Abstract
% Based on the boundary integral equation reduced from the sound-soft high frequency scattering problem in two dimensions, we firstly convert the density to slow oscillatory from high oscillatory by using the asymptotic behaviour as $k \\to +\\infty$ of density in a strictly convex bounded region. Furthermore, the equivalence of second Fredholm integral equation is obtained. Using the composite mesh with a polynomial grading on the illuminated zone and a geometric grading on the shadow zone, we obtain that the convergence is $O(k^{-\\frac {1} {24}}) $ $ (k\\gg 1)$. Moreover, examples are given for demonstrating that our method has good accuracy.
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