A fast direct solver for boundary integral equations in two dimensions

An algorithm is presented that rapidly solves systems of linear algebraic equations associated with the discretization of boundary integral equations in two dimensions. The algorithm is "fast" in the sense that its asymptotic complexity is $O(N\log N)$, where $N$ is the number of nodes in the discretization. Unlike previous fast techniques based on iterative solvers, the present algorithm constructs a sparse factorization of the inverse of the matrix; thus it is suitable for problems involving relatively ill-conditioned matrices, and is particularly efficient in situations involving multiple right hand sides. The performance of the scheme is illustrated with several numerical examples.