Fourier–Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ₃û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H¹(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H¹(Ωˆ) is proved to be 𝒪(h+N⁻¹) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.