AbstractWe examine the numerical approximation of the integral equation (λ −K)u=f, whereKis the double layer (harmonic) potential operator on a closed polyhedral surface in ℝ3and λ, ∣λ∣≥1, is a complex constant. The solution is approximated by Galerkin's method, which is based on piecewise polynomials of arbitrary degree on graded triangulations. By utilizing spline spaces which are modified in that the trial functions vanish on some of the triangles closest to the vertices and edges, we investigate the stability of this method inL2. Furthermore, the use of suitably graded meshes leads to the same quasioptimal error estimates as in the case of a sm