Spectral analysis and spectral symbol for the 2D curl-curl (stabilized) operator with applications to the related iterative solutions
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- by Mariarosa Mazza, Ahmed Ratnani and Stefano Serra-Capizzano HTML | PDF
- Math. Comp. 88 (2019), 1155-1188 Request permission
Abstract:
In this paper, we study structural and spectral features of linear systems of equations arising from Galerkin approximations of $H(\mathrm {curl})$ elliptic variational problems, based on the Isogeometric Analysis (IgA) approach. Such problems arise in Time Harmonic Maxwell and Magnetostatic problems, as well in the preconditioning of MagnetoHydroDynamics equations, and lead to large linear systems, with different and severe sources of ill-conditioning.
First, we consider a compatible B-splines discretization based on a discrete de Rham sequence and we study the structure of the resulting matrices $\mathcal {A}_{\mathbfit {n}}$. It turns out that $\mathcal {A}_{\mathbfit {n}}$ shows a two-by-two pattern and is a principal submatrix of a two-by-two block matrix, where each block is two-level banded, almost Toeplitz, and where the bandwidths grow linearly with the degree of the B-splines.
Looking at the coefficients in detail and making use of the theory of the Generalized Locally Toeplitz (GLT) sequences, we identify the symbol of each of these blocks, that is, a function describing asymptotically, i.e., for $\mathbfit {n}$ large enough, the spectrum of each block. From this spectral knowledge and thanks to some new spectral tools we retrieve the symbol of $\{\mathcal {A}_{\mathbfit {n}}\}_{\mathbfit {n}}$ which as expected is a two-by-two matrix-valued bivariate trigonometric polynomial. In particular, there is a nice elegant connection with the continuous operator, which has an infinite dimensional kernel, and in fact the symbol is a dyad having one eigenvalue like the one of the IgA Laplacian, and one identically zero eigenvalue; as a consequence, we prove that one half of the spectrum of $\mathcal {A}_{\mathbfit {n}}$, for $\mathbfit {n}$ large enough, is very close to zero and this represents the discrete counterpart of the infinite dimensional kernel of the continuous operator. From the latter information, showing that the considered problem has an ill-posed nature, we are able to give a detailed spectral analysis of the matrices $\mathcal {A}_{\mathbfit {n}}$ and of the corresponding zero-order term stabilized matrices, which is fully confirmed by several numerical evidences.
Finally, by taking into consideration the GLT theory and making use of the spectral results, we furnish indications on the convergence features of known iterative solvers and we suggest a further stabilization technique and proper iterative procedures for the numerical solution of the involved linear systems.
References
- Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
- A. Buffa, G. Sangalli, and R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 17-20, 1143–1152. MR 2594830, DOI 10.1016/j.cma.2009.12.002
- L. Chacón, An optimal, parallel, fully implicit Newton-Krylov solver for three-dimensional viscoresistive magnetohydrodynamics, Phys. Plasmas 15 (2008), no. 5, 056103.
- Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers, Robust and optimal multi-iterative techniques for IgA Galerkin linear systems, Comput. Methods Appl. Mech. Engrg. 284 (2015), 230–264. MR 3310285, DOI 10.1016/j.cma.2014.06.001
- Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers, Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, SIAM J. Numer. Anal. 55 (2017), no. 1, 31–62. MR 3592079, DOI 10.1137/140988590
- Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680
- Carlo Garoni, Carla Manni, Francesca Pelosi, Stefano Serra-Capizzano, and Hendrik Speleers, On the spectrum of stiffness matrices arising from isogeometric analysis, Numer. Math. 127 (2014), no. 4, 751–799. MR 3229992, DOI 10.1007/s00211-013-0600-2
- Carlo Garoni and Stefano Serra-Capizzano, Generalized locally Toeplitz sequences: theory and applications. Vol. I, Springer, Cham, 2017. MR 3674485, DOI 10.1007/978-3-319-53679-8
- C. Garoni and S. Serra-Capizzano, The Theory of Multilevel Generalized Locally Toeplitz Sequences: Theory and Applications - Vol II, Springer Monographs, 2017, in preparation. Preliminary version in: Technical Report, N. 2, February 2017, Department of Information Technology, Uppsala University.
