Boundary element method for Laplace equation in a ring domain

In this paper, we present a rapid procedure for approximating the solution to the Laplace equation in a ring domain. We develop a new formulation of boundary integral operators and implement an efficient solution approach using the Galerkin boundary element method for Dirichlet and mixed boundary value problems. The corresponding matrix entries are computed efficiently and accurately, and the resulting circulant block structure allows the matrices of the discrete boundary integral operators to be expressed as a product of sparse matrices. Therefore the fast Fourier transform (FFT) has significantly accelerated matrix-to-vector product. Moreover, the discrete Fourier transform (DFT) enables the construction of efficient preconditioners for conjugate gradient algorithms, and provides a robust direct approach for solving the Dirichlet problem. Numerical experiments for both Dirichlet and mixed problems demonstrate the exceptional efficiency and accuracy of the proposed algorithms.