Startseite Diffusion of tangential tensor fields: numerical issues and influence of geometric properties
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Diffusion of tangential tensor fields: numerical issues and influence of geometric properties

  • Elena Bachini , Philip Brandner , Thomas Jankuhn , Michael Nestler , Simon Praetorius EMAIL logo , Arnold Reusken und Axel Voigt
Veröffentlicht/Copyright: 15. August 2023

Abstract

We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n-tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance of geometric properties and their increasing influence as the tensorial degree changes from n = 0 to n ⩾ 1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution.

JEL Classification: 65M60; 58J35

Funding statement: The authors wish to thank the German Research Foundation (DFG) for financial support within the Research Unit ‘Vector- and Tensor-Valued Surface PDEs’ (FOR 3013) with projects No. RE 1461/11-1 and VO 899/22-1. We further acknowledge computing resources provided by ZIH at TU Dresden and within project PFAMDIS at FZ Jülich.

References

[1] E. Bachini, P. Brandner, M. Nestler, and S. Praetorius, Code for the numerical experiments with SFEM, ISFEM, DI, and TraceFEM, Zenodo, http://dx.doi.org/10.5281/zenodo.7096487, 2022.Suche in Google Scholar

[2] E. Bachini, M. W. Farthing, and M. Putti, Intrinsic finite element method for advection–diffusion–reaction equations on surfaces, J. Comput. Phys. 424 (2021), 109827.10.1016/j.jcp.2020.109827Suche in Google Scholar

[3] P. Bastian, M. Blatt, A. Dedner, N.-A. Dreier, C. Engwer, R. Fritze, C. Gräser, D. Kempf, R. Klöfkorn, M. Ohlberger, and O. Sander, The Dune framework: Basic concepts and recent developments, Computers & Mathematics with Applications 81 (2021), 75–112.10.1016/j.camwa.2020.06.007Suche in Google Scholar

[4] A. Bonito, A. Demlow, and M. Licht, A divergence-conforming finite element method for the surface Stokes equation, SIAM J. Numer. Anal. 58 (2020), No. 5, 2764–2798.10.1137/19M1284592Suche in Google Scholar

[5] A. Bonito, A. Demlow, and R. H. Nochetto, Finite element methods for the Laplace–Beltrami operator, Handbook of Numerical Analysis (Eds. A. Bonito and R. H. Nochetto), Vol. 21, Elsevier, 2020, pp. 1–103.10.1016/bs.hna.2019.06.002Suche in Google Scholar

[6] P. Brandner, T. Jankuhn, S. Praetorius, A. Reusken, and A. Voigt, Finite element discretization methods for velocity–pressure and stream function formulations of surface Stokes equations, SIAM J. Sci. Comput. 44 (2022), A1807–A1832.10.1137/21M1403126Suche in Google Scholar

[7] F. Bürger, Interaction of Mean Curvature Flow and a Diffusion Equation, Ph.D. thesis, Universität Regensburg, December 2021.Suche in Google Scholar

[8] E. Burman, P. Hansbo, M. G. Larson, and A. Massing, Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions, ESAIM: M2AN 52 (2018), No. 6, 2247–2282.10.1051/m2an/2018038Suche in Google Scholar

[9] C. G. Claudel and A. M. Bayen, Lax–Hopf based incorporation of internal boundary conditions into Hamilton–Jacobi equation. Part II: Computational methods, IEEE Trans. Automat. Contr. 55 (2010), No. 5, 1158–1174.10.1109/TAC.2010.2045439Suche in Google Scholar

[10] K. Crane, C. Weischedel, and M. Wardetzky, Geodesics in heat: A new approach to computing distance based on heat flow, ACM Trans. Graph. 32 (2013), No. 5, 1–11.10.1145/2516971.2516977Suche in Google Scholar

[11] A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47 (2009), No. 2, 805–827.10.1137/070708135Suche in Google Scholar

[12] G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer. 22 (2013), 289–396.10.1017/S0962492913000056Suche in Google Scholar

[13] J. Faraudo, Diffusion equation on curved surfaces, I. Theory and application to biological membranes, J. Chem. Phys. 116 (2002), No. 13, 5831–5841.10.1063/1.1456024Suche in Google Scholar

[14] E. S. Gawlik, High-order approximation of Gaussian curvature with Regge finite elements, SIAM J. Numer. Anal. 58 (2020), No. 3, 1801–1821.10.1137/19M1255549Suche in Google Scholar

[15] J. Grande, C. Lehrenfeld, and A. Reusken, Analysis of a high-order trace finite element method for PDEs on level set surfaces, SIAM J. Numer. Anal. 56 (2018), No. 1, 228–255.10.1137/16M1102203Suche in Google Scholar

[16] H. P. McKean, Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differ. Geom. 1 (1967), No. 1-2, 43–69.10.4310/jdg/1214427880Suche in Google Scholar

[17] P. Hansbo, M. G. Larson, and K. Larsson, Analysis of finite element methods for vector Laplacians on surfaces, IMA J. Numer. Anal. 40 (2020), No. 3, 1652–1701.10.1093/imanum/drz018Suche in Google Scholar

[18] H. Hardering and S. Praetorius, Tangential errors of tensor surface finite elements, IMA J. Numer. Anal. 43 (2023), No. 3, 1543–1585.Suche in Google Scholar

[19] T. Jankuhn and A. Reusken, Trace finite element methods for surface vector-Laplace equations, IMA J. Numer. Anal. 41 (2020), No. 1, 48–83.10.1093/imanum/drz062Suche in Google Scholar

