The Poisson equation in axisymmetric domains with conical points

Abstract

This paper analyzes the effects of conical points on the rotation axis of axisymmetric domains Ω ^ ⊂ R 3 on the regularity of the Fourier coefficients u n ( n ∈ Z ) of the solution u ^ of the Dirichlet problem for the Poisson equation - Δ u ^ = f ^ in Ω ^ . The asymptotic behavior of the coefficients u n near the conical points is carefully described and for f ^ ∈ L 2 ( Ω ^ ) , it is proved that if the interior opening angle θ c at the conical point is greater than a certain critical angle θ * , then the regularity of the coefficient u 0 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 u n , 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 W 2 1 ( Ω ^ ) is given.