Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation. (English) Zbl 1142.65448
Summary: In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of R. Hiptmair [SIAM J. Numer. Anal., 36, No. 1, 204–225 (1998; Zbl 0922.65081)] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
Nédélec vector finite elements; kernel of the rotor operator; multilevel iterative solvers; hierarchical preconditioners; domain decompositionCitations:
Zbl 0922.65081References:
| [1] | Hiptmair, R., Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal., 36, 1, 204-225 (1999) · Zbl 0922.65081 |
| [2] | Bossavit, A., Computational electromagnetism. Variational formulations, complementarity, edge elements, (Electromagnetism (1998), Academic Press Inc.: Academic Press Inc. San Diego, CA) · Zbl 0945.78001 |
| [3] | Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford University Press · Zbl 1024.78009 |
| [4] | Chen, Z.; Du, Q.; Zou, J., Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal., 37, 5, 1542-1570 (2000), Electronic · Zbl 0964.78017 |
| [5] | Nédélec, J.-C., Mixed finite elements in \(R^3\), Numer. Math., 35, 3, 315-341 (1980) · Zbl 0419.65069 |
| [6] | Nédélec, J.-C., A new family of mixed finite elements in \(R^3\), Numer. Math., 50, 1, 57-81 (1986) · Zbl 0625.65107 |
| [7] | Hiptmair, R., Finite elements in computational electromagnetism, Acta Numer., 11, 237-339 (2002) · Zbl 1123.78320 |
| [8] | Igarashi, H., On the property of the curl-curl matrix in finite element analysis with edge elements, IEEE Trans. Magn., 37, 5 (part 1), 3129-3132 (2001) |
| [9] | Igarashi, H.; Honma, T., On convergence of iccg applied to finite-element equation for quasi-static fields, IEEE Trans. Magn., 38, 2 (part 1), 565-568 (2002) |
| [10] | Perugia, I., A mixed formulation for 3D magnetostatic problems: Theoretical analysis and face-edge finite element approximation, Numer. Math., 84, 2, 305-326 (1999) · Zbl 1033.78010 |
| [11] | Beck, R.; Hiptmair, R., Multilevel solution of the time-harmonic Maxwell’s equations based on edge elements, Internat. J. Numer. Methods Engrg., 45, 7, 901-920 (1999) · Zbl 0930.35167 |
| [12] | Arnold, D. N.; Falk, R. S.; Winther, R., Multigrid in \(H(div)\) and \(H(curl)\), Numer. Math., 85, 2, 197-217 (2000) · Zbl 0974.65113 |
| [13] | Gopalakrishnan, J.; Pasciak, J. E.; Demkowicz, L. F., Analysis of a multigrid algorithm for time harmonic Maxwell equations, SIAM J. Numer. Anal., 42, 1, 90-108 (2004), Electronic · Zbl 1079.78025 |
| [14] | Elman, H. C.; Ernst, O. G.; O’Leary, D. P., A multigrid method enhanced by Krylov subspace iteration for discrete Helmhotz equations, SIAM J. Sci. Comput., 23, 4, 1291-1315 (2001), Electronic · Zbl 1004.65134 |
| [15] | Erlangga, Y. A.; Oosterlee, C. W.; Vuik, C., A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput., 27, 4, 1471-1492 (2006), Electronic · Zbl 1095.65109 |
| [16] | Bank, R. E., Hierarchical bases and the finite element method, Acta Numer., 5, 1-43 (1996) · Zbl 0865.65078 |
| [17] | Golub, G. H.; Overton, M. L., The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems, Numer. Math., 53, 571-593 (1988) · Zbl 0661.65033 |
| [18] | Giladi, E.; Golub, G. H.; Keller, J. B., Inner and outer iterations for the Chebyshev algorithm, SIAM J. Numer. Anal., 35, 1, 300-319 (1998) · Zbl 0911.65024 |
| [19] | Bai, Z.-Z.; Golub, G. H.; Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24, 3, 603-626 (2003) · Zbl 1036.65032 |
| [20] | van der Vorst, H. A., BiCGSTAB: A fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13, 2, 631-644 (1992) · Zbl 0761.65023 |
| [21] | van der Vorst, H. A., Iterative Krylov Methods for Large Linear Systems (2003), Cambridge University Press · Zbl 1023.65027 |
| [22] | Y. Saad, Iterative methods for sparse linear systems. Book out of print, 2000. Available at URL: http://www-users.cs.umn.edu/ saad/books.html; Y. Saad, Iterative methods for sparse linear systems. Book out of print, 2000. Available at URL: http://www-users.cs.umn.edu/ saad/books.html |
| [23] | van der Vorst, H. A.; Vuik, C., GMRESR: A family of nested GMRES methods, Numer. Lin. Alg. Appl., 1, 369-386 (1994) · Zbl 0839.65040 |
| [24] | Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469 (1993) · Zbl 0780.65022 |
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