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

Toselli, Andrea; Vasseur, Xavier

doi:10.1051/m2an:2006004

A numerical study on Neumann-Neumann methods for

h p

approximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Toselli, Andrea ; Vasseur, Xavier

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 99-122.

Résumé

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 $h p$ 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.

MR Zbl

DOI : 10.1051/m2an:2006004

Classification : 65N22, 65N35, 65N55
Mots clés : domain decomposition, preconditioning, $hp$ finite elements, spectral elements, anisotropic meshes

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     author = {Toselli, Andrea and Vasseur, Xavier},
     title = {A numerical study on {Neumann-Neumann} methods for $hp$ approximations on geometrically refined boundary layer meshes {II.} {Three-dimensional} problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {99--122},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     doi = {10.1051/m2an:2006004},
     mrnumber = {2223506},
     zbl = {1094.65121},
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}

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Toselli, Andrea; Vasseur, Xavier. A numerical study on Neumann-Neumann methods for $hp$ approximations on geometrically refined boundary layer meshes II. Three-dimensional problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 99-122. doi : 10.1051/m2an:2006004. http://www.numdam.org/articles/10.1051/m2an:2006004/

Bibliographie
Cité par

[1] Y. Achdou, P. Le Tallec, F. Nataf and M. Vidrascu, A domain decomposition preconditioner for an advection-diffusion problem. Comput. Methods Appl Mech. Engrg. 184 (2000) 145-170. | Zbl

[2] M. Ainsworth, A preconditioner based on domain decomposition for $h p$ -FE approximation on quasi-uniform meshes. SIAM J. Numer. Anal. 33 (1996) 1358-1376. | Zbl

[3] B. Andersson, U. Falk, I. Babuška and T. Von Petersdorff, Reliable stress and fracture mechanics analysis of complex aircraft components using a $h p$ -version FEM. Int. J. Numer. Meth. Eng. 38 (1995) 2135-2163. | Zbl

[4] O. Axelsson, Iterative Solution Methods. Cambridge University Press (1994). | MR | Zbl

[5] I. Babuška and B. Guo, Approximation properties of the $h p$ -version of the finite element method. Comput. Methods Appl. Mech. Engrg. 133 (1996) 319-346. | Zbl

[6] 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). | MR | Zbl

[7] M. Benzi, Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182 (2002) 418-477. | Zbl

[8] M. Benzi and M. Tuma, A parallel solver for large-scale Markov chains. Appl. Numer. Math. 41 (2002) 135-153. | Zbl

[9] C. Bernardi and Y. Maday, Spectral methods. In Handbook of Numerical Analysis, North-Holland, Amsterdam Vol. V, Part 2 (1997) 209-485.

[10] S. Beuchler, Multigrid solver for the inner problem in domain decomposition methods for $p$ -fem. SIAM J. Numer. Anal. 40 (2002) 928-944. | Zbl

[11] A. Björck, Numerical methods for least-squares problems. SIAM (1996). | MR | Zbl

[12] R. Bridson and W.-P. Tang, Refining an approximate inverse. J. Comput. Appl. Math. 123 (2000) 293-306. | Zbl

[13] P. Brown and H. Walker, GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18 (1997) 37-51. | Zbl

[14] W. Cecot, W. Rachowicz and L. Demkowicz, An $h p$ -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. | Zbl

[15] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21 (2000) 1804-1822. | Zbl

[16] M. Dryja and O.B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48 (1995) 121-155. | Zbl

[17] M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313-348. | Zbl

[18] 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. | Zbl

[19] C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engng. 32 (1991) 1205-1227. | Zbl

[20] D.R. Fokkema, G.L.G. Sleijpen and H.A. Van Der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20 (1998) 94-125. | Zbl

[21] P. Frauenfelder and C. Lage, An object oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937-951. | Numdam | Zbl

[22] 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).

[23] G. Golub and C. Van Loan, Matrix Computations. The John Hopkins University Press (1996). Third edition. | MR | Zbl

[24] G. Golub and Q. Ye, Inexact preconditioned conjugate gradient method with inner-outer iterations. SIAM J. Sci. Comput. 21 (1999) 1305-1320. | Zbl

[25] M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18 (1997) 838-853. | Zbl

[26] W.Z. Gui and I. Babuška, The $h$ -, $p$ - and $h p$ -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 $h p$ -version, III: The adaptive $h p$ -version. Numer. Math. 49 (1986) 577-683. | Zbl

[27] B. Guo and W. Cao, Additive Schwarz methods for the $h p$ version of the finite element method in two dimensions. SIAM J. Scientific Comput. 18 (1997) 1267-1288. | Zbl

