Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

Jay Gopalakrishnan; Michael Neunteufel; Joachim Schöberl; Max Wardetzky

doi:10.5802/smai-jcm.98

Jay Gopalakrishnan ¹; Michael Neunteufel ²; Joachim Schöberl ²; Max Wardetzky ³

¹ Portland State University, PO Box 751, Portland OR 97207, USA
² Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
³ Institute of Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 151-195.

Abstract

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

Published online: 2023-09-11

DOI: 10.5802/smai-jcm.98

Classification: 65N30, 53A70, 83C27
Keywords: Gauss curvature, Regge calculus, finite element method, differential geometry

Author's affiliations:

Jay Gopalakrishnan ¹; Michael Neunteufel ²; Joachim Schöberl ²; Max Wardetzky ³

¹ Portland State University, PO Box 751, Portland OR 97207, USA
² Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
³ Institute of Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Analysis of curvature approximations via covariant curl and incompatibility for {Regge} metrics},
     journal = {The SMAI Journal of computational mathematics},
     pages = {151--195},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Jay Gopalakrishnan; Michael Neunteufel; Joachim Schöberl; Max Wardetzky. Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 151-195. doi : 10.5802/smai-jcm.98. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.98/

References
Cited by

[1] Samuel Amstutz; Nicolas Van Goethem Analysis of the Incompatibility Operator and Application in Intrinsic Elasticity with Dislocations, SIAM J. Math. Anal., Volume 48 (2016) no. 1, pp. 320-348 | DOI | MR | Zbl

[2] Samuel Amstutz; Nicolas Van Goethem Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity, Discrete Contin. Dyn. Syst., Ser. B, Volume 25 (2020) no. 10, pp. 3769-3805 | MR | Zbl

[3] Douglas N. Arnold Finite element exterior calculus, Society for Industrial and Applied Mathematics, 2018

[4] Douglas N. Arnold; Gerard Awanou; Ragnar Winther Finite elements for symmetric tensors in three dimensions, Math. Comput., Volume 77 (2008) no. 263, pp. 1229-1251 | DOI | MR | Zbl

[5] Douglas N. Arnold; Richard S. Falk; Ragnar Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155 | DOI | MR | Zbl

[6] Douglas N. Arnold; Richard S. Falk; Ragnar Winther Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc., Volume 47 (2010) no. 2, pp. 281-354 | DOI | MR | Zbl

[7] Douglas N. Arnold; Kaibo Hu Complexes from Complexes, Found. Comput. Math., Volume 21 (2021) no. 6, pp. 1739-1774 | DOI | MR | Zbl

[8] Douglas N. Arnold; Shawn W. Walker The Hellan–Herrmann–Johnson Method with Curved Elements, SIAM J. Numer. Anal., Volume 58 (2020) no. 5, pp. 2829-2855 | DOI | MR | Zbl

[9] John W. Barrett; Daniele Oriti; Ruth M. Williams Tullio Regge’s legacy: Regge calculus and discrete gravity (2018) | arXiv

[10] Yakov Berchenko-Kogan; Evan S. Gawlik Finite element approximation of the Levi-Civita connection and its curvature in two dimensions, Found. Comput. Math. (2022) | DOI

[11] Daniele Boffi; Franco Brezzi; Michel Fortin Mixed finite element methods and applications, 44, Springer, 2013 | DOI

[12] Vincent Borrelli; Frederic Cazals; Jean-Marie Morvan On the angular defect of triangulations and the pointwise approximation of curvatures, Comput. Aided Geom. Des., Volume 20 (2003) no. 6, pp. 319-341 | DOI | MR | Zbl

[13] Franco Brezzi; Jim Douglas; Luisa D. Marini Two families of mixed finite elements for second order elliptic problems, Numer. Math., Volume 47 (1985) no. 2, pp. 217-235 | DOI | MR | Zbl

[14] Manfredo P. do Carmo Riemannian Geometry, Birkhäuser, 1992 | DOI

[15] Jeff Cheeger; Werner Müller; Robert Schrader On the curvature of piecewise flat spaces, Commun. Math. Phys., Volume 92 (1984) no. 3, pp. 405-454 | DOI | MR | Zbl

[16] Snorre H. Christiansen A characterization of second-order differential operators on finite element spaces, Math. Models Methods Appl. Sci., Volume 14 (2004) no. 12, pp. 1881-1892 | DOI | MR | Zbl

[17] Snorre H. Christiansen On the linearization of Regge calculus, Numer. Math., Volume 119 (2011) no. 4, pp. 613-640 | DOI | MR | Zbl

[18] Snorre H. Christiansen Exact formulas for the approximation of connections and curvature (2015) | arXiv

[19] Snorre H. Christiansen; Jay Gopalakrishnan; Johnny Guzmán; Kaibo Hu A discrete elasticity complex on three-dimensional Alfeld splits (2020) | arXiv

[20] C. J. S. Clarke; Tevian Dray Junction conditions for null hypersurfaces, Class. Quant. Grav., Volume 4 (1987) no. 2, pp. 265-275 | DOI | MR | Zbl

[21] Maria I. Comodi The Hellan–Herrmann–Johnson Method: Some New Error Estimates and Postprocessing, Math. Comput., Volume 52 (1989) no. 185, pp. 17-29 | DOI | MR | Zbl

[22] Michel Crouzeix; Vidar Thomée The Stability in $L_{p}$ and $W_{p}^{1}$ of the $L_{2}$ -projection onto Finite Element Function Spaces, Math. Comput., Volume 48 (1987) no. 178, pp. 521-532

