A fast solver for a hypersingular boundary integral equation

In this paper we develop a fast fully discrete Galerkin method for a hypersingular boundary integral equation based on trigonometric polynomials. Usually, the Galerkin method for this equation leads to a discrete linear system with a dense coefficient matrix. When the order of the linear system is large, the complexity for generating the fully discrete linear system and then solving the corresponding linear system is huge. For this purpose, we propose a truncation strategy to compress the dense coefficient matrix into a sparse matrix, and then we use a numerical integration method to generate the fully discrete truncated linear system, which will be solved by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating and solving the fully discrete truncated linear system is estimated to be linear up to a logarithmic factor. Numerical examples complete the paper.