Analysis of finite element methods for surface vector-Laplace eigenproblems

Reusken, Arnold

doi:10.1090/mcom/3728

Analysis of finite element methods for surface vector-Laplace eigenproblems
HTML articles powered by AMS MathViewer

by Arnold Reusken;

Math. Comp. 91 (2022), 1587-1623

DOI: https://doi.org/10.1090/mcom/3728

Published electronically: May 16, 2022

HTML | PDF | Request permission

Abstract:

In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.

References

Netgen/NGSolve, 17 April 2019, https://ngsolve.org/.
Paola F. Antonietti, Annalisa Buffa, and Ilaria Perugia, Discontinuous Galerkin approximation of the Laplace eigenproblem, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25-28, 3483–3503. MR 2220929, DOI 10.1016/j.cma.2005.06.023
Marino Arroyo and Antonio DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E (3) 79 (2009), no. 3, 031915, 17. MR 2497175, DOI 10.1103/PhysRevE.79.031915
O. Azencot, M. Ben-Chen, F. Chazal, and M. Ovsjanikov, An operator approach to tangent vector field processing, Comput. Graph. Forum (2013), 73–82.
O. Azencot, M. Ovsjanikov, F. Chazal, and M. Ben-Chen, Discrete derivatives of vector fields on surfaces – an operator approach, ACM Trans. Graph. 34 (2015), 29:1–29:13.
I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
M. Ben-Chen, A. Butscher, J. Solomon, and L. Guibas, On discrete Killing fields and patterns on surfaces, Eurographics Symposium on Geometry Processing, Vol. 29, 2010, pp. 1701–1711.
Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. MR 2652780, DOI 10.1017/S0962492910000012
Andrea Bonito, Alan Demlow, and Martin Licht, A divergence-conforming finite element method for the surface Stokes equation, SIAM J. Numer. Anal. 58 (2020), no. 5, 2764–2798. MR 4155235, DOI 10.1137/19M1284592
Andrea Bonito, Alan Demlow, and Ricardo H. Nochetto, Finite element methods for the Laplace-Beltrami operator, Geometric partial differential equations. Part I, Handb. Numer. Anal., vol. 21, Elsevier/North-Holland, Amsterdam, [2020] ©2020, pp. 1–103. MR 4378426, DOI 10.1007/s
Andrea Bonito, Alan Demlow, and Justin Owen, A priori error estimates for finite element approximations to eigenvalues and eigenfunctions of the Laplace-Beltrami operator, SIAM J. Numer. Anal. 56 (2018), no. 5, 2963–2988. MR 3860898, DOI 10.1137/17M1163311
Eugene G. D′yakonov, Optimization in solving elliptic problems, CRC Press, Boca Raton, FL, 1996. Translated from the 1989 Russian original; Translation edited and with a preface by Steve McCormick. MR 1396083
Gerhard Dziuk and Charles M. Elliott, Finite element methods for surface PDEs, Acta Numer. 22 (2013), 289–396. MR 3038698, DOI 10.1017/S0962492913000056
David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102–163. MR 271984, DOI 10.2307/1970699
Thomas-Peter Fries, Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds, Internat. J. Numer. Methods Fluids 88 (2018), no. 2, 55–78. MR 3846120, DOI 10.1002/fld.4510
Jörg Grande, Christoph Lehrenfeld, and Arnold 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. MR 3747467, DOI 10.1137/16M1102203
Sven Gross, Thomas Jankuhn, Maxim A. Olshanskii, and Arnold Reusken, A trace finite element method for vector-Laplacians on surfaces, SIAM J. Numer. Anal. 56 (2018), no. 4, 2406–2429. MR 3840893, DOI 10.1137/17M1146038
Peter Hansbo, Mats G. Larson, and Karl Larsson, Analysis of finite element methods for vector Laplacians on surfaces, IMA J. Numer. Anal. 40 (2020), no. 3, 1652–1701. MR 4122487, DOI 10.1093/imanum/drz018
Thomas Jankuhn, Maxim A. Olshanskii, and Arnold Reusken, Incompressible fluid problems on embedded surfaces: modeling and variational formulations, Interfaces Free Bound. 20 (2018), no. 3, 353–377. MR 3875687, DOI 10.4171/IFB/405
