Fourier series and integral equation method for the exterior Stokes problem

Abstract

The exterior Stokes problem between two parallel planes that are separated by a prismatic cylinder is extended to the interior of the prism by requiring the continuity of the velocity across the lateral faces. The well-posedness of the exterior-interior problem is proved in suitable weighted Sobolev spaces. The solution is represented by Fourier series in the z-variable. The Fourier coefficients, solutions of auxiliary two-dimensional exterior-interior problems, are analyzed by viewing them as boundary integral equations of potential theory and global regularity of the densities, is established in weighted Sobolev spaces of traces. A boundary element method, with suitably refined mesh size, is implemented for the numerical treatment of the Fourier coefficients. This provides optimal convergent semi-and fully-discrete spectral methods of Fourier-Galerkin type.