Conclusion

For all the reasons mentioned above, spectral methods are very attractive, and their high accuracy extends to all types of partial differential equations. The only drawback is perhaps the difficulty of dealing with complex geometries: at present, there is no automatic meshing software for quadrilaterals, let alone hexahedra, suitable for approximation by the spectral element method, and this has to be done by hand. One way of overcoming this is to couple these spectral approximations with finite elements on a small part of the domain, see section IV of