[23] Arthur E. Fischer; Jerrold E. Marsden Deformations of the scalar curvature, Duke Math. J. (1975) | MR | Zbl

[24] Hans Fritz Isoparametric finite element approximation of Ricci curvature, IMA J. Numer. Anal., Volume 33 (2013) no. 4, pp. 1265-1290 | DOI | MR | Zbl

[25] Hans Fritz Numerical Ricci–DeTurck flow, Numer. Math., Volume 131 (2015) no. 2, pp. 241-271 | DOI | MR | Zbl

[26] Evan S. Gawlik Finite Element Methods for Geometric Evolution Equations, Geometric Science of Information (Frank Nielsen; Frédéric Barbaresco, eds.), Springer (2019), pp. 532-540 | DOI | MR | Zbl

[27] Evan S. Gawlik High-Order Approximation of Gaussian Curvature with Regge Finite Elements, SIAM J. Numer. Anal., Volume 58 (2020) no. 3, pp. 1801-1821 | DOI | MR | Zbl

[28] Patrice Hauret; Frédéric Hecht A Discrete Differential Sequence for Elasticity Based upon Continuous Displacements, SIAM J. Sci. Comput., Volume 35 (2013) no. 1, p. B291-B314 | DOI | MR | Zbl

[29] W. Israel Singular hypersurfaces and thin shells in general relativity, Il Nuovo Cimento B Series, Volume 44 (1966) no. 1, pp. 1-14 | DOI

[30] Nikolaĭ N. Kosovskiĭ Gluing of Riemannian manifolds of curvature $\geq κ$ , Algebra Anal., Volume 14 (2002) no. 3, pp. 140-157 | MR

[31] Nikolaĭ N. Kosovskiĭ Gluing with branching of Riemannian manifolds of curvature $\leq κ$ , Algebra Anal., Volume 16 (2004) no. 4, pp. 132-145 | MR

[32] John M. Lee Riemannian manifolds: an introduction to curvature, Springer, 1997 | DOI

[33] John M. Lee Introduction to Smooth Manifolds, Springer, 2012

[34] Lizao Li Regge Finite Elements with Applications in Solid Mechanics and Relativity, Ph. D. Thesis, University of Minnesota (2018) http://hdl.handle.net/11299/199048

[35] Dan Liu; Guoliang Xu Angle deficit approximation of Gaussian curvature and its convergence over quadrilateral meshes, Comput.-Aided Des., Volume 39 (2007) no. 6, pp. 506-517 https://www.sciencedirect.com/science/article/pii/s0010448507000267 | DOI

[36] Michael Neunteufel Mixed Finite Element Methods for Nonlinear Continuum Mechanics and Shells, Ph. D. Thesis, TU Wien (2021) | DOI

[37] Michael Neunteufel; Joachim Schöberl Avoiding Membrane Locking with Regge Interpolation, Comput. Methods Appl. Mech. Eng., Volume 373 (2021), 113524 | DOI | MR | Zbl

[38] Peter Petersen Riemannian Geometry, Springer, 2016 | DOI

[39] Pierre-Arnaud Raviart; Jean-Marie Thomas A mixed finite element method for 2-nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Volume 66, Springer, 1977, pp. 292-315 | DOI

[40] Tullio Regge General relativity without coordinates, Il Nuovo Cimento (1955-1965), Volume 19 (1961) no. 3, pp. 558-571 | DOI | MR

[41] Tullio Regge; Ruth M. Williams Discrete structures in gravity, J. Math. Phys., Volume 41 (2000) no. 6, pp. 3964-3984 | DOI | MR | Zbl

[42] Joachim Schöberl NETGEN An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., Volume 1 (1997) no. 1, pp. 41-52 | DOI | Zbl

[43] Joachim Schöberl C++ 11 implementation of finite elements in NGSolve, 2014 https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/ngs-cpp11.pdf (Institute for Analysis and Scientific Computing, Vienna University of Technology)

[44] Rafael Sorkin Time-evolution problem in Regge calculus, Phys. Rev. D, Volume 12 (1975), pp. 385-396 | DOI | MR

[45] Robert S. Strichartz Defining Curvature as a Measure via Gauss–Bonnet on Certain Singular Surfaces, J. Geom. Anal., Volume 30 (2020) no. 1, pp. 153-160 | DOI | MR | Zbl

[46] John M. Sullivan Curvatures of Smooth and Discrete Surfaces, Birkhäuser (2008), pp. 175-188 | DOI | MR

[47] Loring W. Tu Differential Geometry: Connections, Curvature, Characteristic Classes, Springer, 2017

[48] Shawn W. Walker The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method, IMA J. Numer. Anal., Volume 42 (2022), pp. 3094-3134 | DOI | MR | Zbl

[49] Max Wardetzky Discrete differential operators on polyhedral surfaces—convergence and approximation, Ph. D. Thesis, Freie Universität Berlin (2006)

[50] Hassler Whitney Geometric integration theory, Princeton University Press, 1957 | DOI

[51] Ruth M. Williams; Philip A. Tuckey Regge calculus: a brief review and bibliography, Class. Quant. Grav., Volume 9 (1992) no. 5, pp. 1409-1422 | DOI | MR | Zbl

[52] Guoliang Xu Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces, Comput. Aided Geom. Des., Volume 23 (2006) no. 2, pp. 193-207 https://www.sciencedirect.com/science/article/pii/s0167839605000865 | DOI | MR | Zbl

[53] Zhiqiang Xu; Guoliang Xu Discrete schemes for Gaussian curvature and their convergence, Comput. Math. Appl., Volume 57 (2009) no. 7, pp. 1187-1195 | DOI | MR | Zbl

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