An analysis of discretizations of the Helmholtz equation in L2L2 and in negative norms

For a model Helmholtz problem at high wavenumber kk we present a wavenumber-explicit error analysis in the L2L2- and H−1H−1-norms for the Galerkin FEM. For the convergence in L2L2, we show that the lowest order case p=1p=1 is special in that the relative error in L2L2 scales at best with kk whereas it does not for higher order discretizations. An alternative to the Galerkin method with better dispersion properties is the optimally blended spectral–finite element scheme of Ainsworth and Wajid (2010). For this method, we present an error analysis in L2L2 for the lowest order case p=1p=1 in one dimension, showing that the L2L2-error is improved by a factor kk compared to the lowest order Galerkin FEM.