On accuracy conditions for the numerical computation of waves

The Helmholtz equation ( Δ+ 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, 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.