Abstract
We study the approximation of functions that map a Euclidean domain \(\Omega \subset {\mathbb {R}}^{d}\) into an n-dimensional Riemannian manifold (M, g) minimizing an elliptic, semilinear energy in a function set \(H\subset W^{1,2}(\Omega ,M)\). The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations \(S_{h}\subset H\). We provide a set of conditions on \(S_{h}\) such that we can prove a priori \(W^{1,2}\)- and \(L^{2}\)-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrinsic framework, independently of embeddings of the manifold or the choice of coordinates.
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References
Alouges, F.: A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34(5), 1708–1726 (1997)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Springer, Berlin (2006)
Bartels, S.: Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal. 43(1), 220–238 (2005)
Bartels, S., Prohl, A.: Constraint preserving implicit finite element discretization of harmonic map flow into spheres. Math. Comput. 76(260), 1847–1859 (2007)
Casciaro, B., Francaviglia, M.: A new variational characterization of Jacobi fields along geodesics. Annali di Matematica pura ed applicata 172(4), 219–228 (1997)
Ciarlet, P .G.: The Finite Element Method for Elliptic Problems. Elsevier, Amsterdam (1978)
Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)
Dobrowolski, M., Rannacher, R.: Finite element methods for nonlinear elliptic systems of second order. Math. Nachr. 94, 155–172 (1980)
Dziuk, G., Elliott, C.: \({L}^2\)-estimates for the evolving surface finite element method. Math. Comput. 82(281), 1–24 (2013)
Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)
Dziuk, G., Elliott, C.M.: A fully discrete evolving surface finite element method. SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)
Eichmair, M., Metzger, J.: Large isopserimetric surfaces in initial data sets. J. Differ. Geom. 94(1), 159–186 (2013)
Evans, L .C.: Partial Differential Equations. AMS, Providence (1998)
Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements. Found. Comput. Math. 15(6), 1357–1411 (2015)
Hajłasz, P.: Sobolev mappings between manifolds and metric spaces. In: Maz’ya, V. (ed.) Sobolev Spaces in Mathematics I, Volume 8 of International Mathematical Series, pp. 185–222. Springer, Berlin (2009)
Hardering, H.: Intrinsic Discretization Error Bounds for Geodesic Finite Elements. PhD thesis, Freie Universität Berlin (2015)
Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, second edn. Cambridge University Press, Cambridge (2002)
Hélein, F., Wood, J.C.: Harmonic maps. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, pp. 417–491. Elsevier, Amsterdam (2008)
Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2008)
Karcher, H.: Mollifier smoothing and Riemannian center of mass. Commun. Pure Appl. Math. 30, 509–541 (1977)
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)
Melcher, C.: Chiral skyrmions in the plane. Proc. R. Soc. Lond. A: Mater. 470(2172), 20140394 (2014)
Münch, I: Ein geometrisch und materiell nichtlineares Cosserat-Model—Theorie, Numerik und Anwendungsmöglichkeiten. PhD thesis, Universität Karlsruhe (2007)
Münch, I., Neff, P., Wagner, W.: Transversely isotropic material: nonlinear Cosserat versus classical approach. Contin. Mech. Therm. 23(1), 27–34 (2011)
Münch, I., Wagner, W., Neff, P.: Theory and FE-analysis for structures with large deformation under magnetic loading. Comput. Mech. 44(1), 93–102 (2009)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(1), 20–63 (1956)
Neff, P.: A geometrically exact planar cosserat shell-model with microstructure: existence of minimizers for zero cosserat couple modulus. Math. Models Methods Appl. Sci. 17(03), 363–392 (2007)
Palais, R .S.: Foundations of Global Non-linear Analysis. Benjamin, New York (1968)
Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175(2), 197–226 (1995)
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 (2015)
Sander, O.: Test Function Spaces for Geometric Finite Elements. arXiv:1607.07479 (2016)
Sander, O., Neff, P., Bîrsan, M.: Numerical treatment of a geometrically nonlinear planar Cosserat shell model. Comput. Mech. 57(5), 817–841 (2016)
Schmeißer, H.-J., Sickel, W.: Vector-valued Sobolev spaces and Gagliardo–Nirenberg inequalities. In: Brezis, H., Chipot, M., Escher, J. (eds.) Nonlinear Elliptic and Parabolic Problems, pp. 463–472. Birkhäuser, Boston (2005)
Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18, 253–268 (1983)
Simo, J.C., Fox, D.D., Rifai, M.S.: 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)
Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. Part II: computational aspects. Comput. Methods Appl. Mech. Eng. 58(1), 79–116 (1986)
Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(1), 558–581 (1985)
Vese, L.A., Osher, S.J.: Numerical methods for p-harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40(6), 2085–2104 (2002)
Wriggers, P., Gruttmann, F.: Thin shells with finite rotations formulated in Biot stresses: theory and finite element formulation. Int. J. Num. Methods Eng. 36, 2049–2071 (1993)
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Hardering, H. \(L^{2}\)-discretization error bounds for maps into Riemannian manifolds. Numer. Math. 139, 381–410 (2018). https://doi.org/10.1007/s00211-017-0941-3
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DOI: https://doi.org/10.1007/s00211-017-0941-3