Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation
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Abstract
<p style='text-indent:20px;'>We study the boundary observability of the 1-D homogeneous wave equation when using a Legendre-Galerkin semi-discretization method. It is already known that spurious high frequencies are responsible for its lack of uniformity with respect to the discretization parameter [<xref ref-type="bibr" rid="b4">4</xref>] which may prevent convergence in the approximation of the associated controllability problem. A classical remedy is to filter out the highest frequency components but this comes with a high computational cost in several space dimensions. We present here three remedies: a spectral filtering method, a mixed formulation (already used in the context of finite element method [<xref ref-type="bibr" rid="b14">14</xref>]) and a Nitsche's method. Our numerical results show that the uniform boundary observability inequalities are recovered. On the other hand, surprisingly, none of them seem to provide the trace (or direct) inequality uniformly, a property used to prove the convergence of the numerical controls [<xref ref-type="bibr" rid="b11">11</xref>]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.
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