Thomas Jankuhn and Arnold Reusken, Trace finite element methods for surface vector-Laplace equations, IMA J. Numer. Anal. 41 (2021), no. 1, 48–83. MR 4205052, DOI 10.1093/imanum/drz062
Andrew V. Knyazev, New estimates for Ritz vectors, Math. Comp. 66 (1997), no. 219, 985–995. MR 1415802, DOI 10.1090/S0025-5718-97-00855-7
Andrew V. Knyazev and John E. Osborn, New a priori FEM error estimates for eigenvalues, SIAM J. Numer. Anal. 43 (2006), no. 6, 2647–2667. MR 2206452, DOI 10.1137/040613044
Hajime Koba, Chun Liu, and Yoshikazu Giga, Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no. 2, 359–389. MR 3614501, DOI 10.1090/qam/1452
Philip L. Lederer, Christoph Lehrenfeld, and Joachim Schöberl, Divergence-free tangential finite element methods for incompressible flows on surfaces, Internat. J. Numer. Methods Engrg. 121 (2020), no. 11, 2503–2533. MR 4156753, DOI 10.1002/nme.6317
C. Lehrenfeld, ngsxfem, 17 April 2019, https://github.com/ngsxfem.
Marius Mitrea and Michael Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann. 321 (2001), no. 4, 955–987. MR 1872536, DOI 10.1007/s002080100261
Tatsu-Hiko Miura, On singular limit equations for incompressible fluids in moving thin domains, Quart. Appl. Math. 76 (2018), no. 2, 215–251. MR 3769895, DOI 10.1090/qam/1495
I. Nitschke, S. Reuther, and A. Voigt, Hydrodynamic interactions in polar liquid crystals on evolving surfaces, Phys. Rev. Fluids, 4 (2019), p. 044002.
I. Nitschke, A. Voigt, and J. Wensch, A finite element approach to incompressible two-phase flow on manifolds, J. Fluid Mech. 708 (2012), 418–438. MR 2975450, DOI 10.1017/jfm.2012.317
Maxim A. Olshanskii, Annalisa Quaini, Arnold Reusken, and Vladimir Yushutin, A finite element method for the surface Stokes problem, SIAM J. Sci. Comput. 40 (2018), no. 4, A2492–A2518. MR 3841618, DOI 10.1137/18M1166183
Maxim A. Olshanskii, Arnold Reusken, and Alexander Zhiliakov, Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor-Hood elements for surface PDEs, Math. Comp. 90 (2021), no. 330, 1527–1555. MR 4273108, DOI 10.1090/mcom/3551
Maxim A. Olshanskii and Vladimir Yushutin, A penalty finite element method for a fluid system posed on embedded surface, J. Math. Fluid Mech. 21 (2019), no. 1, Paper No. 14, 18. MR 3911729, DOI 10.1007/s00021-019-0420-y
Peter Petersen, Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016. MR 3469435, DOI 10.1007/978-3-319-26654-1
Arnold Reusken, Stream function formulation of surface Stokes equations, IMA J. Numer. Anal. 40 (2020), no. 1, 109–139. MR 4050536, DOI 10.1093/imanum/dry062
S. Reuther and A. Voigt, The interplay of curvature and vortices in flow on curved surfaces, Multiscale Model. Simul. 13 (2015), no. 2, 632–643. MR 3359666, DOI 10.1137/140971798
S. Reuther and A. Voight, Solving the incompressible surface Navier-Stokes equation by surface finite elements, Phys. Fluids 30 (2018), 012107.
SciPy, http://www.scipy.org.
J. Solomon, M. Ben-Chen, A. Butscher, and L. Guibas, Discovery of intrinsic primitives on triangle meshes, Comput. Graph. Forum 30 (2011), 365–374.
M. Tao, J. Solomon, and A. Butscher, Near-isometric level set tracking, Eurographics Symposium on Geometry Processing, Vol. 35, 2016.
Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI 10.1080/03605309208820892
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
Harry Yserentant, A short theory of the Rayleigh-Ritz method, Comput. Methods Appl. Math. 13 (2013), no. 4, 495–502. MR 3107358, DOI 10.1515/cmam-2013-0013

AMS Website Down

Mathematics of Computation

Analysis of finite element methods for surface vector-Laplace eigenproblems
HTML articles powered by AMS MathViewer

Abstract: