A convergent finite element scheme for a wave equation with a moving boundary

Abstract

We wish to consider in this paper the numerical approximation of the solution of a wave equation when the boundaries of the spatial domain are moving. This problem has many practical applications in engineering science. One encounters wave systems in evolving domains in widely disseminated situations, such that rolling or unrolling antennas of space satellites, decoding the sound waves emitted by moving underwater objects or simulating the displacement of crane cables. In order to obtain computer simulations of this situations, one may try to make use of the following idea : a first discretization of the partial differential equation with respect to the space variable leads to a second order ordinary differential system M(h)$\ddot q$(t)+K(h)q(t)=F(t). The discretization parameter h gives typically the size of a cell, the number of such cells being held constant during the simulation. When the domain evolves with the time, the parameter h is allowed to vary, and one has to solve M(h(t))$\ddot q$(t)+K(h(t))q=F(t).

We shall give evidence in this paper that the results given by such methods are false, as opposed to those obtained by using the concept of convected dense family defined in [2]. One may find in this reference a new proof of the existence of solution for the continuous problem which generalizes the Galerkin method on basis convected from Ω_t to Ω₀. This approach gives a practical way to generate the convergent numerical solutions we are looking for.