论文标题
用于高波数的Helmholtz方程的可杂交不连续的盖金方法。第一部分：线性案例
Hybridizable Discontinuous Galerkin Methods for Helmholtz Equation with High Wave Number. Part I: Linear case
论文作者
论文摘要
本文介绍了针对Helmholtz方程的线性杂交不连续的Galerkin方法（HDG）的几个方面，该方程在高频下具有阻抗边界条件。首先，对于精确解决方案$ u $及其负梯度$ \ mathbf {q} = - \ nabla u $，对HDG近似近似的波浪号$ K $具有明确依赖的错误估计。结果表明，$ k \ vert u -u_h \ vert_ {l^2（ω）} + \ vert \ vert \ mathbf {q} - \ \ mathbf {q} _h \ vert_ \ vert_ {l^2（Ω） $τ\ eqsim k $，其中$ h $是网格尺寸。请注意，$ \ mathbf {q} _h $中的收敛顺序已满，污染错误为$ O（k^4H^3）$，可以改善存在的结果。其次，通过使用HDG方法的标准后处理过程解决椭圆问题，可以获得分段二次函数$ u_h^*$，以便$ k \ vert u-u-u_h^*\ vert_ \ vert_ {l^2（ω）} = o（k^3H^3H^3+k^4H^3+k^4H^3）$。请注意，后处理过程仅改善插值错误（从$ O（k^2h^2）$到$ O（k^3h^3）$），但留下污染错误$ O（k^4H^3）$不变。第三，分散分析和广泛的数值测试表明，在1D情况下可以完全消除污染效应，并通过选择适当的惩罚参数在2D情况下大大减少。在第二部分中研究了具有高波数的Helmholtz方程的高阶HDG方法的较高级HDG方法的误差分析。
This paper addresses several aspects of the linear Hybridizable Discontinuous Galerkin Method (HDG) for the Helmholtz equation with impedance boundary condition at high frequency. First, error estimates with explicit dependence on the wave number $k$ for the HDG approximations to the exact solution $u$ and its negative gradient $\mathbf{q}=-\nabla u$ are derived. It is shown that $k\Vert u - u_h \Vert_{L^2(Ω)} + \Vert \mathbf{q} -\mathbf{q}_h \Vert_{L^2(Ω)} = O(k^2h^2+k^4h^3)$ under the conditions that $k^3h^2$ is sufficiently small and that the penalty parameter $τ\eqsim k$, where $h$ is the mesh size. Note that the convergence order in $\mathbf{q}_h$ is full and the pollution error is $O(k^4h^3)$, which improve the existent results. Secondly, by using a standard postprocessing procedure from the HDG method for elliptic problems, a piecewise quadratic function $u_h^*$ is obtained so that $k\Vert u-u_h^*\Vert_{L^2(Ω)}=O(k^3h^3+k^4h^3)$. Note that the postprocessing procedure improves only the interpolation error (from $O(k^2h^2)$ to $O(k^3h^3)$) but leaves the pollution error $O(k^4h^3)$ unchanged. Thirdly, dispersion analyses and extensive numerical tests show that the pollution effect can be eliminated completely in 1D case and reduced greatly in 2D case by selecting appropriate penalty parameters. The preasymptotic error analysis of the higher order HDG method for the Helmholtz equation with high wave number is studied in Part II.
