On the linearization of Regge calculus. (English) Zbl 1269.83022
Summary: We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the curl t curl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasticity complex. We insert Regge metrics in a discrete version of this complex, equipped with densely defined and commuting interpolators. We show that the eigenpairs of the curl t curl operator, approximated using the quadratic part of the Regge action on Regge metrics, converge to their continuous counterparts, interpreting the computation as a non-conforming finite element method.
MSC:
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |
| 83C30 | Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C40 | Gravitational energy and conservation laws; groups of motions |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 83-08 | Computational methods for problems pertaining to relativity and gravitational theory |
| 35L15 | Initial value problems for second-order hyperbolic equations |
References:
| [1] | Alcubierre, M.: Introduction to 3 + 1 numerical relativity. In: International Series of Monographs on Physics, vol. 140. Oxford University Press, Oxford (2008) · Zbl 1140.83002 |
| [2] | Arnold D.N., Awanou G., Winther R.: Finite elements for symmetric tensors in three dimensions. Math. Comput. 77(263), 1229–1251 (2008) · Zbl 1285.74013 · doi:10.1090/S0025-5718-08-02071-1 |
| [3] | Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods. II. The elasticity complex. In: Compatible Spatial Discretizations. IMA Vol. Math. Appl., vol. 142, pp. 47–67. Springer, New York (2006) · Zbl 1119.65399 |
| [4] | Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006) · Zbl 1185.65204 · doi:10.1017/S0962492906210018 |
| [5] | Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2), 281–354 (2010) · Zbl 1207.65134 · doi:10.1090/S0273-0979-10-01278-4 |
| [6] | Babuška I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1971) · Zbl 0214.42001 · doi:10.1007/BF02165003 |
| [7] | Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, Vol. II, Handb. Numer. Anal., II, pp. 641–787. North-Holland, Amsterdam (1991) · Zbl 0875.65087 |
| [8] | Bahr B., Dittrich B.: Regge calculus from a new angle. New J. Phys. 12(3), 033010 (2010) · Zbl 1360.83016 · doi:10.1088/1367-2630/12/3/033010 |
| [9] | Barrett J.W., Galassi M., Miller W.A., Sorkin R.D., Tuckey P.A., Williams R.M.: Parallelizable implicit evolution scheme for Regge calculus. Int. J. Theoret. Phys. 36(4), 815–839 (1997) · Zbl 0871.53067 · doi:10.1007/BF02435787 |
| [10] | Barrett J.W., Williams R.M.: The convergence of lattice solutions of linearised Regge calculus. Class. Quantum Gravity 5(12), 1543–1556 (1988) · doi:10.1088/0264-9381/5/12/007 |
| [11] | Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity, Solving Einstein’s Equations on the Computer. Cambridge University Press (2010) · Zbl 1198.83001 |
| [12] | Boffi D., Brezzi F., Gastaldi L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69(229), 121–140 (2000) · Zbl 0938.65126 |
| [13] | Boffi D., Fernandes P., Gastaldi L., Perugia I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36(4), 1264–1290 (1999) · Zbl 1025.78014 · doi:10.1137/S003614299731853X |
| [14] | Bossavit, A.: Mixed finite elements and the complex of Whitney forms. In: The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987), pp. 137–144. Academic Press, London (1988) |
| [15] | Brewin, L.C.: Fast algorithms for computing defects and their derivatives in the Regge calculus. Preprint. arxiv:1011.1885:1–38 (2010) |
| [16] | Brewin L.C., Gentle A.P.: On the convergence of Regge calculus to general relativity. Class. Quantum Gravity 18(3), 517–525 (2001) · Zbl 0971.83018 · doi:10.1088/0264-9381/18/3/311 |
| [17] | Brezzi F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R2), 129–151 (1974) |
