Abstract:
We propose a new robust method for the computation of scattering of high-frequency acoustic plane waves by smooth convex objects in 2D.
We formulate this problem by  the direct boundary integral method, using  the classical  combined potential approach.
 By exploiting the known asymptotics  of the solution,  we devise particular expansions, valid in various zones of the boundary,  which express the solution of the integral equation as a  product of  explicit oscillatory functions and  more slowly varying unknown  amplitudes.  The amplitudes are approximated by polynomials (of minimum degree $d$) in each zone using  a Galerkin scheme. We prove that the underlying  bilinear form is continuous  in $L_2$,  with a  continuity constant that grows mildly in the wavenumber $k$. We also show that the bilinear form    is uniformly $L_2$-coercive, independent of $k$, for all  $k$ sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to  circular domains,  but it  also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as $d \rightarrow \infty$ for fixed $k$. We also prove that, as $k \rightarrow \infty$, $d$  has  to increase only very modestly to maintain a fixed error bound ($d \sim k^{1/9}$ is a typical behaviour). Numerical experiments show that the method suffers minimal loss of  accuracy as   $k \rightarrow\infty$, for a fixed number of degrees of freedom. Numerical solutions with a relative  error of about $10^{-5}$ are obtained on domains of size $\cO(1)$ for  $k$ up to $800$ using about $60$ degrees of freedom.