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The quadrature error associated with a regular quadrature rule for evaluationof a layer potential increases rapidly when the evaluation point approaches thesurface and the integral becomes nearly singular. Error estimates are needed todetermine when the accuracy is insufficient and a more costly specialquadrature method should be utilized.  The final result of this paper are such quadrature error estimates for thecomposite Gauss-Legendre rule and the global trapezoidal rule, when applied toevaluate layer potentials defined over smooth curved surfaces in R^3. Theestimates have no unknown coefficients and can be efficiently evaluated giventhe discretization of the surface, invoking a local one-dimensionalroot-finding procedure. They are derived starting with integrals over curves,using complex analysis involving contour integrals, residue calculus and branchcuts. By complexifying the parameter plane, the theory can be used to deriveestimates also for curves in in R^3. These results are then used in thederivation of the estimates for integrals over surfaces. In this procedure, wealso obtain error estimates for layer potentials evaluated over curves in R^2.Such estimates combined with a local root-finding procedure for theirevaluation were earlier derived for the composite Gauss-Legendre rule for layerpotentials written on complex form [4]. This is here extended to providequadrature error estimates for both complex and real formulations of layerpotentials, both for the Gauss-Legendre and the trapezoidal rule.  Numerical examples are given to illustrate the performance of the quadratureerror estimates. The estimates for integration over curves are in many casesremarkably precise, and the estimates for curved surfaces in R^3 are alsosufficiently precise, with sufficiently low computational cost, to bepractically useful.

Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions