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A numerical study on Neumann-Neumann methods for hp approximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Published online by Cambridge University Press:  23 February 2006

Andrea Toselli  and
Xavier Vasseur
Andrea Toselli
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Xavier.Vasseur@cerfacs.fr
Xavier Vasseur
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Xavier.Vasseur@cerfacs.fr
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Abstract

In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal.24 (2004) 123–156]. They confirm that the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. Good results are also obtained for certain singularly perturbed problems. The condition numbers only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes [Pavarino, RAIRO: Modél. Math. Anal. Numér.31 (1997) 471–493]. This paper follows [Toselli and Vasseur, Comput. Methods Appl. Mech. Engrg.192 (2003) 4551–4579] on two dimensional problems.

Keywords

Domain decompositionpreconditioninghp finite elementsspectral elementsanisotropic meshes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 1 , January 2006 , pp. 99 - 122
Copyright
© EDP Sciences, SMAI, 2006

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References

Achdou, Y., Le Tallec, P., Nataf, F. and Vidrascu, M., A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl Mech. Engrg. 184 (2000) 145–170. CrossRef
Ainsworth, M., A preconditioner based on domain decomposition for hp–FE approximation on quasi–uniform meshes. SIAM J. Numer. Anal. 33 (1996) 1358–1376. CrossRef
Andersson, B., Falk, U., Babuška, I. and von Petersdorff, T., Reliable stress and fracture mechanics analysis of complex aircraft components using a hp–version FEM. Int. J. Numer. Meth. Eng. 38 (1995) 2135–2163. CrossRef
O. Axelsson, Iterative Solution Methods. Cambridge University Press (1994).
Babuška, I. and Guo, B., Approximation properties of the hp–version of the finite element method. Comput. Methods Appl. Mech. Engrg. 133 (1996) 319–346. CrossRef
R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edition. SIAM, Philadelphia, PA (1994).
Benzi, M., Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182 (2002) 418–477. CrossRef
Benzi, M. and Tuma, M., A parallel solver for large-scale Markov chains. Appl. Numer. Math. 41 (2002) 135–153. CrossRef
C. Bernardi and Y. Maday, Spectral methods. In Handbook of Numerical Analysis, North-Holland, Amsterdam Vol. V, Part 2 (1997) 209–485.
Beuchler, S., Multigrid solver for the inner problem in domain decomposition methods for p-fem. SIAM J. Numer. Anal. 40 (2002) 928–944. CrossRef
A. Björck, Numerical methods for least-squares problems. SIAM (1996).
Bridson, R. and Tang, W.-P., Refining an approximate inverse. J. Comput. Appl. Math. 123 (2000) 293–306. CrossRef
Brown, P. and Walker, H., GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18 (1997) 37–51. CrossRef
Cecot, W., Rachowicz, W. and Demkowicz, L., An hp-adaptive finite element method for electromagnetics. III. a three-dimensional infinite element for Maxwell's equations. Internat. J. Numer. Methods Engrg. 57 (2003) 899–921. CrossRef
Chow, E., A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21 (2000) 1804–1822. CrossRef
Dryja, M. and Widlund, O.B., Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48 (1995) 121–155. CrossRef
Dryja, M., Sarkis, M.V. and Widlund, O.B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313–348. CrossRef
C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics, in Computational Mechanics Advances, J. Tinsley Oden Ed. North-Holland 2 (1994) 1–124.
Farhat, C. and Roux, F.-X., A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engng. 32 (1991) 1205–1227.
Fokkema, D.R., Sleijpen, G.L.G. and Van der Vorst, H.A., Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20 (1998) 94–125. CrossRef
Frauenfelder, P. and Lage, C., An object oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937–951. CrossRef
R. Geus, The Jacobi-Davidson algorithm for solving large sparse symmetric eigenvalue problems with application to the design of accelerator cavities. Ph.D. thesis, ETH, Zürich, Institut für Wissenschaftliches Rechnen (2002).
G. Golub and C. Van Loan, Matrix Computations. The John Hopkins University Press (1996). Third edition.
Golub, G. and Inexact, Q. Ye preconditioned conjugate gradient method with inner-outer iterations. SIAM J. Sci. Comput. 21 (1999) 1305–1320. CrossRef
Grote, M. and Huckle, T., Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18 (1997) 838–853. CrossRef
Gui, W.Z. and Babuška, I., The h-, p- and hp-version of the Finite Element Method in one dimension, I: The error analysis of the p-version, II: The error analysis of the h- and hp-version, III: The adaptive hp-version. Numer. Math. 49 (1986) 577–683. CrossRef