- Carlo Garoni, Stefano Serra-Capizzano, and Debora Sesana, Spectral analysis and spectral symbol of $d$-variate $\Bbb {Q}_p$ Lagrangian FEM stiffness matrices, SIAM J. Matrix Anal. Appl. 36 (2015), no. 3, 1100–1128. MR 3376130, DOI 10.1137/140976480
- Carlo Garoni, Stefano Serra-Capizzano, and Debora Sesana, Tools for determining the asymptotic spectral distribution of non-Hermitian perturbations of Hermitian matrix-sequences and applications, Integral Equations Operator Theory 81 (2015), no. 2, 213–225. MR 3299836, DOI 10.1007/s00020-014-2157-6
- Leonid Golinskii and Stefano Serra-Capizzano, The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory 144 (2007), no. 1, 84–102. MR 2287378, DOI 10.1016/j.jat.2006.05.002
- Ulf Grenander and Gábor Szegő, Toeplitz forms and their applications, 2nd ed., Chelsea Publishing Co., New York, 1984. MR 890515
- Ralf Hiptmair and Jinchao Xu, Nodal auxiliary space preconditioning in $\textbf {H}(\textbf {curl})$ and $\textbf {H}(\textrm {div})$ spaces, SIAM J. Numer. Anal. 45 (2007), no. 6, 2483–2509. MR 2361899, DOI 10.1137/060660588
- Ralf Hiptmair and Weiying Zheng, Local multigrid in $\textbf {H}(\bf {curl})$, J. Comput. Math. 27 (2009), no. 5, 573–603. MR 2536903, DOI 10.4208/jcm.2009.27.5.012
- T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39-41, 4135–4195. MR 2152382, DOI 10.1016/j.cma.2004.10.008
- Tzanio V. Kolev and Panayot S. Vassilevski, Auxiliary space AMG for $H$(curl) problems, Domain decomposition methods in science and engineering XVII, Lect. Notes Comput. Sci. Eng., vol. 60, Springer, Berlin, 2008, pp. 147–154. MR 2436078, DOI 10.1007/978-3-540-75199-1_{1}3
- T. V. Kolev and P. S. Vassilevski, Parallel auxiliary space AMG for H(curl) problems, J. Comput. Math. (2009), 604–623.
- M. Mazza, C. Manni, A. Ratnani, S. Serra-Capizzano, and H. Speleers, Isogeometric analysis for 2D and 3D curl-div problems: Spectral symbols and fast iterative solvers, ArXiv:1805.10058, 2018.
- Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
- S. Serra, Multi-iterative methods, Comput. Math. Appl. 26 (1993), no. 4, 65–87. MR 1223858, DOI 10.1016/0898-1221(93)90035-T
- Stefano Serra Capizzano, An ergodic theorem for classes of preconditioned matrices, Linear Algebra Appl. 282 (1998), no. 1-3, 161–183. MR 1648324, DOI 10.1016/S0024-3795(98)80002-5
- Stefano Serra Capizzano, Test functions, growth conditions and Toeplitz matrices, Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, Vol. II (Potenza, 2000), 2002, pp. 791–795. MR 1975486
- S. Serra Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl. 366 (2003), 371–402. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000). MR 1987730, DOI 10.1016/S0024-3795(02)00504-9
- Stefano Serra-Capizzano, The GLT class as a generalized Fourier analysis and applications, Linear Algebra Appl. 419 (2006), no. 1, 180–233. MR 2263117, DOI 10.1016/j.laa.2006.04.012
- P. Tilli, Locally Toeplitz sequences: spectral properties and applications, Linear Algebra Appl. 278 (1998), no. 1-3, 91–120. MR 1637331, DOI 10.1016/S0024-3795(97)10079-9
- Paolo Tilli, A note on the spectral distribution of Toeplitz matrices, Linear and Multilinear Algebra 45 (1998), no. 2-3, 147–159. MR 1671591, DOI 10.1080/03081089808818584
Additional Information
- Mariarosa Mazza
- Affiliation: Max-Planck Institut für Plasmaphysik, Boltzmannstraße 2, 87548 Garching bei, München, Germany
- MR Author ID: 1079112
- Email: mariarosa.mazza@ipp.mpg.de
- Ahmed Ratnani
- Affiliation: Max-Planck Institut für Plasmaphysik, Boltzmannstraße 2, 87548 Garching bei, München, Germany; and Technische Universität München, Boltzmannstraße 3, 87548 Garching bei München, Germany
- Email: ahmed.ratnani@ipp.mpg.de
- Stefano Serra-Capizzano
- Affiliation: Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy; and Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
- MR Author ID: 332436
- Email: stefano.serrac@uninsubria.it
- Received by editor(s): April 14, 2017
- Received by editor(s) in revised form: November 8, 2017, and February 11, 2018
- Published electronically: July 6, 2018
- Additional Notes: The work of the first and third authors was partly supported by GNCS-INDAM (Italy).
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1155-1188
- MSC (2010): Primary 15A18, 15B05, 41A15, 15A69, 35Q61
- DOI: https://doi.org/10.1090/mcom/3366
- MathSciNet review: 3904142