On eigenmode approximation for Dirac equations: Differential forms and fractional Sobolev spaces

Christiansen, Snorre

doi:10.1090/mcom/3233

On eigenmode approximation for Dirac equations: Differential forms and fractional Sobolev spaces
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Math. Comp. 87 (2018), 547-580 Request permission

Abstract:

We comment on the discretization of the Dirac equation using finite element spaces of differential forms. In order to treat perturbations by low order terms, such as those arising from electromagnetic fields, we develop some abstract discretization theory and provide estimates in fractional order Sobolev spaces for finite element systems. Eigenmode convergence is proved, as well as optimal convergence orders, assuming a flat background metric on a periodic domain.

References

Douglas N. Arnold and Gerard Awanou, Finite element differential forms on cubical meshes, Math. Comp. 83 (2014), no. 288, 1551–1570. MR 3194121, DOI 10.1090/S0025-5718-2013-02783-4
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297. MR 962210, DOI 10.1090/S0025-5718-1989-0962210-8
I. Babuška and J. E. Osborn, Corrigendum: “Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems” [Math. Comp. 52 (1989), no. 186, 275–297; MR0962210 (89k:65132)], Math. Comp. 63 (1994), no. 208, 831–832. MR 1260125, DOI 10.1090/S0025-5718-1994-1260125-3
Claudio Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni differenziali a derivate parziali, Ann. Mat. Pura Appl. (4) 73 (1966), 233–251 (Italian). MR 205108, DOI 10.1007/BF02415089
Uday Banerjee and John E. Osborn, Estimation of the effect of numerical integration in finite element eigenvalue approximation, Numer. Math. 56 (1990), no. 8, 735–762. MR 1035176, DOI 10.1007/BF01405286
P. Becher and H. Joos, The Dirac-Kähler equation and fermions on the lattice, Z. Phys. C 15 (1982), no. 4, 343–365. MR 688640, DOI 10.1007/BF01614426
Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. MR 2652780, DOI 10.1017/S0962492910000012
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Annalisa Buffa, Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations, SIAM J. Numer. Anal. 43 (2005), no. 1, 1–18. MR 2177953, DOI 10.1137/S003614290342385X
A. Buffa and S. H. Christiansen, The electric field integral equation on Lipschitz screens: definitions and numerical approximation, Numer. Math. 94 (2003), no. 2, 229–267. MR 1974555, DOI 10.1007/s00211-002-0422-0
Françoise Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. MR 716134
S. H. Christiansen, Résolution des équations intégrales pour la diffraction d’ondes acoustiques et électromagnétiques - Stabilisation d’algorithmes itératifs et aspects de l’analyse numérique, PhD thesis, École Polytechnique, HAL Id: tel-00004520 (2002).
Snorre H. Christiansen, Discrete Fredholm properties and convergence estimates for the electric field integral equation, Math. Comp. 73 (2004), no. 245, 143–167. MR 2034114, DOI 10.1090/S0025-5718-03-01581-3
Snorre H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension, Numer. Math. 107 (2007), no. 1, 87–106. MR 2317829, DOI 10.1007/s00211-007-0081-2
Snorre H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739–757. MR 2413036, DOI 10.1142/S021820250800284X
Snorre H. Christiansen, Foundations of finite element methods for wave equations of Maxwell type, Applied wave mathematics, Springer, Berlin, 2009, pp. 335–393. MR 2553283, DOI 10.1007/978-3-642-00585-5_{1}7
Snorre H. Christiansen, On the div-curl lemma in a Galerkin setting, Calcolo 46 (2009), no. 3, 211–220. MR 2533750, DOI 10.1007/s10092-009-0008-7
Snorre H. Christiansen, Éléments finis mixtes minimaux sur les polyèdres, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 217–221 (French, with English and French summaries). MR 2600081, DOI 10.1016/j.crma.2010.01.017
Snorre H. Christiansen, On the linearization of Regge calculus, Numer. Math. 119 (2011), no. 4, 613–640. MR 2854122, DOI 10.1007/s00211-011-0394-z
Snorre H. Christiansen and Andrew Gillette, Constructions of some minimal finite element systems, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 833–850. MR 3507275, DOI 10.1051/m2an/2015089
Snorre H. Christiansen, Hans Z. Munthe-Kaas, and Brynjulf Owren, Topics in structure-preserving discretization, Acta Numer. 20 (2011), 1–119. MR 2805152, DOI 10.1017/S096249291100002X
Snorre H. Christiansen and Francesca Rapetti, On high order finite element spaces of differential forms, Math. Comp. 85 (2016), no. 298, 517–548. MR 3434870, DOI 10.1090/mcom/2995
Snorre H. Christiansen and Claire Scheid, Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 4, 739–760. MR 2804657, DOI 10.1051/m2an/2010100
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, DOI 10.1090/S0025-5718-07-02081-9
Snorre H. Christiansen and Ragnar Winther, On variational eigenvalue approximation of semidefinite operators, IMA J. Numer. Anal. 33 (2013), no. 1, 164–189. MR 3020954, DOI 10.1093/imanum/drs002
Bernardo Cockburn and Weifeng Qiu, Commuting diagrams for the TNT elements on cubes, Math. Comp. 83 (2014), no. 286, 603–633. MR 3143686, DOI 10.1090/S0025-5718-2013-02729-9
J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1–52 (1977). MR 488179
Jerome Droniou, Robert Eymard, Thierry Gallouet, and Raphaele Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations, Math. Models Methods Appl. Sci. 23 (2013), no. 13, 2395–2432. MR 3109434, DOI 10.1142/S0218202513500358
Norbert Heuer, On the equivalence of fractional-order Sobolev semi-norms, J. Math. Anal. Appl. 417 (2014), no. 2, 505–518. MR 3194499, DOI 10.1016/j.jmaa.2014.03.047
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
Ralf Hiptmair, Jingzhi Li, and Jun Zou, Real interpolation of spaces of differential forms, Math. Z. 270 (2012), no. 1-2, 395–402. MR 2875840, DOI 10.1007/s00209-010-0803-5
Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
Mathieu Lewin and Éric Séré, Spectral pollution and how to avoid it (with applications to Dirac and periodic Schrödinger operators), Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, 864–900. MR 2640293, DOI 10.1112/plms/pdp046
P. Leopardi and A. Stern, The abstract Hodge-Dirac operator and its stable discretization, arXiv:1401.1576 (2014).
T. Regge, General relativity without coordinates, Nuovo Cimento (10) 19 (1961), 558–571 (English, with Italian summary). MR 127372
Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, DOI 10.1090/S0025-5718-07-02030-3
Luc Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin; UMI, Bologna, 2007. MR 2328004
Michael E. Taylor, Partial differential equations. II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. Qualitative studies of linear equations. MR 1395149, DOI 10.1007/978-1-4757-4187-2