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An error estimate for the Laplace problem with edge singularities

In a neighborhood of concave edge through the origin in a cylindrical region Q the solution of the Laplace equation with zero Dirichlet data can have a decomposition of the form u=u r +(c*E)Xr π/ω sin[(π/ω)(θ-ω₁)] with u r ∈ H²(Q) and c ∈ H 1-π/ω (?) where ω = ω₂-ω₁ ＞ π is the opening angle, χ is a cutoff function and * is the convolution on the edge. We define the approximations u r,h and c h for the regular part ur and the edge flux intensity function c by applying the inverse Fourier transform to the finite element solution of the Laplace problem with parameter. We derive the error estimate ∥u r -u r,h ∥ I,Q +∥c-c h ∥ 1-π/ω,R ≤ Ch 1-e ∥f∥ 0,Q , where e is an arbitrarily small positive number, which shows the rate of convergence faster than that expected by the regularity of the function c.