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

Abstract

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 -pΔ 3 u = f in three-dimensional axisymmetric domains Ω. Here, p 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 H 1 (Ω), if the interface is smooth or if it meets the boundary of Ω at some edge. In general, the solution u 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 Ω. The rate of convergence of the combined approximation in H 1 (Ω) 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.