A full numerical Galerkin method for solving the sound-soft acoustic scattering by smooth convex curves in 2D

Authors: Víctor Domínguez and Ivan G. Graham.

In this work we study the practical implementation of a Galerkin scheme for solving a boundary integral equation appearing in the simulation of high-frequency acoustic plane wave scattering by a 2D convex object. The method, proposed by V. Dominguez, I.G. Graham and V.P. Smyshliaev in 2006, was designed by using all the asymptotic information on the form of the solution and incorporating it in the discrete space.

The method is not robust in $k$, the wave-number, but suffers from a very weak deterioration for increasing values of $k$. Hence, typically an increase of the degrees of freedom as $k^{1/9}$ is enough to preserve the accuracy of the solution as $k$ grows up to infinite.

Regarding the numerical implementation, the number of degree of freedom is usually very small, but any entry of the corresponding matrix requires the evaluation of a double integral of a highly oscillatory function. Hence standard quadrature formulas are prohibitively expensive since the cost of such rules is proportional to $k^2$. In this talk we show how these integrals can be efficiently approximated. Roughly speaking, we proceed as follows. By using suitable changes of variables, we rewrite the oscillating term in a simpler form, namely as a complex exponential which depends only on the variable of the outer integral. Therefore standard quadrature rules can be applied for evaluating the inner integral. The initial problem is then reduced to approximate a single integral of a fast oscillatory function, which is computed using Filon-type quadrature rules. These strategies makes the Galerkin method with quadrature competitive and efficient in the high frequency regimen.