Solvability of discrete Helmholtz equations

Abstract We study the unique solvability of discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin finite element method. Well-posedness discrete equations is typically investigated by applying compact perturbation argument to continuous so that `sufficiently rich' discretization results in small' and well-posedness inherited via Fredholm’s alternative. The qualitative notion rich', however, involves unknown constants only asymptotic nature. Our paper focussed on fully approach mimicking tools for proving directly level. In this way, computable criterion derived, which certifies without relying an argument. By novel we obtain (a) new existence uniqueness $hp$-FEM problem, (b) examples meshes such becomes unstable (Galerkin matrix singular) (c) simple checking Algorithm MOTZ `marching-of-the-zeros', guarantees posteriori way given mesh certified well-posed discretization.