| [18] | Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) · Zbl 0788.73002 |
| [19] | Buffa A., Perugia I.: Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44(5), 2198–2226 (2006) · Zbl 1344.65110 · doi:10.1137/050636887 |
| [20] | Caorsi S., Fernandes P., Raffetto M.: Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. Math. Model. Numer. Anal. 35(2), 331–354 (2001) · Zbl 0993.78016 · doi:10.1051/m2an:2001118 |
| [21] | Chatelin, F.: 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 · Zbl 0517.65036 |
| [22] | Cheeger J., Müller W., Schrader R.: On the curvature of piecewise flat spaces. Commun. Math. Phys. 92(3), 405–454 (1984) · Zbl 0559.53028 · doi:10.1007/BF01210729 |
| [23] | Christiansen S.H.: A characterization of second-order differential operators on finite element spaces. Math. Models Methods Appl. Sci. 14(12), 1881–1892 (2004) · Zbl 1076.83008 · doi:10.1142/S0218202504003854 |
| [24] | Christiansen S.H.: Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numer. Math. 107(1), 87–106 (2007) · Zbl 1127.65085 · doi:10.1007/s00211-007-0081-2 |
| [25] | Christiansen S.H., Halvorsen T.G.: Convergence of lattice gauge theory for Maxwell’s equations. BIT 49(4), 645–667 (2009) · Zbl 1184.78079 · doi:10.1007/s10543-009-0242-z |
| [26] | Christiansen, S.H., Winther, R.: On variational eigenvalue approximation of semidefinite operators. Preprint. arXiv:1005.2059v4 (2010) · Zbl 1269.65121 |
| [27] | Ciarlet, P.G., Lions, J.-L. (eds.): Handbook of numerical analysis. Handbook of Numerical Analysis, vol. II. Finite Element Methods, Part 1. North-Holland, Amsterdam (1991) · Zbl 0712.65091 |
| [28] | Ciarlet P.G., Ciarlet P. Jr: Direct computation of stresses in planar linearized elasticity. Math. Models Methods Appl. Sci. 19(7), 1043–1064 (2009) · Zbl 1169.49035 · doi:10.1142/S0218202509003711 |
| [29] | Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. Preprint. arXiv:math/0508341v2:1–53 (2005) |
| [30] | Frauendiener J.: Discrete differential forms in general relativity. Class. Quantum Gravity 23(16), S369–S385 (2006) · Zbl 1191.83018 · doi:10.1088/0264-9381/23/16/S05 |
| [31] | Friedberg R., Lee T.D.: Derivation of Regge’s action from Einstein’s theory of general relativity. Nuclear Phys. B 242(1), 145–166 (1984) · doi:10.1016/0550-3213(84)90137-8 |
| [32] | Gentle A.P.: Regge calculus: a unique tool for numerical relativity. Gen. Relativ. Gravit. 34(10), 1701–1718 (2002) · Zbl 1019.83006 · doi:10.1023/A:1020128425143 |
| [33] | Gentle A.P., Miller W.A.: A fully (3 + 1)-dimensional Regge calculus model of the Kasner cosmology. Class. Quantum Gravity 15(2), 389–405 (1998) · Zbl 0909.53058 · doi:10.1088/0264-9381/15/2/013 |
| [34] | Geymonat G., Krasucki F.: Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Commun. Pure Appl. Anal. 8(1), 295–309 (2009) · Zbl 1156.58004 |
| [35] | Grandclément, P., Novak, J.: Spectral methods for numerical relativity. Living Rev. Relativ. 12(1) (2009) · Zbl 1166.83004 |
| [36] | Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006). Structure-preserving algorithms for ordinary differential equations · Zbl 1094.65125 |
| [37] | Hauret P., Kuhl E., Ortiz M.: Diamond elements: a finite element/discrete-mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity. Int. J. Numer. Methods Eng. 72(3), 253–294 (2007) · Zbl 1194.74406 · doi:10.1002/nme.1992 |
| [38] | Hiptmair R.: Canonical construction of finite elements. Math. Comput. 68(228), 1325–1346 (1999) · Zbl 0938.65132 · doi:10.1090/S0025-5718-99-01166-7 |
| [39] | Holst, M., Stern, A.: Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Preprint. arXiv:1005.4455v1 (2010) |
| [40] | Kikuchi F.: On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(3), 479–490 (1989) · Zbl 0698.65067 |
