Abstract
This is an introduction to the spatial Galerkin discretization of Maxwell’s equations on bounded domains covering both modeling in the framework of exterior calculus, the construction of discrete differential forms, and a glimpse of a priori discretization error estimates. The presentation focuses on central ideas, skipping technical details for the sake of lucid presentation.
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References
M. Ainsworth, J. Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes. Int. J. Numer. Methods Eng. 58, 2103–2130 (2003)
M. Ainsworth, G. Andriamaro, O. Davydov, A Bernstein-Bézier basis for arbitrary order Raviart-Thomas finite elements. Report NI12079-AMM, Newton Institute, Cambridge (2012)
C. Amrouche, C. Bernardi, M. Dauge, V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
D. Arnold, A. Logg, Periodic table of the finite elements. SIAM News 47(9) (2014)
D. Arnold, R. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)
D. Arnold, D. Boffi, F. Bonizzoni, Finite element differential forms on curvilinear cubic meshes and their approximation properties. Preprint (2014). arXiv:1212.6559v4 [math.NA]
B. Auchmann, S. Kurz, de Rham currents in discrete electromagnetism. COMPEL 26, 743–757 (2007)
D. Baldomir, P. Hammond, Geometry of Electromagnetic Systems (Clarendon Press, Oxford, 1996)
M. Bergot, M. Duruflé, Approximation of \(H(\mathop{\mathrm{div}}\nolimits )\) with high-order optimal finite elements for pyramids, prisms and hexahedra. Commun. Comput. Phys. 14, 1372–1414 (2013)
M. Bergot, M. Duruflé, High-order optimal edge elements for pyramids, prisms and hexahedra. J. Comput. Phys. 232, 189–213 (2013)
D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44 (Springer, Heidelberg, 2013)
A. Bossavit, Computational Electromagnetism: Variational Formulation, Complementarity, Edge Elements. Electromagnetism Series, vol. 2 (Academic, San Diego, 1998)
A. Bossavit, On the geometry of electromagnetism I: affine space. J. Jpn. Soc. Appl. Electromagn. Mech. 6, 17–28 (1998)
A. Bossavit, On the geometry of electromagnetism II: geometrical objects. J. Jpn. Soc. Appl. Electromagn. Mech. 6, 114–123 (1998)
A. Bossavit, On the geometry of electromagnetism III: integration, Stokes’, Faraday’s law. J. Jpn. Soc. Appl. Electromagn. Mech. 6, 233–240 (1998)
A. Bossavit, On the geometry of electromagnetism IV: “Maxwell’s house”. J. Jpn. Soc. Appl. Electromagn. Mech. 6, 318–326 (1998)
A. Bossavit, On the Lorenz gauge. COMPEL 18, 323–336 (1999)
A. Bossavit, Applied Differential Geometry: A Compendium. Unpublished Lecture Notes (2002)
A. Bossavit, Discretization of electromagnetic problems: the “generalized finite differences”, in Numerical Methods in Electromagnetics, ed. by W. Schilders, W. ter Maten. Handbook of Numerical Analysis, vol. XIII (Elsevier, Amsterdam, 2005), pp. 443–522
A. Buffa, R. Hiptmair, Galerkin boundary element methods for electromagnetic scattering, in Topics in Computational Wave Propagation: Direct and Inverse Problems, ed. by M. Ainsworth, P. Davis, D. Duncan, P. Martin, B. Rynne. Lecture Notes in Computational Science and Engineering, vol. 31 (Springer, Berlin, 2003), pp. 83–124
W. Burke, Applied Differential Geometry (Cambridge University Press, Cambridge, 1985)
H. Cartan, Formes Différentielles (Hermann, Paris, 1967)
S.H. Christiansen, R. Winther, Smoothed projections in finite element exterior calculus. Math. Comput. 77, 813–829 (2008)
P. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4 (North-Holland, Amsterdam, 1978)
P. Ciarlet Jr., J. Zou, Fully discrete finite element approaches for time-dependent Maxwell equations. Numer. Math. 82, 193–219 (1999)
D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 2nd edn. (Springer, Heidelberg, 2013)
M. Costabel, A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265, 297–320 (2010)
L. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)
R.S. Falk, R. Winther, Local bounded cochain projection. Preprint (2012). arXiv:1211.5893
R.S. Falk, R. Winther, Double complexes and local cochain projections. Numer. Methods Partial Differ. Equ. 31, 541–551 (2015)
T. Frankel, The Geometry of Physics, 2nd edn. (Cambridge University Press, Cambridge, 2004)
R. Hiptmair, Canonical construction of finite elements. Math. Comput. 68, 1325–1346 (1999)
R. Hiptmair, Discrete Hodge operators. Numer. Math. 90, 265–289 (2001)
R. Hiptmair, Higher order Whitney forms, in Geometric Methods for Computational Electromagnetics, ed. by F. Teixeira. Progress in Electromagnetics Research, vol. 32 (EMW Publishing, Cambridge, 2001), pp. 271–299
R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
R. Hiptmair, C. Schwab, Natural boundary element methods for the electric field integral equation on polyhedra. SIAM J. Numer. Anal. 40, 66–86 (2002)
S. Kurz, B. Auchmann, Differential forms and boundary integral equations for Maxwell-type problems, in Fast Boundary Element Methods in Engineering and Industrial Applications, ed. by U. Langer, M. Schanz, O. Steinbach, W.L. Wendland. Lecture Notes in Applied and Computational Mechanics, vol. 63 (Springer, Berlin/Heidelberg, 2012), pp. 1–62
S. Lang, Differential and Riemannian Manifolds. Graduate Texts in Mathematics, vol. 160 (Springer, New York, 1995)
J.M. Lee, Manifolds and Differential Geometry. Graduate Studies in Mathematics, vol. 107 (American Mathematical Society, Providence, 2009)
P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon Press, Oxford, 2003)
N. Nigam, J. Phillips, High-order conforming finite elements on pyramids. IMA J. Numer. Anal. 32, 448–483 (2012)
P. Oswald, Multilevel Finite Element Approximation. Teubner Skripten zur Numerik (B.G. Teubner, Stuttgart, 1994)
F. Rapetti, High-order edge elements on simplicial meshes. M2AN Math. Model. Numer. Anal. 41(6), 1001–1020 (2007)
M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations. Texts in Applied Mathematics, vol. 13, 2nd edn. (Springer, New York, 2004)
S. Sauter, C. Schwab, Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39 (Springer, Heidelberg, 2010)
J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Texas A&M University, College Station (2001)
T. Tarhasaari, L. Kettunen, A. Bossavit, Some realizations of a discrete Hodge: a reinterpretation of finite element techniques. IEEE Trans. Magn. 35, 1494–1497 (1999)
S. Zaglmayr, High order finite element methods for electromagnetic field computation, Ph.D. thesis, Johannes Kepler Universität Linz, 2006
J. Zhao, Analysis of finite element approximation for time-dependent Maxwell problems. Math. Comput. 73, 1089–1105 (2004)
L. Zhong, S. Shu, G. Wittum, J. Xu, Optimal error estimates for Nedelec edge elements for time-harmonic Maxwell’s equations. J. Comput. Math. 27, 563–572 (2009)
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Hiptmair, R. (2015). Maxwell’s Equations: Continuous and Discrete. In: Bermúdez de Castro, A., Valli, A. (eds) Computational Electromagnetism. Lecture Notes in Mathematics(), vol 2148. Springer, Cham. https://doi.org/10.1007/978-3-319-19306-9_1
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