On accuracy conditions for the numerical computation of waves (Q1104715)

The Helmholtz equation \((\Delta +K^ 2n^ 2)u=f\) 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 \(K^ 3h^ 2\) is shown to determine the accuracy in the \(L^ 2\) norm for a second-order discretization method applied to several propagation models.