An optimal change of variables scheme for single scattering problems

Abstract

In this work we are concentrated on the direct obstacle scattering problem for convex bodies in two dimensions. In order to calculate the scattered eld, we rst need to compute the normal derivative of the total eld on the object's surface. This quantity is the unique solution of a combined eld integral equation which we solve using Galerkin method wherein the approximation spaces depend on the wave number and the geometry of the scatterer. We are particularly focused on the large wave numbers in which the solution has highly oscillating behavior. In order to analyze this solution accurately, we separate the highly oscillating part of it and then study the derivatives of the acquired function. This derivative study gives us the information about the smoothness of the solution and an idea about how to approximate it. As for the geometry of the scatterer, we divide the boundary of the object into subregions regarding where we expect high oscillations. In each region, in order to achieve improved approximations, we choose di erent polynomial bases. In various scenarios, we examine the polynomial bases such as monomial, Lagrange, and Chebyshev. As the wave number increases, in order to obtain better results one needs to formulate these approximation spaces with higher polynomial degrees. However, it includes enormous computational cost and the condition numbers of Galerkin matrices elevate dramatically. The goal of this research is to optimize the choice of approximation spaces so as to improve accuracy of numerical solutions while keeping the number of degrees of freedom independent of frequency, and reduce the condition numbers of the related Galerkin matrices.