[28] R. Henderson, Dynamic refinement algorithms for spectral element methods. Comput. Methods Appl. Mech. Engrg. 175 (1999) 395-411. | Zbl

[29] I.C.F. Ipsen and C.D. Meyer, The idea behind Krylov methods. Amer. Math. Monthly 105 (1998) 889-899. | Zbl

[30] G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD. Oxford University Press (1999). | MR | Zbl

[31] V. Korneev, J.E. Flaherty, J.T. Oden and J. Fish, Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes. Appl. Numer. Math 43 (2002) 399-421. | Zbl

[32] V. Korneev, U. Langer and L.S. Xanthis, On fast domain decomposition solving procedures for hp-discretizations of 3d elliptic problems. Comput. Methods Appl. Math. 3 (2003) 536-559. | Zbl

[33] P. Le Tallec and A. Patra, Non-overlapping domain decomposition methods for adaptive $h p$ approximations of the Stokes problem with discontinuous pressure fields. Comput. Methods Appl. Mech. Engrg. 145 (1997) 361-379. | Zbl

[34] 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). | Zbl

[35] J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65 (1996) 1387-1401. | Zbl

[36] J.M. Melenk and C. Schwab, $h p$ -FEM for reaction-diffusion equations. I: Robust exponential convergence. SIAM J. Numer. Anal. 35 (1998) 1520-1557. | Zbl

[37] M. Melenk, $h p$ -finite element methods for singular perturbations. Springer Verlag. Lect. Notes Math. 1796 (2002). | MR | Zbl

[38] P. Monk, Finite element methods for Maxwell's equations. Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, 2003. | Zbl

[39] R. Nicolaides, Deflation of conjugate gradients with application to boundary value problems. SIAM J. Numer. Anal. 24 (1987) 355-36. | Zbl

[40] J.T. Oden, A. Patra and Y. Feng, Parallel domain decomposition solver for adaptive $h p$ finite element methods. SIAM J. Numer. Anal. 34 (1997) 2090-2118. | Zbl

[41] L.F. Pavarino, Neumann-Neumann algorithms for spectral elements in three dimensions. RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471-493. | Numdam | Zbl

[42] L.F. Pavarino and O.B. Widlund, Balancing Neumann-Neumann algorithms for incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 55 (2002) 302-335. | Zbl

[43] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR | Zbl

[44] J. Ruge and K. Stüben, Algebraic multigrid, in Multigrid Methods, S. Mc Cormick Ed. SIAM Philadelphia (1987) 73-130.

[45] Y. Saad, A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14 (1993) 461-469. | Zbl

[46] Y. Saad and M. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear system. SIAM J. Sci. Statist. Comput. 7 (1986) 856-869. | Zbl

[47] Y. Saad and B. Suchomel, Arms: an algebraic recursive multilevel solver for general sparse linear systems. Numer. Linear Algebra Appl. 9 (2002) 359-378. | Zbl

[48] 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.

[49] D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $h p$ -version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38 (2000) 837-875. | Zbl

[50] C. Schwab, $p$ - and $h p$ - Finite Element Methods. Oxford Science Publications (1998). | Zbl

[51] C. Schwab and M. Suri, The $p$ and $h p$ version of the finite element method for problems with boundary layers. Math. Comp. 65 (1996) 1403-1429. | Zbl

[52] C. Schwab, M. Suri and C.A. Xenophontos, The $h p$ -FEM for problems in mechanics with boundary layers. Comput. Methods Appl. Mech. Engrg. 157 (1998) 311-333. | Zbl

[53] B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996). | MR | Zbl

[54] P. Solin, K. Segeth and I. Dolezel, Higher-order finite element methods. Studies in Advanced Mathematics, Chapman and Hall, 2004. | MR | Zbl

[55] A. Toselli, FETI domain decomposition methods for scalar advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5759-5776. | Zbl

[56] A. Toselli and X. Vasseur, Domain decomposition methods of Neumann-Neumann type for $h p$ -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.

[57] A. Toselli and X. Vasseur, A numerical study on Neumann-Neumann and FETI methods for $h p$ -approximations on geometrically refined boundary layer meshes in two dimensions. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4551-4579. | Zbl

[58] A. Toselli and X. Vasseur, Domain decomposition methods of Neumann-Neumann type for $h p$ -approximations on boundary layer meshes in three dimensions. IMA J. Numer. Anal. 24 (2004) 123-156. | Zbl

[59] A. Toselli and O. Widlund, Domain Decomposition methods - Algorithms and Theory. Springer Series on Computational Mathematics, Springer 34 (2004). | Zbl

[60] U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid. Academic Press, London (2000). Guest contribution by Klaus Stüben: “An Introduction to Algebraic Multigrid”. | Zbl

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