Robust high-frequency solvers

This work concerns the numerical solutions of the direct obstacle scattering problem in $\mathbb{R}^2$. To this end, we formulate the problem as an equivalent integral equation. We then review the fundamental numerical methods such as Nyström, collocation and Galerkin methods, for integral equations of the second kind. We establish convergence results and error estimates for these methods, and incorporate numerical examples considering different integral equations. Although these methods are very efficient for low frequencies, they can not be utilized for high frequency scenarios as the computational cost grows linearly with the wave number $k$. In this connection, we propose a robust convergent algorithm based on a Galerkin formulation utilizing the geometrical optics ansatz to adapt the approximation spaces to high frequency scattering (by convex obstacles) which overcomes this type of growth in complexity requiring only $\mathcal{O}(k^{\varepsilon})$, $\varepsilon>0$, increase in the degrees of freedom to maintain a given accuracy. Numerical experiments demonstrate the efficiency of our method by exhibiting numerical errors and condition numbers in several scenarios.