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Efficient spectral element methodsfor partial differential equations

In this thesis we applied a spectral element approximation to some elliptic partial differential 
equations. We demonstrated the difficulties related to the approximation of a discontinuous function in which the discontinuity is not fitted to the computational 
mesh. Such a situation gives rise to the Gibbs phenomenon. A h− p spectral element equivalent of the eXtended Finite Element Method (XFEM), which we termed 
the eXtended Spectral Element Method (XSEM) was developed. This was applied to some model problems. XSEM removes some of the oscillations caused by Gibbs 
phenomenon. We then explained that when approximating a discontinuous function, XSEM is able to capture the discontinuity precisely. We derive spectral element error 
estimates. The convergence of the approximations is studied. 
We have introduced several enrichment functions with the purpose of improving the approximation of discontinuous functions. In particular we have considered the twodimensional Poisson equation. Unfortunately, this implementation of XSEM was not able to recover spectral convergence. An alternative idea in which the discontinuity is constrained within a spectral element produces accurate SEM approximation. 
The Stokes problem was considered and solved using SEM coupled with an iterative PCG method. The zero volume condition on the pressure is satisfied identicaly using an alternative formulation of the continuity equation. Furthermore, we investigated the dependence of the accurency of the spectral element approximation on the 
weighting factor as well as the convergence properties of the preconditioner. An efficient and robust preconditioner is constructed for the Stokes problem. Exponential convergence was attained.