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 𝐿₂(Ω) ×𝐿₂(Ω)^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 𝑝^∼of the finite element test space grows proportionally with max(log 𝜅, 𝑝²), with 𝑝being the order at trial side. Numerical results also on other domains show a much better accuracy than for the Galerkin method.