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Integration of \(hp\)-adaptivity and a two-grid solver for elliptic problems. (English) Zbl 1093.65112
Summary: We present implementation details and analyze convergence of a two-grid solver forming the core of the fully automatic \(hp\)-adaptive strategy for elliptic problems. The solver delivers a solution for a fine grid obtained from an arbitrary coarse \(hp\)-grid by a global \(hp\)-refinement. The classical V-cycle algorithm combines an overlapping block Jacobi smoother with optimal relaxation, and a direct solved on the coarse grid. A simple theoretical analysis is illustrated with extensive numerical experimentation.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations
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[1] Ainsworth, M., A hierarchical domain decomposition preconditioner for hp finite element approximation on locally refined meshes, SIAM J. sci. comput., 17, 6, 1395-1414, (1996) · Zbl 0860.65111
[2] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, Society for Industrial and Applied Mathematics, Version 3.0, 1999.
[3] Axelsson, O., Iterative solution methods, (1993), Cambridge University Press Cambridge
[4] Barragy, E.; Carey, G.F., A parallel element-by-element solution scheme, Int. J. numer. methods engrg., 26, 2367-2382, (1988) · Zbl 0662.73051
[5] Barrett, R.; Berry, M.; Chan, T.F.; Demmel, J.; Donato, J.M.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; Van der Vost, H., Templates for the solution of linear systems: building blocks for iterative methods, (1994), SIAM Philadelphia
[6] E.B. Becker, T.H. Miller, G.A. Collingwood, User’s Manual for the TEXVISC Computer Program, NASA Report MSFC-RPT-1568, 1988.
[7] Bramble, J.H., Multigrid methods, (1995), Longman Scientific & Technical
[8] (2004)
[9] Carey, G.F.; Barragy, E.; McLay, R.; Sharma, M., Element-by-element vector and parallel computations, Commun. appl. numer. methods, 4, 299-307, (1988) · Zbl 0638.73042
[10] L. Demkowicz, 2D hp-Adaptive Finite Element Package (2Dhp90). Version 2.0, TICAM Report 02-06, January 02. · Zbl 1093.65113
[11] Demkowicz, L.; Gerdes, K.; Schwab, Ch.; Bajer, A.; Walsh, T., HP90: a general and flexible Fortran 90 hp-FE code, Comput. visual. sci., 1, 145-163, (1998) · Zbl 0912.68014
[12] L. Demkowicz, P. Monk, L. Vardapetyan, W. Rachowicz, The Rham diagram for hp finite element spaces, Ticam Report 99-07, 1999. · Zbl 0955.65084
[13] L. Demkowicz, I. Babuska, p Interpolation error estimates for edge finite elements of variable order in 2D, SIAM J. Numer. Anal. 41(4) 1195-1208. · Zbl 1067.78016
[14] Demkowicz, L.; Rachowicz, W.; Devloo, Ph., A fully automatic hp-adaptivity, J. scient. comput., 17, 1-3, 127-155, (2002) · Zbl 0999.65121
[15] L. Demkowicz, D. Pardo, W. Rachowicz, 3D hp-Adaptive Finite Element Package (3Dhp90), Version 2.0. The ultimate (?) data structure for three-dimensional, anisotropic hp-refinements, Ticam Report 02-24, June 2002.
[16] Demmel, J.W.; Gilbert, J.R.; Li, X.S., Superlu users’ guide, (1999), University of California Berkeley
[17] Duff, I.; Grimes, R.; Lewis, J., Sparse matrix test problems, ACM trans. math. soft., 15, 1-14, (1989) · Zbl 0667.65040
[18] Chemnitz, M.J.; Nepomnyaschikh, S.V., Variable additive preconditioning procedures, Computing, 62, 109-128, (1999) · Zbl 0932.65125
[19] Mandel, J., Hybrid domain decomposition with unstructured subdomains, (), 103-112 · Zbl 0796.65127
[20] McLay, R.T.; Swift, S.; Carey, G.F., Maximizing sparse matrix-vector product performance on RISC based MIMD computers, J. parallel distr. comput., 37, 146-158, (1996)
[21] Oden, J.T.; Demkowicz, L.F., Applied functional analysis for science and engineering, (1996), CRC Press Boca Raton
[22] W. Rachowicz, L. Demkowicz, A. Bajer, T. Walsh, A two-grid iterative solver for stationary Maxwell’s equations, in: D. Kincaid et al. (Eds.), Iterative Methods in Scientific Computations II, IMACS, 1999.
[23] Shewchuck, J.R., An introduction to the conjugate gradient algorithm without the agonizing pain, (1994), School of Computer Science, Carnegie Mellon University Pittsburgh
[24] Smith, B.F.; Bjorstad, P.E.; Gropp, W.D., Domain decomposition. parallel multilevel methods for elliptic partial differential equations, (1996), Cambridge University Press New York · Zbl 0857.65126
[25] Van de Geijn, R., Using PLAPACK: parallel linear algebra package, (1997), The MIT Press Cambridge
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