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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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.
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Additional Information
  • Christian Lubich
  • Affiliation: Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany
  • MR Author ID: 116445
  • 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
  • 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”
  • © Copyright 2014 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 513-542
  • MSC (2010): Primary 65M12, 65M15, 65M60
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02882-2
  • MathSciNet review: 3290953