Various aspects that may help to enhance the implementation of the spectral/hp element method have been considered. A first challenge encountered is to implement the method and the corresponding algorithms in a digestible, generic and coherent manner. Therefore, we first of all demonstrate how the mathematical structure of a spectral/ hp element discretisation can be encapsulated in an object-oriented environment, leading to a generic and flexible spectral/hp software library. Secondly, we

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... nt a generic framework for time-stepping partial differential equations. Based upon the unifying concept of General Linear Methods, we have designed an object-oriented framework that allows the user to apply a broad range of time-stepping schemes in a unified fashion. The spectral/hp element method can be considered as bridging the gap between the - traditionally low-order - finite element method on one side and spectral methods on the other side. Consequently, a second challenge which arises in implementing the spectral/hp element methods is to design algorithms that perform efficiently for both low- and high-order spectral/hp discretisations, as well as discretisations in the intermediate regime. In this thesis, we describe how the judicious use of different implementation strategies for the evaluation of spectral/hp operators can be employed to achieve high efficiency across a wide range of polynomial orders. Furthermore, we explain how the multi-level static condensation technique can be applied as an efficient direct solution technique for solving linear systems that arise in the spectral/hp element method. Finally, based upon such an efficient implementation of the spectral/hp element method, we analyse which spectral/hp discretisation (that is, which specific combination of mesh size h and polynomial order P) minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy. We investigate this question for a set of both smooth and non-smooth problems.