Spectral Element Methods on Unstructured Meshes: Comparisons and Recent Advances

The celebrated Spectral Element Method (SEM) has appeared to be of great interest to extend the capabilities of spectral methods to complex geometries. However, using quadrangular (hexahedral in 3D) elements may be a severe restriction for the most complex geometries, because requiring a structured mesh. Thus, Finite element meshes are usually based on triangles (tetrahedra in 3D). Efforts have been made in the late 1990s to implement spectral methods based on simplices. In [11] (and related works) it is proposed to use a “collapsed coordinate system”, so that the integration rules for quadrangles may be used for triangles. Such an approach follows ideas developed earlier in the context of integration rules for simplices (see, e.g., [13] and references herein). Its main drawback is that the quadrature points are not symmetrically located in the triangle (tetrahedron), and a priori useless accumulations of quadrature points, also used as approximation points [11], arise in some vertices.