Quadrature error estimates for kernels with logarithmic singularity

Boundary integral methods are a powerful tool to solve partial differential equations by reformulating them as integral equations over the boundary of the domain. When dealing with boundary integral methods, and in particular with the numerical integration of layer potentials, it is essential to estimate the magnitude of the error associated with the underlying quadrature rule. As the evaluation point approaches the boundary, the integral becomes nearly-singular and the associated quadrature error increases rapidly. Being able to estimate such error is needed to identify when the accuracy becomes inadequate, and the use of a specialized quadrature method is required. In this work we provide accurate quadrature error estimates for the Gauss-Legendre and the trapezoidal rules in computing two-dimensional layer potentials with logarithmic singularities.