Convergence of a two-grid algorithm for the control of the wave equation
Convergence of a two-grid algorithm for the control of the wave equation
复制标题
波动方程控制的二网格算法的收敛性
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
E. Zuazua
中科院分区：
文献类型：
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作者：
L. Ignat;E. Zuazua
本文分析了二维波动方程有限差分空间半离散的边界可观测性问题。我们证明了由Glowinski引用{0763.76042}引入的双网格预条件所得到的解的一致（相对于网格大小）边界可观测性。我们的方法使用以前已知的低频解的均匀可观测性不等式和二进谱时间分解。结果证明了计算波动方程边界控制的双网格算法的收敛性。该方法可应用于任何空间维度，适用于更一般的域和其他离散化方案。
We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski cite{0763.76042}. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.