First-order system least-squares for the Helmholtz equation

This paper develops a multilevel least-squares approach for the numerical s olution of the complex scalar exterior Helmholtz equation. This second-orde r equation is rst recast into an equivalent first-order system by introduci ng several field variables. A combination of scaled L-2 and H-1 norms is th en applied to the residual of this system to create a least-squares functio nal. It is shown that, in an appropriate Hilbert space, the homogeneous par t of this functional is equivalent to a squared graph norm, that is, a prod uct norm on the space of individual variables. This equivalence to a norm t hat decouples the variables means that standard finite element discretizati on techniques and standard multigrid solvers can be applied to obtain optim al performance. However, this equivalence is not uniform in the wavenumber k, which can signal degrading performance of the numerical solution process as k increases. To counter this difficulty, we obtain a result that charac terizes the error components causing performance degradation. We do this by defining a finite-dimensional subspace of these components on whose orthog onal complement k-uniform equivalence is proved for this functional and an analogous functional that is based only on L-2 norms. This subspace equival ence motivates a nonstandard multigrid method that attempts to achieve opti mal convergence uniformly in k. We report on numerical experiments that emp irically confirm k-uniform optimal performance of this multigrid solver. We also report on tests of the error in our discretization that seem to con r m optimal accuracy that is free of the so-called pollution effect.