Optimal convergence recovery for the Fourier-finite-element approximation of Maxwellʹs equations in nonsmooth axisymmetric domains Original Research Article

Three-dimensional time-harmonic Maxwellʹs problems in axisymmetric domains View the MathML sourceΩˆ with edges and conical points on the boundary are treated by means of the Fourier-finite-element method. The Fourier-fem combines the approximating Fourier series expansion of the solution with respect to the rotational angle using trigonometric polynomials of degree View the MathML sourceN(N→∞), with the finite element approximation of the Fourier coefficients on the plane meridian domain View the MathML sourceΩa⊂R+2 of View the MathML sourceΩˆ with mesh size View the MathML sourceh(h→0). The singular behaviors of the Fourier coefficients near angular points of the domain ΩaΩa are fully described by suitable singular functions and treated numerically by means of the singular function method with the finite element method on graded meshes. It is proved that the rate of convergence of the mixed approximations in View the MathML sourceH1(Ωˆ)3 is of the order O(h+N−1)O(h+N−1) as known for the classical Fourier-finite-element approximation of problems with regular solutions.