A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

The main purpose of this paper is to develop a fast numerical method for solving the first kind boundary integral equation, arising from the two-dimensional interior Dirichlet boundary value problem for the Helmholtz equation with a smooth boundary. This method leads to a fully discrete linear system with a sparse coefficient matrix. We observe that it requires a nearly linear computational cost to produce and then solve this system, and the corresponding approximate solution obtained by this proposed method preserves the optimal convergence order. One numerical example demonstrates the efficiency and accuracy of the proposed method.