Taking the cue from stabilized Galerkin methods for scalar advection problems, we adapt the technique to boundary value problems modeling the advection of magnetic fields. We provide rigorous a priori error estimates for both fully discontinuous piecewise polynomial trial functions and -conforming finite elements.
Keywords: magnetic advection, lie derivative, Friedrichs system, stabilized Galerkin method, upwinding, edge elements
@article{M2AN_2013__47_6_1713_0, author = {Heumann, Holger and Hiptmair, Ralf}, title = {Stabilized {Galerkin} methods for magnetic advection}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1713--1732}, publisher = {EDP-Sciences}, volume = {47}, number = {6}, year = {2013}, doi = {10.1051/m2an/2013085}, mrnumber = {3123373}, zbl = {1293.76088}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013085/} }
TY - JOUR AU - Heumann, Holger AU - Hiptmair, Ralf TI - Stabilized Galerkin methods for magnetic advection JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1713 EP - 1732 VL - 47 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013085/ DO - 10.1051/m2an/2013085 LA - en ID - M2AN_2013__47_6_1713_0 ER -
%0 Journal Article %A Heumann, Holger %A Hiptmair, Ralf %T Stabilized Galerkin methods for magnetic advection %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1713-1732 %V 47 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013085/ %R 10.1051/m2an/2013085 %G en %F M2AN_2013__47_6_1713_0
Heumann, Holger; Hiptmair, Ralf. Stabilized Galerkin methods for magnetic advection. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 6, pp. 1713-1732. doi : 10.1051/m2an/2013085. http://www.numdam.org/articles/10.1051/m2an/2013085/
[1] 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). | MR | Zbl
,[2] UFC: a Finite Element Code Generation Interface, Chapt. 16. Springer (2012). | MR
, and ,[3] Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1-155. | MR | Zbl
, and ,[4] 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. | MR | Zbl
,[5] On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 (2000) 121-140. | MR | Zbl
, and ,[6] Extrusion, contraction: Their discretization via Whitney forms. COMPEL 22 (2004) 470-480. | MR | Zbl
,[7] Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Mod. Meth. Appl. Sci. 14 (2004) 1893-1903. | MR | Zbl
, and ,[8] P. Castillo, B. Cockburn and I. Perugi and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676-1706. | MR | Zbl
[9] Advanced FI2TD algorithms for transient eddy current problems. COMPEL 20 (2001) 365-379. | Zbl
, and ,[10] Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753-778. | MR | Zbl
and ,[11] Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1999) 935-952 (electronic). | MR | Zbl
and ,[12] Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958) 333-418. | MR | Zbl
,[13] Stable upwind schemes for the magnetic induction equation. ESAIM: M2AN 43 (2009) 825-852. | Numdam | MR | Zbl
, , and ,[14] Upwind 3-d vector potential formulation for electromagnetic braking simulations. IEEE Trans. Magn. 46 (2010) 2835-2838.
, , and ,[15] Eulerian and Semi-Lagrangian Methods for Advection-Diffusion of Differential Forms, Ph.D. thesis, ETH Zürich, Switzerland (2011).
,[16] Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete Contin. Dyn. Syst. 29 (2011) 1471-1495. | MR | Zbl
and ,[17] Finite elements in computational electromagnetism. Acta Numer. 11 237-339 (2002). | MR | Zbl
,[18] Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485-518. | MR | Zbl
, , and ,[19] Mixed discontinuous Galerkin approximation of the Maxwell operator: the indefinite case. ESAIM: M2AN 39 (2005) 727-753. | Numdam | MR | Zbl
, , and ,[20] Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42 (2004) 434-459. | MR | Zbl
, and ,[21] Mixed discontinuous Galerkin approximation of the Maxwell operator: non-stabilized formulation. J. Sci. Comput. 22/23 (2005) 315-346. | MR | Zbl
, and ,[22] Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133-2163. | MR | Zbl
, and ,[23] 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. | MR | Zbl
and ,[24] 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. | MR | Zbl
, and ,[25] Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions. Ph.D. thesis, University of Oxford, England (2005).
,[26] 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. | MR | Zbl
,[27] A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399 (electronic). | MR | Zbl
and ,[28] 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. | MR | Zbl
and ,[29] DOLFIN: a C++/Python Finite Element Library, Chapt. 10. Springer (2012). | MR
, and ,[30] Discrete Lie advection of differential forms. Foundations of Computational Mathematics 11 (2011) 131-149. | MR | Zbl
, , , , , , and ,[31] Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | MR | Zbl
,[32] A new family of mixed finite elements in R3. Numer. Math. 50 (1986) 57-81. | MR | Zbl
,[33] A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28 (1991) 133-140. | MR | Zbl
,[34] Triangular mesh methods for the neutron transport equation. Tech. Rep. LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM (1973).
and ,[35] An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50 (1988) 75-88. | MR | Zbl
,[36] 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). | MR | Zbl
, and ,[37] How accurate is the streamline diffusion finite element method? Math. Comput. 66 (1997) 31-44. | MR | Zbl
,Cited by Sources: