A High-Order Method for Evaluating Derivatives of Harmonic Functions in Planar Domains

Abstract

We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions that are specified by their Dirichlet data in planar domains. The tangential derivative of the given Dirichlet data is used to form a complementary Neumann problem, whose solution is a harmonic conjugate of the function whose derivatives we seek. We use a high-order Nyström method to compute the Dirichlet trace of the harmonic conjugate on the domain boundary. The tangential derivative of this harmonic conjugate, effected via an FFT, is the normal derivative of the original function. Because the original and conjugate harmonic functions are the real and imaginary parts of a complex analytic function, we are able to use Cauchy's integral formulas to compute function values and derivatives inside the domain. Several numerical experiments, on smooth domains and domains with corners, illustrate the rapid convergence and high accuracy of the proposed approach.