Spectral Element and hp Methods

Spectral/hp element methods provide high-order discretization, which is essential in the longtime integration of advection-diffusion systems. Here we review the main formulations for simulations of incompressible, compressible, and plasma viscous flows, and present several examples with some emphasis on moving domains. The first generation of (nodal) spectral elements was limited to relatively simple geometries and smooth solutions. However, the new generation of spectral/hp elements, consisting of both nodal and modal forms, can handle very complex geometries by using unstructured grids and can capture strong shocks by employing discontinuous Galerkin methods. New implementation approaches have also been developed on the basis of multilevel parallel algorithms that allow dynamic p-refinement at constant wall clock time. After two decades of intense developments, spectral element and hp methods are mature and efficient to be used effectively in applications of industrial complexity. They provide the capabilities that standard finite element and finite volume methods do, but, in addition, they exhibit high-order accuracy and error control. Keywords: spectral methods; spectral elements; hp finite elements; complex-geometry; high-order; unstructured grids