𝐿²-estimates for the evolving surface finite element method

Dziuk, Gerhard; Elliott, Charles

doi:10.1090/S0025-5718-2012-02601-9

$L^2$-estimates for the evolving surface finite element method
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by Gerhard Dziuk and Charles M. Elliott PDF

Math. Comp. 82 (2013), 1-24 Request permission

Abstract:

In this paper we consider the evolving surface finite element meth- od for the advection and diffusion of a conserved scalar quantity on a moving surface. In an earlier paper using a suitable variational formulation in time dependent Sobolev space we proposed and analysed a finite element method using surface finite elements on evolving triangulated surfaces. An optimal order $H^1$-error bound was proved for linear finite elements. In this work we prove the optimal error bound in $L^2(\Gamma (t))$ uniformly in time.

References

David Adalsteinsson and J. A. Sethian, Transport and diffusion of material quantities on propagating interfaces via level set methods, J. Comput. Phys. 185 (2003), no. 1, 271–288. MR 2010161, DOI 10.1016/S0021-9991(02)00057-8
Marcelo Bertalmío, Li-Tien Cheng, Stanley Osher, and Guillermo Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2001), no. 2, 759–780. MR 1868103, DOI 10.1006/jcph.2001.6937
Donna A. Calhoun and Christiane Helzel, A finite volume method for solving parabolic equations on logically Cartesian curved surface meshes, SIAM J. Sci. Comput. 31 (2009/10), no. 6, 4066–4099. MR 2566584, DOI 10.1137/08073322X
Klaus Deckelnick, Gerhard Dziuk, and Charles M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer. 14 (2005), 139–232. MR 2168343, DOI 10.1017/S0962492904000224
Klaus Deckelnick, Gerhard Dziuk, Charles M. Elliott, and Claus-Justus Heine, An $h$-narrow band finite-element method for elliptic equations on implicit surfaces, IMA J. Numer. Anal. 30 (2010), no. 2, 351–376. MR 2608464, DOI 10.1093/imanum/drn049
Alan 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. MR 2485433, DOI 10.1137/070708135
Alan Demlow and Gerhard Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45 (2007), no. 1, 421–442. MR 2285862, DOI 10.1137/050642873
Gerhard Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial differential equations and calculus of variations, Lecture Notes in Math., vol. 1357, Springer, Berlin, 1988, pp. 142–155. MR 976234, DOI 10.1007/BFb0082865
G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI 10.1093/imanum/drl023
G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comput. Math. 25 (2007), no. 4, 385–407. MR 2337402
G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces, Interfaces Free Bound. 10 (2008), no. 1, 119–138. MR 2383539, DOI 10.4171/IFB/182
G. Dziuk and C. M. Elliott, An Eulerian approach to transport and diffusion on evolving implicit surfaces, Comput. Vis. Sci. 13 (2010), no. 1, 17–28. MR 2565107, DOI 10.1007/s00791-008-0122-0
C. Eilks and C. M. Elliott, Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method, J. Comput. Phys. 227 (2008), no. 23, 9727–9741. MR 2469030, DOI 10.1016/j.jcp.2008.07.023
Charles M. Elliott and Björn Stinner, Modeling and computation of two phase geometric biomembranes using surface finite elements, J. Comput. Phys. 229 (2010), no. 18, 6585–6612. MR 2660322, DOI 10.1016/j.jcp.2010.05.014
Charles M. Elliott and Björn Stinner, Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface, Math. Models Methods Appl. Sci. 19 (2009), no. 5, 787–802. MR 2531040, DOI 10.1142/S0218202509003620
Charles M. Elliott, Björn Stinner, Vanessa Styles, and Richard Welford, Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal. 31 (2011), no. 3, 786–812. MR 2832780, DOI 10.1093/imanum/drq005
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1998.
John B. Greer, An improvement of a recent Eulerian method for solving PDEs on general geometries, J. Sci. Comput. 29 (2006), no. 3, 321–352. MR 2272322, DOI 10.1007/s10915-005-9012-5
John B. Greer, Andrea L. Bertozzi, and Guillermo Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys. 216 (2006), no. 1, 216–246. MR 2223442, DOI 10.1016/j.jcp.2005.11.031
Morton E. Gurtin, Configurational forces as basic concepts of continuum physics, Applied Mathematical Sciences, vol. 137, Springer-Verlag, New York, 2000. MR 1735282
S. Hildebrandt, Analysis 2, Springer-Verlag, Berlin, Heidelburg, New York, 1982.
Lili Ju and Qiang Du, A finite volume method on general surfaces and its error estimates, J. Math. Anal. Appl. 352 (2009), no. 2, 645–668. MR 2501909, DOI 10.1016/j.jmaa.2008.11.022
Lili Ju, Li Tian, and Desheng Wang, A posteriori error estimates for finite volume approximations of elliptic equations on general surfaces, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 5-8, 716–726. MR 2498524, DOI 10.1016/j.cma.2008.10.007
Martin Lenz, Simplice Firmin Nemadjieu, and Martin Rumpf, Finite volume method on moving surfaces, Finite volumes for complex applications V, ISTE, London, 2008, pp. 561–568. MR 2451453
Colin B. Macdonald and Steven J. Ruuth, Level set equations on surfaces via the closest point method, J. Sci. Comput. 35 (2008), no. 2-3, 219–240. MR 2429939, DOI 10.1007/s10915-008-9196-6
Colin B. Macdonald and Steven J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces, SIAM J. Sci. Comput. 31 (2009/10), no. 6, 4330–4350. MR 2594984, DOI 10.1137/080740003
Maxim A. Olshanskii, Arnold Reusken, and Jörg Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3339–3358. MR 2551197, DOI 10.1137/080717602
Maxim A. Olshanskii and Arnold Reusken, A finite element method for surface PDEs: matrix properties, Numer. Math. 114 (2010), no. 3, 491–520. MR 2570076, DOI 10.1007/s00211-009-0260-4
Stanley Osher and Ronald Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, vol. 153, Springer-Verlag, New York, 2003. MR 1939127, DOI 10.1007/b98879
Andreas Rätz and Axel Voigt, PDE’s on surfaces—a diffuse interface approach, Commun. Math. Sci. 4 (2006), no. 3, 575–590. MR 2247931
Steven J. Ruuth and Barry Merriman, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys. 227 (2008), no. 3, 1943–1961. MR 2450979, DOI 10.1016/j.jcp.2007.10.009
J. A. Sethian, Level set methods and fast marching methods, 2nd ed., Cambridge Monographs on Applied and Computational Mathematics, vol. 3, Cambridge University Press, Cambridge, 1999. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. MR 1700751
Jian-Jun Xu and Hong-Kai Zhao, An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput. 19 (2003), no. 1-3, 573–594. Special issue in honor of the sixtieth birthday of Stanley Osher. MR 2028859, DOI 10.1023/A:1025336916176