In this paper, a generalized finite element method (GFEM) with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, $(i)$ wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains; $(ii)$ a quasi-optimal convergence of the method is established by assuming that the size of subdomains is $O(1/k)$ ($k$ is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as $h\rightarrow 0$ ($h$ denoting the mesh size) towards the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [I. Babuska and J. M. Melenk, Int.\;J.\;Numer.\;Methods Eng., 40 (1997), pp.~727--758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient non-iterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.
翻译:在本文中,正在系统地研究一种通用的有限元素方法(GFEM),该方法具有解决高频混杂 Helmholtz 问题的最佳本地近似空间;当地空间是用在通用协调空间中定义的、经过仔细设计的本地二元值问题所选择的精密源数构建的。在连续和离散的层次上,获得当地和全球近似差错的明显和近乎指数衰减率率,而没有假定子焦距值大小;如果假设子焦距值的大小为O(1/k)美元(美元为波数),则该方法的准最佳离差值趋同度就会被确定为该方法的连续本地差值(美元为$hrightrorrow 0美元,注意到离差值的离差值值值值为美元),那么在连续水平上将统一方法的平面波偏差与本地差的尺寸之间的新异共振效果(E.I\\\\\Serma) 标准方法。