Optimal convergence recovery for the Fourier-finite-element approximation of Maxwell's equations in nonsmooth axisymmetric domains

Abstract

Three-dimensional time-harmonic Maxwell's problems in axisymmetric domains Ω ˆ 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 N ( N → ∞ ) , with the finite element approximation of the Fourier coefficients on the plane meridian domain Ω a ⊂ R + 2 of Ω ˆ with mesh size h ( h → 0 ) . The singular behaviors of the Fourier coefficients near angular points of the domain Ω a 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 H 1 ( Ω ˆ ) 3 is of the order O ( h + N −1 ) as known for the classical Fourier-finite-element approximation of problems with regular solutions.