Multigrid for Time-Harmonic Eddy Currents without Gauge
Part of the
Mathematics in Industry
book series (MATHINDUSTRY, volume 4)
The application of multigrid (MG) methods for the solution of electromagnetic problems has attracted attention in recent years (e.g. ). These problems are related to bilinear forms (curl·, curl·)L Ω 2 +α (·, ·)L Ω 2 , α ∈ ℝ, which require special smoothers as presented in [1,7]. This paper shows by numerical experiments that these ideas also work for the time-harmonic eddy currents, i.e. for complex bilinear forms (curl·, curl·)L Ω 2 +iα (·, ·)L Ω 2 . Furthermore, an approximate projection procedure is presented that allows the application of multigrid to an un-gauged electric formulation even if there are regions with zero conductivity. Numerical results are shown for the TEAM Workshop problem 7.
KeywordsBilinear Form Benchmark Problem Multigrid Method Weak Divergence Projection Step
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D. Arnold, R. Falk, and R. Winther. Multigrid in H
(div) and H
(curl). Numer. Math.
, 85(2):175–195, 2000.MathSciNetCrossRefGoogle Scholar
P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert, and C. Wieners. UG—a flexible software toolbox for solving partial differential equations. Computing and Visualization in Science
, 1:27–40, 1997.CrossRefzbMATHGoogle Scholar
R. Beck, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Residual based a-posteriori error estimators for eddy current computation. M2
AN, 34(1):159–182, 2000.MathSciNetzbMATHGoogle Scholar
R. Freund. Transpose-free quasi-minimal residual methods for non-Hermitian linear systems. IMA Volumes in Mathematics and its Applications
, 60(69):69–93, 1994.MathSciNetCrossRefGoogle Scholar
K. Fujiwara and T. Nakata. Results for benchmark problem 7. COMPEL
, 9(3):137–154, 1990.CrossRefGoogle Scholar
W. Hackbusch. Multi-grid Methods and Applications
. Springer-Verlag, Berlin, 1985.CrossRefzbMATHGoogle Scholar
R. Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal.
, 36(1):204–225, 1999.MathSciNetCrossRefGoogle Scholar
M. Kaltenbacher, S. Reitzinger, M. Schinnerl, J. Schöberl, and H. Landes. Multigrid methods for the computation of 3D electromagnetic field problems. COMPEL
, 20(2):581–594, 2001.CrossRefzbMATHGoogle Scholar
E. Martensen. Potentialtheorie
. Teubner, Stuttgart, 1968.zbMATHGoogle Scholar
H. van der Vorst. BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.
, 13(2):631–644, 1992.CrossRefzbMATHGoogle Scholar
C. Wieners. Local multigrid methods on hierarchical meshes. In Proceedings of the 17th GAMM-Seminar
, pages 1–8, Leipzig, 2001.Google Scholar
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