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Stabilized Galerkin methods for magneticadvection

Published online by Cambridge University Press:  07 October 2013

Holger Heumann  and
Ralf Hiptmair
Holger Heumann
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA.. [email protected]
Ralf Hiptmair
Affiliation:
SAM, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland.; [email protected]
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Abstract

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adaptthe technique to boundary value problems modeling the advection of magnetic fields. Weprovide rigorous a priori error estimates for both fully discontinuouspiecewise polynomial trial functions and -conforming finite elements.

Keywords

Magnetic advectionlie derivativeFriedrichs systemstabilized Galerkin methodupwindingedge elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1713 - 1732
Copyright
© EDP Sciences, SMAI 2013

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References

S. Agmon, Lectures on elliptic boundary value problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1965).
M.S. Alnæs, A. Logg and K.-A. Mardal, UFC: a Finite Element Code Generation Interface, Chapt. 16. Springer (2012).
Arnold, D.N., Falk, R.S. and Winther, R., Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. Google Scholar
Boffi, D., Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements. Comput. Meth. Appl. Mech. Eng. 196 (2007) 3672–3681. Google Scholar
Boffi, D., Brezzi, F. and Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 (2000) 121–140. Google Scholar
Bossavit, A., Extrusion, contraction: Their discretization via Whitney forms. COMPEL 22 (2004) 470–480. Google Scholar
Brezzi, F., Marini, L.D. and Süli, E., Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Mod. Meth. Appl. Sci. 14 (2004) 1893–1903. Google Scholar
Castillo, P., Cockburn, B. and Perugi, I. and Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676–1706. Google Scholar
Clemens, M., Wilke, M. and Weiland, T., Advanced FI2TD algorithms for transient eddy current problems. COMPEL 20 (2001) 365–379. Google Scholar
Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753–778. Google Scholar
Falk, R.S. and Richter, G.R., Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1999) 935–952 (electronic). Google Scholar
Friedrichs, K.O., Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333–418. Google Scholar
Fuchs, F.G., Karlsen, K.H., Mishra, S. and Risebro, N.H., Stable upwind schemes for the magnetic induction equation. ESAIM: M2AN 43 (2009) 825–852. Google Scholar
Henrotte, F., Heumann, H., Lange, E. and Haymeyer, K., Upwind 3-d vector potential formulation for electromagnetic braking simulations. IEEE Trans. Magn. 46 (2010) 2835–2838. Google Scholar
H. Heumann, Eulerian and Semi-Lagrangian Methods for Advection-Diffusion of Differential Forms, Ph.D. thesis, ETH Zürich, Switzerland (2011).
Heumann, H. and Hiptmair, R., Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete Contin. Dyn. Syst. 29 (2011) 1471–1495. Google Scholar
Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 237–339 (2002). Google Scholar
Houston, P., Perugia, I., Schneebeli, A. and Schötzau, D., Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485–518. Google Scholar
Houston, P., Perugia, I., Schneebeli, A. and Schötzau, D., Mixed discontinuous Galerkin approximation of the Maxwell operator: the indefinite case. ESAIM: M2AN 39 (2005) 727–753. Google Scholar
Houston, P., Perugia, I. and Schötzau, D., Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42 (2004) 434–459. Google Scholar
Houston, P., Perugia, I. and Schötzau, D., Mixed discontinuous Galerkin approximation of the Maxwell operator: non-stabilized formulation. J. Sci. Comput. 22/23 (2005) 315–346. Google Scholar
Houston, P., Schwab, C. and Süli, E., Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. Google Scholar
T.J.R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion. In Finite Element Methods for Convection Dominated Flows, vol. 34 of AMD, Amer. Soc. Mech. Engrg. New York (1979) 19–35.
Hughes, T.J.R., Franca, L.P. and Hulbert, G.M., A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173–189. Google Scholar
M. Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions. Ph.D. thesis, University of Oxford, England (2005).
M. Jensen, On the discontinuous Galerkin method for Friedrichs systems in graph spaces. In Large-scale scientific computing. Lecture Notes in Comput. Sci., vol. 3743. Springer, Berlin (2006) 94–101.
Karakashian, O.A. and Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399 (electronic). Google Scholar
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Proc. Sympos., Math. Res. Center, Univ. of Wisconsin-Madison vol. 33. Academic Press, New York (1974) 89–123.
A. Logg, G.N. Wells and J. Hake, DOLFIN: a C++/Python Finite Element Library, Chapt. 10. Springer (2012).
Mullen, P., McKenzie, A., Pavlov, D., Durant, L., Tong, Y., Kanso, E., Marsden, J. and Desbrun, M., Discrete Lie advection of differential forms. Foundations of Computational Mathematics 11 (2011) 131–149. Google Scholar
Nédélec, J.-C., Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. Google Scholar
Nédélec, J.-C., A new family of mixed finite elements in R3. Numer. Math. 50 (1986) 57–81. Google Scholar
Peterson, T.E., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28 (1991) 133–140. Google Scholar
W.H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation. Tech. Rep. LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM (1973).
Richter, G.R., An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50 (1988) 75–88. Google Scholar
H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Convection-diffusion-reaction and flow problems, volume 24 of Springer Series in Computational Mathematics. 2nd edition. Springer-Verlag, Berlin (2008).
Zhou, G., How accurate is the streamline diffusion finite element method? Math. Comput. 66 (1997) 31–44. Google Scholar