Abstract
Geometric finite elements (GFE) generalize the idea of Galerkin methods to variational problems for unknowns that map into nonlinear spaces. In particular, GFE methods introduce proper discrete function spaces that are conforming in the sense that values of geometric finite element functions are in the codomain manifold \(\mathcal {M}\) at any point. Several types of such spaces have been constructed, and some are even completely intrinsic, i.e., they can be defined without any surrounding space. GFE spaces enable the elegant numerical treatment of variational problems posed in Sobolev spaces with nonlinear codomain space. Indeed, as GFE spaces are geometrically conforming, such variational problems have natural formulations in GFE spaces. These correspond to the discrete formulations of classical finite element methods. Also, the canonical projection onto the discrete maps commutes with the differential for a suitable notion of the tangent bundle as a manifold, and we therefore also obtain natural weak formulations. Rigorous results exist that show the optimal behavior of the a priori L 2 and H 1 errors under reasonable smoothness assumptions. Although the discrete function spaces are no vector spaces, their elements can nevertheless be described by sets of coefficients, which live in the codomain manifold. Variational discrete problems can then be reformulated as algebraic minimization problems on the set of coefficients. These algebraic problems can be solved by established methods of manifold optimization. This text will explain the construction of several types of GFE spaces, discuss the corresponding test function spaces, and sketch the a priori error theory. It will also show computations of the harmonic maps problem, and of two example problems from nanomagnetics and plate mechanics.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abatzoglou, T.J.: The minimum norm projection on C 2-manifolds in \(\mathbb {R}^n\). Trans. Am. Math. Soc. 243, 115–122 (1978)
Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)
Absil, P.A., Mahony, R., Trumpf, J.: An extrinsic look at the Riemannian Hessian. In: Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 361–368. Springer, Berlin (2013)
Absil, P.A., Gousenbourger, P.Y., Striewski, P., Wirth, B.: Differentiable piecewise-Bézier surfaces on Riemannian manifolds. SIAM J. Imaging Sci. 9(4), 1788–1828 (2016)
Alouges, F.: A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34(5), 1708–1726 (1997)
Alouges, F., Jaisson, P.: Convergence of a finite element discretization for the landau–lifshitz equations in micromagnetism. Math. Models Methods Appl. Sci. 16(2), 299–316 (2006)
Ambrosio, L.: Metric space valued functions of bounded variation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17(3), 439–478 (1990)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Springer, Berlin (2006)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29(1), 328–347 (2007)
Bartels, S., Prohl, A.: Constraint preserving implicit finite element discretization of harmonic map flow into spheres. Math. Comput. 76(260), 1847–1859 (2007)
Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity – Solving Einstein’s Equations on the Computer. Cambridge University Press, Cambridge (2010)
Belavin, A., Polyakov, A.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett. 22(10), 245–247 (1975)
Bergmann, R., Laus, F., Persch, J., Steidl, G.: Processing manifold-valued images. SIAM News 50(8), 1,3 (2017)
Berndt, J., Boeckx, E., Nagy, P.T., Vanhecke, L.: Geodesics on the unit tangent bundle. Proc. R. Soc. Edinb. A Math. 133(06), 1209–1229 (2003)
Bogdanov, A., Hubert, A.: Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994)
Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20, 95–126 (2001)
Cartan, E.: Groupes simples clos et ouverts et géométrie riemannienne. J. Math. Pures Appl. 8, 1–34 (1929)
Chiron, D.: On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Commun. Contemp. Math. 9(04), 473–513 (2007)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Elsevier, Amsterdam (1978)
Convent, A., Van Schaftingen, J.: Intrinsic colocal weak derivatives and Sobolev spaces between manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 16(5), 97–128 (2016)
Convent, A., Van Schaftingen, J.: Higher order weak differentiability and Sobolev spaces between manifolds (2017). arXiv preprint 1702.07171
de Gennes, P., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993)
Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, 2nd edn. Academic, Boston (1990)
Fert, A., Reyren, N., Cros, V.: Magnetic skyrmions: advances in physics and potential applications. Nat. Rev. Mater. 2(17031) (2017)
Focardi, M., Spadaro, E.: An intrinsic approach to manifold constrained variational problems. Ann. Mat. Pura Appl. 192(1), 145–163 (2013)
Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. Henri Poincaré 10(4), 215–310 (1948)