Guo, B. and Cao, W., Additive Schwarz methods for the hp version of the finite element method in two dimensions. SIAM J. Scientific Comput. 18 (1997) 1267–1288. CrossRef
R. Henderson, Dynamic refinement algorithms for spectral element methods. Comput. Methods Appl. Mech. Engrg. 175 (1999) 395–411.
Ipsen, I.C.F. and Meyer, C.D., The idea behind Krylov methods. Amer. Math. Monthly 105 (1998) 889–899. CrossRef
G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD. Oxford University Press (1999).
Korneev, V., Flaherty, J.E., Oden, J.T. and Fish, J., Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes. Appl. Numer. Math 43 (2002) 399–421. CrossRef
Korneev, V., Langer, U. and Xanthis, L.S., On fast domain decomposition solving procedures for hp-discretizations of 3d elliptic problems. Comput. Methods Appl. Math. 3 (2003) 536–559. CrossRef
Le Tallec, P. and Patra, A., Non–overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields. Comput. Methods Appl. Mech. Engrg. 145 (1997) 361–379. CrossRef
J.W. Lottes and P.F. Fischer, Hybrid Multigrid/Schwarz algorithms for the spectral element method. Technical report, Mathematics and Computer Science Division, Argonne National Laboratory (January 2003).
Mandel, J. and Brezina, M., Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65 (1996) 1387–1401. CrossRef
Melenk, J.M. and Schwab, C., hp–FEM for reaction–diffusion equations. I: Robust exponential convergence. SIAM J. Numer. Anal. 35 (1998) 1520–1557. CrossRef
M. Melenk, hp-finite element methods for singular perturbations. Springer Verlag. Lect. Notes Math. 1796 (2002).
P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, 2003.
Nicolaides, R., Deflation of conjugate gradients with application to boundary value problems. SIAM J. Numer. Anal. 24 (1987) 355–36. CrossRef
Oden, J.T., Patra, A. and Feng, Y., Parallel domain decomposition solver for adaptive hp finite element methods. SIAM J. Numer. Anal. 34 (1997) 2090–2118. CrossRef
Pavarino, L.F., Neumann-Neumann algorithms for spectral elements in three dimensions. RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471–493.
Pavarino, L.F. and Widlund, O.B., Balancing Neumann-Neumann algorithms for incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 55 (2002) 302–335. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994).
J. Ruge and K. Stüben, Algebraic multigrid, in Multigrid Methods, S. Mc Cormick Ed. SIAM Philadelphia (1987) 73–130.
Saad, Y., A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14 (1993) 461–469. CrossRef
Saad, Y. and Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear system. SIAM J. Sci. Statist. Comput. 7 (1986) 856–869. CrossRef
Saad, Y. and Suchomel, B., Arms: an algebraic recursive multilevel solver for general sparse linear systems. Numer. Linear Algebra Appl. 9 (2002) 359–378. CrossRef
M.V. Sarkis, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. Ph.D. thesis, Courant Institute, New York University, September (1994). TR671, Department of Computer Science, New York University, URL: file://cs.nyu.edu/pub/tech-reports/tr671.ps.Z.
Schötzau, D. and Schwab, C., Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38 (2000) 837–875. CrossRef
C. Schwab, p– and hp– Finite Element Methods. Oxford Science Publications (1998).
Schwab, C. and Suri, M., The p and hp version of the finite element method for problems with boundary layers. Math. Comp. 65 (1996) 1403–1429. CrossRef
Schwab, C., Suri, M. and Xenophontos, C.A., The hp–FEM for problems in mechanics with boundary layers. Comput. Methods Appl. Mech. Engrg. 157 (1998) 311–333. CrossRef
B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996).
P. Solin, K. Segeth and I. Dolezel, Higher-order finite element methods. Studies in Advanced Mathematics, Chapman and Hall, 2004.
A. Toselli, FETI domain decomposition methods for scalar advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5759–5776.
A. Toselli and X. Vasseur, Domain decomposition methods of Neumann-Neumann type for hp-approximations on geometrically refined boundary layer meshes in two dimensions. Technical Report 02–15, Seminar für Angewandte Mathematik, ETH, Zürich (September 2002). Submitted to Numerische Mathematik.
A. Toselli and X. Vasseur, A numerical study on Neumann-Neumann and FETI methods for hp-approximations on geometrically refined boundary layer meshes in two dimensions. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4551–4579.
Toselli, A. and Vasseur, X., Domain decomposition methods of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions. IMA J. Numer. Anal. 24 (2004) 123–156. CrossRef
A. Toselli and O. Widlund, Domain Decomposition methods – Algorithms and Theory. Springer Series on Computational Mathematics, Springer 34 (2004).
U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid. Academic Press, London (2000). Guest contribution by Klaus Stüben: “An Introduction to Algebraic Multigrid”.