| [41] | Marsden J.E., West M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001) · Zbl 1123.37327 · doi:10.1017/S096249290100006X |
| [42] | McDonald J.R., Miller W.A.: A geometric construction of the Riemann scalar curvature in Regge calculus. Class. Quantum Gravity 25(19), 195017 (2008) · Zbl 1151.83317 · doi:10.1088/0264-9381/25/19/195017 |
| [43] | Misner C.W., Thorne K.S., Wheeler J.A.: Gravitation. W. H. Freeman and Co., San Francisco (1973) |
| [44] | Nédélec J.-C.: Mixed finite elements in R 3. Numer. Math. 35(3), 315–341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415 |
| [45] | Peuker, F.: Regge calculus and the finite element method. PhD thesis, Friedrich Schiller Universität Jena (2009) |
| [46] | Porter J.: A new approach to the Regge calculus. I. Formalism. Class. Quantum Gravity 4(2), 375–389 (1987) · Zbl 0609.53034 · doi:10.1088/0264-9381/4/2/017 |
| [47] | Porter J.: A new approach to the Regge calculus. II. Application to spherically symmetric vacuum spacetimes. Class. Quantum Gravity 4(2), 391–410 (1987) · Zbl 0609.53035 · doi:10.1088/0264-9381/4/2/018 |
| [48] | Pretorius F.: Simulation of binary black hole spacetimes with a harmonic evolution scheme. Class. Quantum Gravity 23(16), S529–S552 (2006) · Zbl 1191.83026 · doi:10.1088/0264-9381/23/16/S13 |
| [49] | Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292–315. Lecture Notes in Math., vol. 606. Springer, Berlin (1977) |
| [50] | Regge T.: General relativity without coordinates. Nuovo Cimento (10) 19, 558–571 (1961) · doi:10.1007/BF02733251 |
| [51] | Regge T., Williams R.M.: Discrete structures in gravity. J. Math. Phys. 41(6), 3964–3984 (2000) · Zbl 0984.83016 · doi:10.1063/1.533333 |
| [52] | Reula O.A.: Strongly hyperbolic systems in general relativity. J. Hyperbolic Differ. Equ. 1(2), 251–269 (2004) · Zbl 1074.58014 · doi:10.1142/S0219891604000111 |
| [53] | Richter R., Frauendiener J.: Discrete differential forms for (1 + 1)-dimensional cosmological space-times. SIAM J. Sci. Comput. 32(3), 1140–1158 (2010) · Zbl 1215.83014 · doi:10.1137/080734583 |
| [54] | Richter, R., Lubich, C.: Free and constrained symplectic integrators for numerical general relativity. Class. Quantum Gravity 25(22), 225018, 21 (2008) · Zbl 1152.83309 |
| [55] | Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Handbook of Numerical Analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991) · Zbl 0875.65090 |
| [56] | Rothe, H.J.: An introduction: Lattice gauge theories. In: World Scientific Lecture Notes in Physics, vol. 74, 3rd edn. World Scientific, Hackensack (2005) · Zbl 1087.81004 |
| [57] | Sathyaprakash, B.S., Schutz, B.F.: Physics, astrophysics and cosmology with gravitational waves. Living Rev. Relativ. 12(2) (2009) · Zbl 1166.85002 |
| [58] | Schöberl J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77(262), 633–649 (2008) · Zbl 1136.78016 |
| [59] | Schöberl, J., Sinwel, A.: Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. Ricam Report 10 (2007) |
| [60] | Stern, A.: Geometric discretization of Lagrangian mechanics and field theories. PhD thesis, California Institute of Technology (2009) |
| [61] | Hooft G. t’: Introduction to General Relativity. Rindon Press, Princeton (2001) · Zbl 1008.83001 |
| [62] | Wald R.M.: General Relativity. University of Chicago Press, Chicago (1984) |
| [63] | Weil A.: Sur les théorèmes de de Rham. Comment. Math. Helv. 26, 119–145 (1952) · Zbl 0047.16702 · doi:10.1007/BF02564296 |
| [64] | Whitney H.: Geometric Integration Theory. Princeton University Press, Princeton (1957) · Zbl 0083.28204 |
| [65] | Zumbusch, G.: Finite element, discontinuous Galerkin, and finite difference evolution schemes in spacetime. Class. Quantum Gravity 26(17), 175011, 15 (2009) · Zbl 1176.83005 |
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