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Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems. (English) Zbl 1446.65178

Summary: In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J20 Variational methods for second-order elliptic equations

Software:

PCBDDC; BDDCML
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Hiptmair, R.; Moiola, A.; Perugia, I., A survey of Trefftz methods for the Helmholtz equation, (Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (2016), Springer), 237-279 · Zbl 1357.65282
[2] Cessenat, O.; Despres, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM J. Numer. Anal., 35, 1, 255-299 (1998) · Zbl 0955.65081
[3] Gamallo, P.; Astley, R. J., A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems, Internat. J. Numer. Methods Engrg., 71, 4, 406-432 (2007) · Zbl 1194.76239
[4] Kovalevsky, L.; Ladevéze, P.; Riou, H., The Fourier version of the variational theory of complex rays for medium-frequency acoustics, Comput. Methods Appl. Mech. Engrg., 225, 142-153 (2012) · Zbl 1253.76118
[5] Ladevéze, P.; Arnaud, L.; Rouch, P.; Blanzé, C., The variational theory of complex rays for the calculation of medium-frequency vibrations, Eng. Comput., 18, 1/2, 193-214 (2001) · Zbl 0997.74025
[6] Monk, P.; Wang, D., A least-squares method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 175, 1, 121-136 (1999) · Zbl 0943.65127
[7] Hu, Q.; Yuan, L., A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations, Int. J. Numer. Anal. Model., 11, 3, 587-607 (2014) · Zbl 1499.65666
[8] Hu, Q.; Yuan, L., A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations, Adv. Comput. Math., 44, 1, 245-275 (2018) · Zbl 1383.78047
[9] Gittelson, C. J.; Hiptmair, R.; Perugia, I., Plane wave discontinuous Galerkin methods: analysis of the h-version, M2AN Math. Model. Numer. Anal., 43, 2, 297-331 (2009) · Zbl 1165.65076
[10] Hiptmair, R.; Moiola, A.; Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49, 1, 264-284 (2011) · Zbl 1229.65215
[11] Toselli, A.; Widlund, O. B., (Domain Decomposition Methods: Algorithms and Theory. Domain Decomposition Methods: Algorithms and Theory, Springer Series in Computational Mathematics, vol. 34 (2005)) · Zbl 1069.65138
[12] C. Farhat, A. Macedo, R. Tezaur, FETI-H: A scalable domain decomposition method for high frequency exterior Helmholtz problems, in: Eleventh International Conference on Domain Decomposition Method, 1999, pp. 231-241.
[13] Farhat, C.; Avery, P.; Tezaur, R.; Li, J., FETI-DPH: a dual-primal domain decomposition method for acoustic scattering, J. Comput. Acoust., 13, 03, 499-524 (2005) · Zbl 1189.76338
[14] Gander, M. J.; Magoules, F.; Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24, 1, 38-60 (2002) · Zbl 1021.65061
[15] Gander, M. J.; Halpern, L.; Magoules, F., An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation, Internat. J. Numer. Methods Fluids, 55, 2, 163-175 (2007) · Zbl 1125.65114
[16] Chen, W.; Liu, Y.; Xu, X., A robust domain decomposition method for the Helmholtz equation with high wave number, ESAIM Math. Model. Numer., 50, 3, 921-944 (2016) · Zbl 1361.65093
[17] Engquist, B.; Ying, L., Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers, Multiscale Model. Simul., 9, 2, 686-710 (2011) · Zbl 1228.65234
[18] Chen, Z.; Xiang, X., A source transfer domain decomposition method for Helmholtz equations in unbounded domain, Part II: Extensions, Numer. Math. Theory Methods, 6, 3, 538-555 (2013) · Zbl 1299.65287
[19] Dohrmann, C. R., A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput., 25, 1, 246-258 (2003) · Zbl 1038.65039
[20] Mandel, J.; Dohrmann, C. R., Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10, 639-659 (2003) · Zbl 1071.65558
[21] Gippert, S.; Klawonn, A.; Rheinbach, O., Analysis of FETI-DP and BDDC for linear elasticity in 3D with almost incompressible components and varying coefficients inside subdomains, SIAM J. Numer. Anal., 50, 5, 2208-2236 (2012) · Zbl 1255.74063
[22] Pavarino, L. F.; Widlund, O. B.; Zampini, S., BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions, SIAM J. Sci. Comput., 32, 6, 3604-3626 (2010) · Zbl 1278.74058
[23] Li, J.; Tu, X., Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems, Numer. Linear Algebra Appl., 16, 9, 745-773 (2009) · Zbl 1224.65248
[24] Tu, X.; Li, J., BDDC for nonsymmetric positive definite and symmetric indefinite problems, (Domain Decomposition Methods in Science and Engineering, Vol. XVIII (2009), Springer), 75-86 · Zbl 1183.65036
