We consider an ultra-weak first order system discretization of the Helmholtz equation. By employing the optimal test norm, the `ideal' method yields the best approximation to the pair of the Helmholtz solution and its scaled gradient w.r.t.~the norm on $L_2(\Omega)\times L_2(\Omega)^d$ from the selected finite element trial space. On convex polygons, the `practical', implementable method is shown to be pollution-free when the polynomial degree of the finite element test space grows proportionally with $\log \kappa$. Numerical results also on other domains show a much better accuracy than for the Galerkin method.

翻译：我们考虑亥姆霍兹方程的超弱一阶系统离散化。通过采用最优检验范数，该“理想”方法从选定的有限元试探空间出发，在$L_2(\Omega)\times L_2(\Omega)^d$范数下，为亥姆霍兹解及其缩放梯度对提供最佳逼近。在凸多边形区域上，当有限元检验空间的多项式阶数随$\log \kappa$成比例增长时，该“实用”、可执行的方法被证明是无污染的。其他区域上的数值结果也显示出比伽辽金方法更高的精度。