Variational discretization of wave equations on evolving surfaces

Lubich, Christian; Mansour, Dhia

doi:10.1090/S0025-5718-2014-02882-2

Variational discretization of wave equations on evolving surfaces
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by Christian Lubich and Dhia Mansour PDF

Math. Comp. 84 (2015), 513-542 Request permission

Abstract:

A linear wave equation on a moving surface is derived from Hamilton’s principle of stationary action. The variational principle is discretized with functions that are piecewise linear in space and time. This yields a discretization of the wave equation in space by evolving surface finite elements and in time by a variational integrator, a version of the leapfrog or Störmer–Verlet method. We study stability and convergence of the full discretization in the natural time-dependent norms under the same CFL condition that is required for a fixed surface. Using a novel modified Ritz projection for evolving surfaces, we prove optimal-order error bounds. Numerical experiments illustrate the behavior of the fully discrete method.

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
Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
P. Bastian, M. Blatt, A. Dedner, C. Engwer, J. Fahlke, C. Gräser, R. Klöfkorn, M. Nolte, M. Ohlberger, and O. Sander, 2012. http://www.dune-project.org.
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
Andreas Dedner, Robert Klöfkorn, Martin Nolte, and Mario Ohlberger, A generic interface for parallel and adaptive discretization schemes: abstraction principles and the DUNE-FEM module, Computing 90 (2010), no. 3-4, 165–196. MR 2735465, DOI 10.1007/s00607-010-0110-3
A. Dedner, R. Klöfkorn, M. Nolte, and M. Ohlberger, 2012. http://dune.mathematik.uni-freiburg.de.
Todd Dupont, $L^{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880–889. MR 349045, DOI 10.1137/0710073
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
Gerhard Dziuk and Charles M. Elliott, $L^2$-estimates for the evolving surface finite element method, Math. Comp. 82 (2013), no. 281, 1–24. MR 2983013, DOI 10.1090/S0025-5718-2012-02601-9
Gerhard Dziuk and Charles M. Elliott, Finite element methods for surface PDEs, Acta Numer. 22 (2013), 289–396. MR 3038698, DOI 10.1017/S0962492913000056
Gerhard Dziuk, Dietmar Kröner, and Thomas Müller, Scalar conservation laws on moving hypersurfaces, Interfaces Free Bound. 15 (2013), no. 2, 203–236. MR 3105772, DOI 10.4171/IFB/301
G. Dziuk, Ch. Lubich, and D. Mansour, Runge-Kutta time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 32 (2012), no. 2, 394–416. MR 2911394, DOI 10.1093/imanum/drr017
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer. 12 (2003), 399–450. MR 2249159, DOI 10.1017/S0962492902000144
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR 1227985
E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506, DOI 10.1007/978-3-642-05221-7
Martin Lenz, Simplice Firmin Nemadjieu, and Martin Rumpf, A convergent finite volume scheme for diffusion on evolving surfaces, SIAM J. Numer. Anal. 49 (2011), no. 1, 15–37. MR 2764419, DOI 10.1137/090776767
Shingyu Leung, John Lowengrub, and Hongkai Zhao, A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion, J. Comput. Phys. 230 (2011), no. 7, 2540–2561. MR 2772929, DOI 10.1016/j.jcp.2010.12.029
Christian Lubich, Dhia Mansour, and Chandrasekhar Venkataraman, Backward difference time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 33 (2013), no. 4, 1365–1385. MR 3119720, DOI 10.1093/imanum/drs044
Dhia Mansour, Numerical analysis of partial differential equations on evolving surfaces, Dissertation (PhD thesis), Univ. Tübingen, 2013.
Dhia Mansour, Gauss-Runge-Kutta time discretization of wave equations on evolving surfaces, published on line in Numer. Math. (2014).
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001), 357–514. MR 2009697, DOI 10.1017/S096249290100006X
Yu. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation, Mat. Model. 2 (1990), no. 4, 78–87 (Russian, with English summary). MR 1064467
A. P. Veselov, Integrable systems with discrete time, and difference operators, Funktsional. Anal. i Prilozhen. 22 (1988), no. 2, 1–13, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 2, 83–93. MR 947601, DOI 10.1007/BF01077598
Morten Vierling, Control-constrained parabolic optimal control problems on evolving surfaces — theory and variational discretization, Preprint arXiv:1106.0622, 2011.
J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR 895589, DOI 10.1017/CBO9781139171755
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