The Poisson equation in axisymmetric domains with conical points

This paper analyzes the effects of conical points on the rotation axis of axisymmetric domains Ω^⊂R3 on the regularity of the Fourier coefficients un(n∈Z) of the solution u^ of the Dirichlet problem for the Poisson equation -Δu^=f^ in Ω^. The asymptotic behavior of the coefficients un near the conical points is carefully described and for f^∈L2(Ω^), it is proved that if the interior opening angle θc at the conical point is greater than a certain critical angle θ*, then the regularity of the coefficient u0 will be lower than expected. Moreover, it is shown that conical points on the rotation axis of the axisymmetric domain do not affect the regularity of the coefficients un,n≠0. An approximation of the critical angle θ* is established numerically and a priori error estimate for the Fourier-finite-element solutions in the norm of W21(Ω^) is given.