Harmonic extension elements : Eigenproblems and error estimation

A non-intrusive extension to the standard p-version of the finite element method, so-called harmonic extension elements, is studied in the context of eigenproblems. The standard polynomial shape functions are replaced where appropriate with harmonic extensions of the boundary restrictions of the standard shape functions or solutions to a local Poisson problem. The reference elements are adapted to include extensions in order to ensure a conforming discretisation even if the meshes are not conforming. The hierarchic structure of the extension basis means that auxiliary space error estimators of the p-version of the finite element method are directly applicable. The additional computational workload in construction of the required extensions can be reduced using symmetries and multimesh techniques. The numerical experiments demonstrate the efficiency of the proposed extension resulting in exponential convergence in the quantities of interest if the mesh is properly graded.