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Variational discretization of wave equations on evolving surfaces


Authors: Christian Lubich and Dhia Mansour
Journal: Math. Comp. 84 (2015), 513-542
MSC (2010): Primary 65M12, 65M15, 65M60
DOI: https://doi.org/10.1090/S0025-5718-2014-02882-2
Published electronically: October 24, 2014
MathSciNet review: 3290953
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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.


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Additional Information

Christian Lubich
Affiliation: Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany
Email: lubich@na.uni-tuebingen.de

Dhia Mansour
Affiliation: Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany
Email: mansour@na.uni-tuebingen.de

DOI: https://doi.org/10.1090/S0025-5718-2014-02882-2
Keywords: Wave equation, evolving surface finite element method, variational integrator, Ritz projection, error analysis.
Received by editor(s): November 23, 2012
Received by editor(s) in revised form: June 14, 2013
Published electronically: October 24, 2014
Additional Notes: This work was supported by DFG, SFB/TR 71 “Geometric Partial Differential Equations”
Article copyright: © Copyright 2014 American Mathematical Society

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