Spectral Element Methods for Axisymmetric Stokes Problems

The approximation of the Stokes problem in axisymmetric geometries using the spectral element method is considered. The presence of the volume element r dr dz in the weak formulation of the problem is shown to be a potential source of difficulty. The discrete equations associated with nodes on the axis of symmetry can lead to a degeneracy in the global system of equations. This difficulty is resolved by incorporating the factor r into the weight function for spectral elements adjacent to the axis of symmetry and using appropriate basis functions in these elements in the radial direction. Properties of the Jacobi polynomials are used to construct the elements of the modified method. Numerical results are presented demonstrating some of the features of the proposed approach.