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 [4] 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 [14]) 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 [11]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.

中文翻译：

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一维波动方程的Legendre-Galerkin公式的统一边界可观性

我们使用Legendre-Galerkin半离散化方法研究一维齐次波动方程的边界可观性。众所周知，杂散高频是造成离散化参数缺乏一致性的原因[4可以防止在相关的可控制性问题的近似中收敛。一种经典的补救方法是滤除最高频率分量，但这会在多个空间维度上带来很高的计算成本。在这里，我们提出三种补救措施：频谱滤波方法，混合公式（已经在有限元方法[14]）和Nitsche方法。我们的数值结果表明，统一的边界可观性不等式得以恢复。另一方面，令人惊讶的是，它们似乎都没有均匀地提供痕量（或直接）不等式，该性质用于证明数控系统的收敛性[11]。但是，我们的数值测试表明，当统一的可观察性不等式成立时，可以确保数控的收敛性。