On accuracy conditions for the numerical computation of waves
The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models.