Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains

Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the boundary if a fixed quadrature rule is used. Firstly, we analyze this error for Laplace's equation with analytic density and the global periodic trapezoid rule, and find an intimate connection to the complexification of the boundary parametrization. Our main result is then a simple and efficient scheme for accurate evaluation up to the boundary for single- and double-layer potentials for the Laplace and Helmholtz equations, using surrogate local expansions about centers placed near the boundary. The scheme---which also underlies the recent QBX Nystr\"om quadrature---is asymptotically exponentially convergent (we prove this in the analytic Laplace case), requires no adaptivity, generalizes simply to three dimensions, and has O(N) complexity when executed via a locally-corrected fast multipole sum. We give an example of high-frequency scattering from an obstacle with perimeter 700 wavelengths long, evaluating the solution at $2\times 10^5$ points near the boundary with 11-digit accuracy in 30 seconds in MATLAB on a single CPU core.