Boundary element method for elliptic differential equations with a small parameter (Q1899328)

For the two-dimensional Helmholtz equation \(\Delta u- \lambda^2 u=0\) with real \(\lambda\) the following numerical method is described: As usual in the direct boundary element method the solution is seeked as the sum of a simple and a double layer potential. Since the parameter \(\varepsilon= 1/\lambda\) is assumed to be very small, the fundamental solution \(w(r)= (i/4) H_0^{(1)} (i\lambda r)\) as well as the derivative \(\partial w/\partial r= (\lambda/4) H_1^{(1)} (i\lambda r)\) is replaced by the first term of the asymptotic expansion of the Hankel functions \(H_0^{(1)}\), \(H_1^{(1)}\) for large arguments. After this substitution both potentials have no jumps and the so derived boundary integral equation is discretized.