preconditioners which are spectrally equivalent to the global stiffness matrix, which only require element-by-element and edge-by-edge operations, and which reduce computational cost. In this way, elimination is avoided when dealing with the interface problem. The domain decomposition algorithms essentially use prolongation operators from the interface boundary inside the subdomains of the decomposition according to the approach initially used for the hp-version finite element methods with quadrilateral elements [S.A. Ivanov, V.G. Korneev, Izv. Vyssh. Uchebn. Zaved. 395 (1995) 62-81; Technische Universität Chemnitz-Zwickau, Preprint SPC 95-35, 1995, 1-15, and Preprint SPC 95-36, 1995, 1-14; Math. Modeling 8 (1996) 63-73].

AB - Highly parallelizable domain decomposition Dirichlet-Dirichlet solvers for hp-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. The edge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes, or hierarchical polynomials of several types. These coordinate functions are defined within elements as being arbitrary or discrete quasi-harmonic coordinate functions. The latter are obtained from explicit and inexpensive prolongation operators. In all situations, we are able to suggest preconditioners which are spectrally equivalent to the global stiffness matrix, which only require element-by-element and edge-by-edge operations, and which reduce computational cost. In this way, elimination is avoided when dealing with the interface problem. The domain decomposition algorithms essentially use prolongation operators from the interface boundary inside the subdomains of the decomposition according to the approach initially used for the hp-version finite element methods with quadrilateral elements [S.A. Ivanov, V.G. Korneev, Izv. Vyssh. Uchebn. Zaved. 395 (1995) 62-81; Technische Universität Chemnitz-Zwickau, Preprint SPC 95-35, 1995, 1-15, and Preprint SPC 95-36, 1995, 1-14; Math. Modeling 8 (1996) 63-73].

UR - http://www.scopus.com/inward/record.url?scp=0036895124&partnerID=8YFLogxK

U2 - 10.1016/S0168-9274(02)00104-6

DO - 10.1016/S0168-9274(02)00104-6

M3 - Article

AN - SCOPUS:0036895124

VL - 43

SP - 399

EP - 421

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 4

ER -