Spectral Elements

Earlier, in Chap. 2 , Sect. 2.4 , we discussed the possibility of using higher-degree piecewise polynomials to improve the order of the finite element approximations. We mentioned cubic polynomials, in particular. But we also cautioned that using equidistant nodal points to define these polynomials can lead to spurious oscillations of the approximate solutions. To avoid such spurious oscillations, alternative approaches have been used, in which the nodal points are spaced non-uniformly within the element. Thus, for a polynomial of degree \(N-1\) , one chooses N unequally spaced nodal points.