A Fully Discrete Fast Fourier–Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

We develop a fully discrete fast Fourier–Galerkin method for solving a boundary integral equation for the biharmonic equation by introducing a quadrature scheme for computing the integrals of non-smooth functions that appear in the Fourier–Galerkin method. A key step in developing the fully discrete fast Fourier–Galerkin method is the design of a fast quadrature scheme for computing the Fourier coefficients of the non-smooth kernel function involved in the boundary integral equation. We prove that with the proposed quadrature algorithm, the total number of additions and multiplications used in generating the compressed coefficient matrix for the proposed method is quasi-linear (linear with a logarithmic factor), and the resulting numerical solution of the equation preserves the optimal convergence order. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed method.