VARIATIONAL DISCRETIZATION OF WAVE EQUATIONS ON EVOLVING SURFACES CHRISTIAN LUBICH AND DHIA MANSOUR Abstract. A linear wave equation on a moving surface is derived from Hamil- ton’s principle of stationary action. The variational principle is discretized with functions that are piecewise linear in space and time. This yields a discretiza- tion of the wave equation in space by evolving surface finite elements and in time by a variational integrator, a version of the leapfrog or St¨ormer–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 sur- faces, we prove optimal-order error bounds. Numerical experiments illustrate the behavior of the fully discrete method. 1. Introduction In recent years, there have been significant advances in the numerical analysis of partial differential equations on fixed and moving surfaces. Concerning the latter, we refer to the review articles by Deckelnick, Dziuk & Elliott [4] and Dziuk & Elliott [11] and to recent papers on linear parabolic equations on time-dependent surfaces discretized by evolving surface finite elements and various time discretizations [9, 10, 13, 21, 27], by finite volume methods [19], by a grid-based particle method [20] and by level set methods [1, 29], and to [12] for conservation laws on time-dependent surfaces. Many more references are found in [11]. In the present paper we consider a linear wave equation on a given time-dependent surface, which is the natural analog of the classical acoustic wave equation on a fixed spatial domain. We have no specific application in mind, but consider the problem as prototypical for dynamical problems on a moving surface that are described by Hamilton’s principle of stationary action, a fundamental principle of mechanics. Just as the numerical analysis of the linear wave equation on a fixed domain has provided much insight into the numerical treatment of more complicated, linear and nonlinear, wave problems in a variety of application areas, we expect similar benefits from a thorough numerical analysis of the linear wave equation on evolving surfaces based on the variational formulation. Among novel analytic techniques developed here is a stability analysis of full discretizations with time-dependent mass and stiffness matrices in the natural time- dependent norms, and the use of appropriately modified Ritz projections to derive optimal-order error bounds. Our stability analysis operates at the matrix-vector 2010 Mathematics Subject Classification. 65M12, 65M15, 65M60 . Key words and phrases. Wave equation, evolving surface finite element method, variational integrator, Ritz projection, error analysis. This work was supported by DFG, SFB/TR 71 “Geometric Partial Differential Equations”. June 14, 2013. 1