Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis
Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis
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DOI：
10.1093/imanum/drv026
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发表时间：
2016-04
影响因子：
2.1
通讯作者：
S. Ervedoza;A. Marica;E. Zuazua
中科院分区：
文献类型：
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作者：
S. Ervedoza;A. Marica;E. Zuazua
我们为一维波动方程的有限差分和有限元逼近建立非均匀数值网格，确保所有数值解都像连续解一样到达边界，因为可以通过边界观察到完整的离散能量测量，相对于网格尺寸均匀。非均匀网格的构造是通过对均匀网格进行凹同态变换而得到的，当网格逼近右边界时，网格会变得越来越细。对于均匀网格，已知高频数值波包传播非常缓慢，从未到达边界。我们的研究结果表明，这种病理可以避免采取适当的非均匀网格。这也使我们能够建立收敛的数值算法的波动方程的边界控制的近似。
We build nonuniform numerical meshes for the finite difference and finite element approximations of the one-dimensional wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a nonuniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without ever getting to the boundary. Our results show that this pathology can be avoided by taking suitable nonuniform meshes. This also allows us to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.