论文标题
$ hp $ -fem的vavenumber-strapic融合，用于全空间异质Helmholtz方程，具有光滑系数
Wavenumber-explicit convergence of the $hp$-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients
论文作者
论文摘要
$ HP $ -FEM的收敛理论应用于各种恒定的Helmholtz问题，在论文中率先[Melenk-Sauter，2010]，[Melenk-Sauter，2011]，[Esterhazy-Melenk，2012]，[Melenk-Parsania-Sauter，2013年]，2013年]。该理论表明，如果解决方案运算符在波数$ k $中是多项式界定的，那么Galerkin方法是quasioptimal的，前提是$ hk/p \ leq c_1 $和$ p \ geq c_2 \ log k $，其中$ c_1 $很小，$ c_2 $，$ c_2 $，$ c_2 $ andife offore $ $ $ $ $ k.该结果的意义是，如果$ hk/p = c_1 $和$ p = c_2 \ log k $，则可以实现准timimation，而自由度的总数与$ k^d $成正比；即，$ HP $ -FEM不会遭受污染效应。本文证明了异质（即可变效率）Helmholtz方程的类似准timigation结果，它以$ \ Mathbb {r}^d $，$ d = 2,3 $呈现，带有sommerfeld Infinity和$ c^\ infty $ ceefficients。在平面波散射问题的特定情况下，我们还证明了Galerkin解决方案的相对误差的束缚。这些是用于helmholtz方程的$ hp $ -FEM的波数义相互收敛的第一个结果，具有可变系数。
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 $p=C_2\log k$, then quasioptimality is achieved with the total number of degrees of freedom proportional to $k^d$; i.e., the $hp$-FEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable-coefficient) Helmholtz equation, posed in $\mathbb{R}^d$, $d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^\infty$ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem. These are the first ever results on the wavenumber-explicit convergence of the $hp$-FEM for the Helmholtz equation with variable coefficients.
