Boundary obeservability for the space semi-discretization for the 1-d wave equation
Boundary obeservability for the space semi-discretization for the 1-d wave equation
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DOI：
10.1051/m2an:1999123
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发表时间：
1999-03
期刊：
影响因子：
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通讯作者：
J. Infante;E. Zuazua
中科院分区：
文献类型：
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作者：
J. Infante;E. Zuazua
我们考虑具有齐次狄利克雷边界条件的有界区间内一维波动方程的空间半离散化。我们分析了边界可观测性问题，即解的总能量是否可以根据净间距 h → 0 时集中在边界上的能量来统一估计的问题。我们证明，由于数值格式在高频引入的杂散模式，不存在这样的统一边界。然而，我们证明了由离散系统的低频生成的解子空间中的统一界限。当 h → 0 时，有限维空间增加并最终覆盖整个空间。因此，当网格尺寸趋于零时，我们恢复了连续系统众所周知的可观测性属性，作为离散可观测性估计的极限。我们考虑有限差分和有限元半离散化。
We consider space semi-discretizations of the 1- d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e. , the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations.