ENSURING WELL-POSEDNESS BY ANALOGY - STOKES PROBLEM AND BOUNDARY CONTROL FOR THE WAVE-EQUATION

In this article we give a comparative discussion of the finite element approximation of two partial differential equation problems. These two problems which are apparantly quite unrelated are the Stokes problem for incompressible viscous flow and an exact boundary controllability problem for the wave equation. We show that straightforward discrete approximations to these problems yield approximate problems which are ill-posed. The analysis of the ill-posedness of the above problems shows an identical cause, namely the strong damping of the high frequency modes, beyond a critical wave number. From this analogy, a well-known cure for the discrete Stokes problem, i.e., using more accurate approximations for velocity than for pressure, provides a simple way to eliminate the ill-posedness of the discrete exact boundary controllability problem. Numerical examples concerning the control problem testify about the soundness of the new approach. To conclude this paper one takes advantage of the previous analysis to give a brief discussion of the wavelet approximation of the Stokes problem, for Dirichlet boundary conditions. © 1992.