[20] F. Knöppel, K. Crane, U. Pinkall, and P. Schröder, Globally optimal direction fields, ACM Trans. Graphics 32 (2013), No. 4, 1–10.10.1145/2461912.2462005Suche in Google Scholar

[21] P. L. Lederer, C. Lehrenfeld, and J. Schöberl, Divergence-free tangential finite element methods for incompressible flows on surfaces, Int. J. Numer. Methods Eng. 121 (2020), No. 11, 2503–2533.10.1002/nme.6317Suche in Google Scholar PubMed PubMed Central

[22] C. Lehrenfeld and A. Reusken, High order unfitted finite element methods for interface problems and PDEs on surfaces. In: Transport Processes at Fluidic Interfaces (Eds. D. Bothe and A. Reusken), Birkhäuser, Cham, 2017, pp. 33–63.10.1007/978-3-319-56602-3_2Suche in Google Scholar

[23] X. Li, J. Lowengrub, K. E. Teigen, A. Voigt, and F. Wang, A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci. 7 (2009), No. 4, 1009–1037.10.4310/CMS.2009.v7.n4.a10Suche in Google Scholar

[24] C. Lubich, D. Mansour, and C. Venkataraman, Backward difference time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 33 (2013), No. 4, 1365–1385.10.1093/imanum/drs044Suche in Google Scholar

[25] M. Nestler, I. Nitschke, S. Praetorius, and A. Voigt, Orientational order on surfaces: the coupling of topology, geometry, and dynamics, J. Nonlinear Sci. 28 (2018), 147–191.10.1007/s00332-017-9405-2Suche in Google Scholar

[26] M. Nestler, I. Nitschke, and A. Voigt, A finite element approach for vector- and tensor-valued surface PDEs, J. Comput. Phys. 389 (2019), 48–61.10.1016/j.jcp.2019.03.006Suche in Google Scholar

[27] M. Nestler and A. Voigt, A diffuse interface approach for vector-valued PDEs on surfaces, arXiv:2303.07135, 2023, March 2023.10.4310/CMS.2024.v22.n6.a13Suche in Google Scholar

[28] M. Neunteufel and J. Schöberl, The Hellan–Herrmann–Johnson method for nonlinear shells, Computers & Structures 225 (2019), 106109.10.1016/j.compstruc.2019.106109Suche in Google Scholar

[29] ngsxfem, An add-on to NGSolve for unfitted finite element discretizations, https://github.com/ngsxfem, 2020.Suche in Google Scholar

[30] M. A. Olshanskii and A. Reusken, Trace finite element methods for PDEs on surfaces, In: Geometrically Unfitted Finite Element Methods and Applications (Eds. S. P. A. Bordas, E. Burman, M. G. Larson, and M. A. Olshanskii), Springer, Cham, 2017, pp. 211–258.10.1007/978-3-319-71431-8_7Suche in Google Scholar

[31] S. Praetorius and F. Stenger, Dune-CurvedGrid – A Dune module for surface parametrization, Arch. Numer. Soft. 6 (2022), No. 1, 1–27.Suche in Google Scholar

[32] A. Rätz and A. Voigt, PDE’s on surfaces—A diffuse interface approach, Commun. Math. Sci. 4 (2006), No. 3, 575–590.10.4310/CMS.2006.v4.n3.a5Suche in Google Scholar

[33] O. Sander, DUNE—The Distributed and Unified Numerics Environment, Lecture Notes in Computational Science and Engineering, Vol. 140, Springer International Publishing, 2020.10.1007/978-3-030-59702-3Suche in Google Scholar

[34] J. Schöberl, NETGEN An advancing front 2D/3D-mesh generator based on abstract rules, Computing and Visualization in Science 1 (1997), No. 1, 41–52.10.1007/s007910050004Suche in Google Scholar

[35] J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, Institute for Analysis and Scientific Computing, Vienna University of Technology, Report, 2014.Suche in Google Scholar

[36] N. Sharp, Y. Soliman, and K. Crane, The vector heat method, ACM Trans. Graph. 38 (2019), No. 3, 1–19.10.1145/3243651Suche in Google Scholar

[37] A. Singer and H.-T. Wu, Vector diffusion maps and the connection Laplacian, Commun. Pure Appl. Math. 65 (2012), No. 8, 1067–1144.10.1002/cpa.21395Suche in Google Scholar PubMed PubMed Central

[38] F. Stenger, Meshconv: a Tool for Various Mesh-Conversions and Mesh-Transformations., https://gitlab.mn.tu-dresden.de/iwr/meshconv 2020, v3.20.Suche in Google Scholar

[39] J. Sun, M. Ovsjanikov, and L. Guibas, A concise and provably informative multi-scale signature based on heat diffusion, In: Proc. Symp. Geom. Process. ’09, Eurographics Association, Goslar, DEU, 2009, pp. 1383–1392,10.1111/j.1467-8659.2009.01515.xSuche in Google Scholar

[40] S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Commun. Pure Appl. Math. 20 (1967), No. 2, 431–455.10.1002/cpa.3160200210Suche in Google Scholar

[41] S. Vey and A. Voigt, AMDiS: Adaptive MultiDimensional Simulations, Comput. Vis. Sci. 10 (2007), No. 1, 57–67.10.1007/s00791-006-0048-3Suche in Google Scholar

[42] T. Witkowski, S. Ling, S. Praetorius, and A. Voigt, Software concepts and numerical algorithms for a scalable adaptive parallel finite element method, Advances in Computational Mathematics 41 (2015), No. 6, 1145–1177.10.1007/s10444-015-9405-4Suche in Google Scholar

Received: 2022-09-20
Revised: 2023-04-17
Accepted: 2023-03-19
Published Online: 2023-08-15
Published in Print: 2024-03-25

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