A pollution-free ultra-weak FOSLS discretization of the Helmholtz equation

We consider an ultra-weak first order system discretization of the Helmholtz equation. When 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 essentially whenever the order $\tilde{p}$ of the finite element test space grows proportionally with $\max(\log \kappa,p^2)$, with $p$ being the order at trial side. Numerical results also on other domains show a much better accuracy than for the Galerkin method.