Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs
Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs
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DOI：
10.2140/paa.2022.4.225
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Thomas Beck;Y. Canzani;J. Marzuola
中科院分区：
文献类型：
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作者：
Thomas Beck;Y. Canzani;J. Marzuola
我们证明了凸多边形域上涉及阻抗边界条件的一族算子族的定量范数界。通过Dirichlet-to-Neumann算子构造可变介质中Helmholtz散射解的稳健数值构造包括将区域分解成一系列不同尺度的矩形，并在每个子域上构造阻抗-阻抗边界算子。我们的估计特别确保了根据这些子域的阻抗到阻抗的算子来重构原始Dirichlet-to-Neumann算子所需的合并算子的可逆性，并且具有频率上的定量界限。证明中的一个关键步骤是获得阻抗问题解的Neumann和Dirichlet边界迹估计，这两个估计是独立的。除了可变的介质设置外，我们还构造了障碍散射问题中类似合并算子的界。
We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.