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

Abstract: 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, thepractical', 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.