In the analysis of the $h$-version of the finite-element method (FEM), with fixed polynomial degree $p$, applied to the Helmholtz equation with wavenumber $k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here $C_{\rm sol}$ is the norm of the Helmholtz solution operator, normalised so that $C_{\rm sol} \sim k$ for nontrapping problems. In the $\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition $\textit{either}$ realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball $\textit{or}$ approximated either by a radial perfectly-matched layer (PML) or an impedance boundary condition. Previously, such bounds for $p>1$ were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalisation of the "elliptic-projection" argument (the argument used to obtain the result for $p=1$) which can be applied to a wide variety of abstract Helmholtz-type problems.

翻译：在应用固定多项式次数 $p$ 的 $h$ 型有限元方法（FEM）分析波数 $k\gg 1$ 的Helmholtz方程时，$\textit{渐近区域}$ 指 $(hk)^p C_{\rm sol}$ 充分小且Galerkin解序列拟最优的情形；这里 $C_{\rm sol}$ 是Helmholtz解算子的范数，对非捕获问题归一化为 $C_{\rm sol} \sim k$。在 $\textit{预渐近区域}$ 中，预期当 $(hk)^{2p}C_{\rm sol}$ 充分小时（对于物理数据），Galerkin解的相对误差可控地小。本文针对变系数Helmholtz方程，在Dirichlet、Neumann或可穿透障碍物（或其组合）外部区域，且辐射条件$\textit{或}$通过球边界上的Dirichlet-to-Neumann映射精确实现，$\textit{或}$采用径向完美匹配层（PML）或阻抗边界条件近似实现的情形下，证明了预渐近区域中的自然误差界。此前，$p>1$ 的情形仅对以阻抗边界条件近似辐射条件的Dirichlet障碍物有效。本文结果通过“椭圆投影”论证（用于得到 $p=1$ 结论的论证方法）的新型推广获得，该推广可广泛适用于各类抽象Helmholtz型问题。