Conditioning of infinite element schemes for wave problems

Abstract

While a number of infinite element schemes have been implemented for time-harmonic unbounded wave problems in two and three dimensions, stability and conditioning of these schemes can limit the radial order at which they can be applied. In this paper, the choice of radial basis functions is shown to influence the condition number of such schemes. This effect is illustrated for three formulations; the Bettess-Burnett formulation, the conjugated Burnett formulation and the Astley-Leis formulation. Calculated values for the condition number are presented for infinite element schemes based on an orthogonal modal decomposition at the finite element/infinite element interface, and for more conventional infinite element schemes based on a transverse finite element discretization. Ill-conditioning can be avoided in the two conjugated formulations by the selection of a suitable radial basis. In the case of the Bettess-Burnett formulation, the condition number increases rapidly irrespective of the radial basis. The effect of the condition number on the convergence of the various schemes is also discussed.