本文考虑在具有解析边界和阻抗边界条件的区域中、高波数 k 下的时谐 Maxwell 方程。作者发展了一种波数显式的稳定性与正则性理论，将解分解为一个在 k 上一致受控、具有有限 Sobolev 正则性的部分，以及一个解析部分。利用该正则性，在 k 显式尺度解析条件下证明了基于网格尺寸 h、阶数为 p 的 Nedelec 元的 Galerkin 离散化的拟最优性；该条件为：a) kh/p 足够小，且 b) p/ln k 有下界。
英文摘要：
The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nedelec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that a) kh/p is sufficient small and b) p/\ln k is bounded from below.
The time-harmonic Maxwell equations at high wavenumber $k$ are discretized by edge elements of degree $p$ on a mesh of width $h$. For the case of a ball and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the $k$-explicit scale resolution condition that a) $kh/p$ is sufficient small and b) $p/\log k$ is bounded from below.
Hyperbolic model for Helmholtz equation with impedance boundary conditions
Ivane Javakhishvili Tbilisi State University（伊万·贾瓦赫什维利第比利斯国立大学）；Institute of Energy and Climate research 8: Troposphere, Research Center Jülich（于利希研究中心能源与气候研究所8：对流层）
A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber $k$, then the Galerkin method is quasioptimal provided that $hk/p \leq C_1$ and $p\geq C_2 \log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent of $k,h,$ and $p$. The significance of this result is that if $hk/p= C_1$ and $p=C_2\l…
An explicit factorization of the Green's function for an acoustic half-space problem with impedance boundary conditions into an oscillatory exponential and a slowly varying function
针对阻抗边界条件下的 d 维声学半空间问题，提出 Green 函数的统一新表示，将其分解为振荡复指数与可高效多项式逼近的缓变函数。
Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular d…
A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM). Wave-number-explicit stability and regularity estimates and the exponential convergence are proved for the nonlinear truncated PML problem. Preasymptotic error estimates are obtained for the FEM, where the logarithmic factors in h required by the previous results for the NLH with impedance boundary condition are removed in the case of two dimensions. Moreover, local quadratic …
We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems with absorption and increasing wavenumber $k$. These problems are discretized using the Galerkin method with nodal conforming finite elements of any (fixed) order on meshes with diameter $h = h(k)$, chosen to maintain accuracy as $k$ increases. The action of the preconditioner requires solution of independent (parallel) subproblems (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $δ\leq H$. The solutions of these subproblems are linked together using prolongation/…
In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k\to\infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods). It is well known that t…
