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：注释： 1. 需要将Helmholtz保留为英文专有名词。 2. 对于$L_2(\Omega)\times L_2(\Omega)^d$中的$L_2$，需要以英语形式保留。 3. 在中文翻译中，“理想”和“实用”这两个词，是对照“ideal”和“practical”中给出的，因为它们在这里需要用作一个专业术语的翻译。