Gawlik, E.S., Leok, M.: Embedding-based interpolation on the special orthogonal group. SIAM J. Sci. Comput. 40(2), A721–A746 (2018)
Gawlik, E.S., Leok, M.: Interpolation on symmetric spaces via the generalized polar decomposition. Found. Comput. Math. 18(3), 757–788 (2018)
Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2004). https://books.google.de/books?id=4NWZdMBH1fsC
Grohs, P.: Quasi-interpolation in Riemannian manifolds. IMA J. Numer. Anal. 33(3), 849–874 (2013)
Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Found. Comput. Math. 15(6), 1357–1411 (2015)
Grohs, P., Hardering, H., Sander, O., Sprecher, M.: Projection-based finite elements for nonlinear function spaces. SIAM J. Numer. Anal. 57(1), 404–428 (2019)
Hajłasz, P.: Sobolev mappings between manifolds and metric spaces. In: Sobolev Spaces in Mathematics I. International Mathematical Series, vol. 8, pp. 185–222. Springer, Berlin (2009)
Hajlasz, P., Tyson, J.: Sobolev peano cubes. Michigan Math. J. 56(3), 687–702 (2008)
Hardering, H.: Intrinsic discretization error bounds for geodesic finite elements. Ph.D. thesis, Freie Universität Berlin (2015)
Hardering, H.: The Aubin–Nitsche trick for semilinear problems (2017). arXiv e-prints arXiv:1707.00963
Hardering, H.: L 2-discretization error bounds for maps into Riemannian manifolds (2018). ArXiv preprint 1612.06086
Hardering, H.: L 2-discretization error bounds for maps into Riemannian manifolds. Numer. Math. 139(2), 381–410 (2018)
Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, 2nd edn. Cambridge University Press, Cambridge (2002)
Hélein, F., Wood, J.C.: Harmonic maps. In: Handbook of Global Analysis, pp. 417–491. Elsevier, Amsterdam (2008)
Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)
Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Springer, New York (2011)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)
Ketov, S.V.: Quantum Non-linear Sigma-Models. Springer, Berlin (2000)
Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1(4), 561–659 (1993)
Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification. Bull. Tokyo Gakugei Univ. 40, 1–29 (1997)
Kružík, M., Prohl, A.: Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48(3), 439–483 (2006)
Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2003)
Melcher, C.: Chiral skyrmions in the plane. Proc. R. Soc. A 470(2172) (2014)
Mielke, A.: Finite elastoplasticity Lie groups and geodesics on SL(d). In: Newton, P., Holmes, P., Weinstein, A. (eds.) Geometry, Mechanics, and Dynamics, pp. 61–90. Springer, New York (2002)
Münch, I.: Ein geometrisch und materiell nichtlineares Cosserat-Modell – Theorie, Numerik und Anwendungsmöglichkeiten
Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. Sib. Mat. Zh. 38(3), 657–675 (1997)
Rubin, M.: Cosserat Theories: Shells, Rods, and Points. Springer, Dordrecht (2000)
Sander, O.: Geodesic finite elements for Cosserat rods. Int. J. Numer. Methods Eng. 82(13), 1645–1670 (2010)
Sander, O.: Geodesic finite elements on simplicial grids. Int. J. Numer. Methods Eng. 92(12), 999–1025 (2012)
Sander, O.: Geodesic finite elements of higher order. IMA J. Numer. Anal. 36(1), 238–266 (2016)
Sander, O.: Test function spaces for geometric finite elements (2016). ArXiv e-prints 1607.07479
Sander, O., Neff, P., Bîrsan, M.: Numerical treatment of a geometrically nonlinear planar Cosserat shell model. Comput. Mech. 57(5), 817–841 (2016)
Shatah, J., Struwe, M.: Geometric Wave Equations. American Mathematical Society, Providence (2000)
Simo, J., Fox, D., Rifai, M.: On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory. Comput. Methods Appl. Mech. Eng. 79(1), 21–70 (1990)
Sprecher, M.: Numerical methods for optimization and variational problems with manifold-valued data. Ph.D. thesis, ETH Zürich (2016)
Stahl, S.: The Poincaré Half-Plane – A Gateway to Modern Geometry. Jones and Bartlett Publishers, Burlington (1993)
Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(1), 558–581 (1985)
Walther, A., Griewank, A.: Getting started with ADOL-C. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing. Computational Science, pp. 181–202. Chapman-Hall CRC, Boca Raton (2012)
Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7(4), 2226–2257 (2014)
Wriggers, P., Gruttmann, F.: Thin shells with finite rotations formulated in Biot stresses: theory and finite element formulation. Int. J. Numer. Methods Eng. 36, 2049–2071 (1993)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. 1. Springer, New York (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hardering, H., Sander, O. (2020). Geometric Finite Elements. In: Grohs, P., Holler, M., Weinmann, A. (eds) Handbook of Variational Methods for Nonlinear Geometric Data. Springer, Cham. https://doi.org/10.1007/978-3-030-31351-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-31351-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-31350-0
Online ISBN: 978-3-030-31351-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)