The Extended Spectral Element Method for the Approximation of Discontinuous Functions

High-order polynomial approximations of discontinuous functions give rise to oscillations in the vicinity of the discontinuity known as Gibbs phenomenon. Enrichment of the basis using discontinuous functions is shown to remove these oscillations and to recover the convergence properties generally associated with the spectral approximation of smooth functions. The convergence properties of the enriched method, known as the extended spectral element method (XSEM) are studied and optimal error estimates are derived. The extension of these ideas to the immersed boundary method (IBM) is considered. The IBM is typically used for problems with an interface or discontinuity that is unfitted to the underlying computational mesh. An extended basis is used to approximate the pressure and the implication of this for the inf-sup for the Stokes problem is investigated.