The spectral/hp element method combines the geometric flexibility of classical h-type finite element techniques with the desirable resolution properties of spectral methods. In this approach a polynomial expansion of order P is applied to every elemental domain of a coarse finite element type mesh. These techniques have been applied in many fundamental studies of fluid mechanics [31] and more recently have gained greater popularity in the modelling of wave-based phenomena such as computational electromagnetics [14] and shallow water problems [4] - particularly when applied within a Discontinuous Galerkin formulation.
There are at least two major challenges which arise in developing an efficient implementation of a spectral/hp element discretisation:
In order to design algorithms which are efficient for both low- and high-order spectral/hp discretisations, it is important clearly define what we mean with low- and high-order. The spectral/hp element method can be considered as bridging the gap between the high-order end of the traditional finite element method and low-order end of conventional spectral methods. However, the concept of high- and low-order discretisations can mean very different things to these different communities. For example, the seminal works by Zienkiewicz & Taylor [36] and Hughes list examples of elemental expansions only up to third or possibly fourth-order, implying that these orders are considered to be high-order for the traditional h-type finite element community. In contrast the text books of the spectral/hp element community typically show examples of problems ranging from a low-order bound of minimally fourth-order up to anything ranging from 10th-order to 15th-order polynomial expansions. On the other end of the spectrum, practitioners of global (Fourier-based) spectral methods [12] would probably consider a 16th-order global expansion to be relatively low-order approximation.
One could wonder whether these different definitions of low- and high-order are just inherent to the tradition and lore of each of the communities or whether there are more practical reasons for this distinct interpretation. Proponents of lower-order methods might highlight that some problems of practical interest are so geometrically complex that one cannot computationally afford to use high-order techniques on the massive meshes required to capture the geometry. Alternatively, proponents of high-order methods highlight that if the problem of interest can be captured on a computational domain at reasonable cost then using high-order approximations for sufficiently smooth solutions will provide a higher accuracy for a given computational cost. If one however probes even further it also becomes evident that the different communities choose to implement their algorithms in different manners. For example the standard h-type finite element community will typically uses techniques such as sparse matrix storage formats (where only the non-zero entries of a global matrix are stored) to represent a global operator. In contrast the spectral/hp element community acknowledges that for higher polynomial expansions more closely coupled computational work takes place at the individual elemental level and this leads to the use of elemental operators rather than global matrix operators. In addition the global spectral method community often make use of the tensor-product approximations where products of one-dimensional rules for integration and differentiation can be applied.