A numerical study on Neumann-Neumann methods for $hp$ approximations on geometrically refined boundary layer meshes II. Three-dimensional problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 99-122

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.

DOI : https://doi.org/10.1051/m2an:2006004
Classification:  65N22,  65N35,  65N55
Keywords: domain decomposition, preconditioning, $hp$ finite elements, spectral elements, anisotropic meshes
@article{M2AN_2006__40_1_99_0,
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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {1},
year = {2006},
pages = {99-122},
doi = {10.1051/m2an:2006004},
zbl = {1094.65121},
mrnumber = {2223506},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_1_99_0}
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 99-122. doi : 10.1051/m2an:2006004. http://www.numdam.org/item/M2AN_2006__40_1_99_0/

[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 0979.76043

[2] M. Ainsworth, A preconditioner based on domain decomposition for $hp$-FE approximation on quasi-uniform meshes. SIAM J. Numer. Anal. 33 (1996) 1358-1376. | Zbl 0855.65044

[3] B. Andersson, U. Falk, I. Babuška and T. Von Petersdorff, Reliable stress and fracture mechanics analysis of complex aircraft components using a $hp$-version FEM. Int. J. Numer. Meth. Eng. 38 (1995) 2135-2163. | Zbl 0834.73064

[4] O. Axelsson, Iterative Solution Methods. Cambridge University Press (1994). | MR 1276069 | Zbl 0795.65014

[5] I. Babuška and B. Guo, Approximation properties of the $hp$-version of the finite element method. Comput. Methods Appl. Mech. Engrg. 133 (1996) 319-346. | Zbl 0882.65096

[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 1247007 | Zbl 0814.65030

[7] M. Benzi, Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182 (2002) 418-477. | Zbl 1015.65018

[8] M. Benzi and M. Tuma, A parallel solver for large-scale Markov chains. Appl. Numer. Math. 41 (2002) 135-153. | Zbl 0997.65006

[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 1030.65125

[11] A. Björck, Numerical methods for least-squares problems. SIAM (1996). | MR 1386889 | Zbl 0847.65023

[12] R. Bridson and W.-P. Tang, Refining an approximate inverse. J. Comput. Appl. Math. 123 (2000) 293-306. | Zbl 0982.65035

[13] P. Brown and H. Walker, GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl. 18 (1997) 37-51. | Zbl 0876.65019

[14] W. Cecot, W. Rachowicz and L. Demkowicz, 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. | Zbl 1034.78017

[15] E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21 (2000) 1804-1822. | Zbl 0957.65023

[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 0824.65106

[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 0857.65131

[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 0805.73062

[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 0758.65075

[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 0924.65027

[21] P. Frauenfelder and C. Lage, An object oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937-951. | Numdam | Zbl 1032.65128

[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 1417720 | Zbl 0865.65009

[24] G. Golub and Q. Ye, Inexact preconditioned conjugate gradient method with inner-outer iterations. SIAM J. Sci. Comput. 21 (1999) 1305-1320. | Zbl 0955.65022

[25] M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18 (1997) 838-853. | Zbl 0872.65031

[26] W.Z. Gui and I. Babuška, 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. | Zbl 0614.65088

[27] B. Guo and W. Cao, Additive Schwarz methods for the $hp$ version of the finite element method in two dimensions. SIAM J. Scientific Comput. 18 (1997) 1267-1288. | Zbl 0892.65072

[28] R. Henderson, Dynamic refinement algorithms for spectral element methods. Comput. Methods Appl. Mech. Engrg. 175 (1999) 395-411. | Zbl 0927.76077

[29] I.C.F. Ipsen and C.D. Meyer, The idea behind Krylov methods. Amer. Math. Monthly 105 (1998) 889-899. | Zbl 0982.65034

[30] G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD. Oxford University Press (1999). | MR 1696933 | Zbl 0954.76001

[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 1018.65126

[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 1038.65133

[33] P. Le Tallec and A. Patra, 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. | Zbl 0891.76053

[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 1078.65570

[35] J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65 (1996) 1387-1401. | Zbl 0853.65129

[36] J.M. Melenk and C. Schwab, $hp$-FEM for reaction-diffusion equations. I: Robust exponential convergence. SIAM J. Numer. Anal. 35 (1998) 1520-1557. | Zbl 0972.65093

[37] M. Melenk, $hp$-finite element methods for singular perturbations. Springer Verlag. Lect. Notes Math. 1796 (2002). | MR 1939620 | Zbl 1021.65055

[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 1024.78009

[39] R. Nicolaides, Deflation of conjugate gradients with application to boundary value problems. SIAM J. Numer. Anal. 24 (1987) 355-36. | Zbl 0624.65028

[40] J.T. Oden, A. Patra and Y. Feng, Parallel domain decomposition solver for adaptive $hp$ finite element methods. SIAM J. Numer. Anal. 34 (1997) 2090-2118. | Zbl 0890.65124

[41] L.F. Pavarino, Neumann-Neumann algorithms for spectral elements in three dimensions. RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471-493. | Numdam | Zbl 0881.65121

[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 1024.76025

[43] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR 1299729 | Zbl 0803.65088

[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 0780.65022

[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 0599.65018

[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 1071.65001

[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 $hp$-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38 (2000) 837-875. | Zbl 0978.65091

[50] C. Schwab, $p$- and $hp$- Finite Element Methods. Oxford Science Publications (1998). | Zbl 0910.73003

[51] C. Schwab and M. Suri, The $p$ and $hp$ version of the finite element method for problems with boundary layers. Math. Comp. 65 (1996) 1403-1429. | Zbl 0853.65115

[52] C. Schwab, M. Suri and C.A. Xenophontos, The $hp$-FEM for problems in mechanics with boundary layers. Comput. Methods Appl. Mech. Engrg. 157 (1998) 311-333. | Zbl 0959.74073

[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 1410757 | Zbl 0857.65126

[54] P. Solin, K. Segeth and I. Dolezel, Higher-order finite element methods. Studies in Advanced Mathematics, Chapman and Hall, 2004. | MR 2000261 | Zbl 1032.65132

[55] A. Toselli, FETI domain decomposition methods for scalar advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 190 (2001) 5759-5776. | Zbl 1017.76048

[56] 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.

[57] 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. | Zbl 1054.65117

[58] A. Toselli and X. Vasseur, 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. | Zbl 1048.65125

[59] A. Toselli and O. Widlund, Domain Decomposition methods - Algorithms and Theory. Springer Series on Computational Mathematics, Springer 34 (2004). | Zbl 1069.65138

[60] U. Trottenberg, C. Oosterlee and A. Schüller, Multigrid. Academic Press, London (2000). Guest contribution by Klaus Stüben: “An Introduction to Algebraic Multigrid”. | Zbl 0976.65106