一种使用威尔逊基求解高频散射问题的高效伽辽金方法
An efficient Galerkin method for high-frequency scattering problems using Wilson bases
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中文总结 AI 辅助
针对波散射问题，提出基于微局部化基函数的伽辽金离散化新方案，以威尔逊基取代伽博框架解决条件数问题，能在大波数下以低自由度实现一致精确，建立了误差估计和条件数界并给出数值示例。
中文摘要 AI 辅助
我们提出了一种基于微局部化基函数的波散射问题伽辽金离散化新方案。对于大波数\(k\)，该方法能以自由度仅按\(k^{d - 1/2}\)缩放的方式实现一致精确，且生成本质上稀疏的线性系统。相比之下，有限元方法需至少按\(k^d\)缩放自由度才能有相同特性。此前作者提出的基于伽博框架的类似方法存在严重条件数问题，本文用威尔逊基取代伽博框架解决了此问题。我们严格建立了该方法的误差估计和条件数界，并给出一维数值示例说明理论结果。
英文摘要
We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers $k$ with a number of degrees of freedom only scaling as $k^{d-1/2}$, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as $k^d$ to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.