[25] Šístek, J.; Sousedík, B.; Burda, P.; Mandel, J.; Novotnỳ, J., Application of the parallel BDDC preconditioner to the Stokes flow, Comput. Fluids, 46, 1, 429-435 (2011) · Zbl 1432.76087
[26] Tu, X., A BDDC algorithm for flow in porous media with a hybrid finite element discretization, Electron. Trans. Numer. Anal., 26, 146-160 (2007) · Zbl 1170.76034
[27] Zampini, S.; Tu, X., Multilevel balancing domain decomposition by constraints deluxe algorithms with adaptive coarse spaces for flow in porous media, SIAM J. Sci. Comput., 39, 4, A1389-A1415 (2017) · Zbl 1426.76582
[28] Da Veiga, L. B.; Pavarino, L. F.; Scacchi, S.; Widlund, O. B.; Zampini, S., Isogeometric BDDC preconditioners with deluxe scaling, SIAM J. Sci. Comput., 36, 3, A1118-A1139 (2014) · Zbl 1320.65047
[29] Kim, H. H.; Chung, E. T., A BDDC algorithm with enriched coarse spaces for two-dimensional elliptic problems with oscillatory and high contrast coefficients, Multiscale Model. Simul., 13, 2, 571-593 (2015) · Zbl 1317.65090
[30] Mandel, J.; Sousedík, B., Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods, Comput. Methods Appl. Mech. Engrg., 196, 8, 1389-1399 (2007) · Zbl 1173.74435
[31] Kim, H. H.; Chung, E.; Wang, J., BDDC and FETI-DP algorithms with a change of basis formulation on adaptive primal constraints, Electron. Trans. Numer. Anal., 49, 64-80 (2018) · Zbl 1391.65068
[32] Klawonn, A.; Radtke, P.; Rheinbach, O., Adaptive coarse spaces for BDDC with a transformation of basis, (Domain Decomposition Methods in Science and Engineering, Vol. XXII (2016), Springer), 301-309 · Zbl 1339.65238
[33] Oh, D. S.; Widlund, O.; Zampini, S.; Dohrmann, C., BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields, Math. Comp., 87, 310, 659-692 (2018) · Zbl 1380.65065
[34] Zampini, S.; Vassilevski, P.; Dobrev, V.; Kolev, T., Balancing domain decomposition by constraints algorithms for Curl-conforming spaces of arbitrary order, (Bjørstad, P. E.; Brenner, S. C.; Halpern, L.; Kim, H. H.; Kornhuber, R.; Rahman, T.; Widlund, O. B., Domain Decomposition Methods in Science and Engineering, Vol. XXIV (2018), Springer International Publishing: Springer International Publishing Cham), 103-116 · Zbl 1442.65435
[35] Peng, J.; Shu, S.; Wang, J., An adaptive BDDC algorithm in variational form for mortar discretizations, J. Comput. Appl. Math., 335, 185-206 (2018) · Zbl 1448.65246
[36] Kim, H. H.; Chung, E. T.; Xu, C., A BDDC algorithm with adaptive primal constraints for staggered discontinuous Galerkin approximation of elliptic problems with highly oscillating coefficients, J. Comput. Appl. Math., 311, 599-617 (2017) · Zbl 1382.65410
[37] Da Veiga, L. B.; Pavarino, L. F.; Scacchi, S.; Widlund, O. B.; Zampini, S., Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners, SIAM J. Sci. Comput., 39, 1, A281-A302 (2017) · Zbl 1360.65090
[38] Brenner, S. C.; Sung, L.-Y., BDDC and FETI-DP without matrices or vectors, Comput. Methods Appl. Mech. Engrg., 196, 8, 1429-1435 (2007) · Zbl 1173.65363
[39] Peng, J.; Wang, J.; Shu, S., Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations, Internat. J. Numer. Methods Engrg., 116, 683-707 (2018)
[40] Zampini, S., PCBDDC: a class of robust dual-primal methods in PETSc, SIAM J. Sci. Comput., 38, 5, S282-S306 (2016) · Zbl 1352.65632
[41] Klawonn, A.; Radtke, P.; Rheinbach, O., A comparison of adaptive coarse spaces for iterative substructuring in two dimensions, Electron. Trans. Numer. Anal., 45, 75-106 (2016) · Zbl 1338.65084
[42] Hu, Q.; Li, X., Efficient multilevel preconditioners for three-dimensional plane wave Helmholtz systems with large wave numbers, Multiscale Model. Simul., 15, 3, 1242-1266 (2017)
[43] Pechstein, C.; Dohrmann, C. R., A unified framework for adaptive BDDC, Electron. Trans. Numer. Anal., 46, 273-336 (2017) · Zbl 1368.65043
[44] Anderson, W. N.; Duffin, R. J., Series and parallel addition of matrices, J. Math. Anal. Appl., 26, 3, 576-594 (1969) · Zbl 0177.04904
[45] Mandel, J.; Sousedík, B.; Dohrmann, C. R., Multispace and multilevel BDDC, Computing, 83, 2-3, 55-85 (2008) · Zbl 1163.65091
[46] Dohrmann, C. R.; Pechstein, C., Constraint and Weight Selection Algorithms for BDDCTech. Rep. (2012), Sandia National Lab (SNL-NM): Sandia National Lab (SNL-NM) Albuquerque, NM, United States
[47] Rixen, D. J.; Farhat, C., A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems, Internat. J. Numer. Methods Engrg., 44, 4, 489-516 (1999) · Zbl 0940.74067
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