Stability Analysis of Quadrature Methods for Two-Dimensional Singular Integral Equations

In this paper we apply a quadrature method based on the tensor product trapezoidal rule to the solution of a singular integral equation over the two-dimensional torus. We prove that this method is stable if and only if a certain numerical symbol does not vanish. For a special kernel function, we present a plot of numerically computed symbol values and, for symmetric kernels (Mikhlin-Giraud kernels), we show that the symbol is di erent from zero if the singular integral operator is invertible. Finally, we prove the convergence of our method and present numerical tests.