Spectral Element Method - Related Methods

Related Methods

• G-NI or SEM-NI: these are the most used spectral methods. The Galerkin formulation of spectral methods or spectral element methods, for G-NI or SEM-NI respectively, is modified and Gaussian numerical integration is used instead of integrals in the definition of the bilinear form and in the functional . These method are a family of Petrov–Galerkin methods their convergence is a consequence of Strang's lemma.
• The spectral element method uses tensor product space spanned by nodal basis functions associated with Gauss–Lobatto points. In contrast, the p-version finite element method spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for numerical stability. Since not all interior basis functions need to be present, the p-version finite element method can create a space that contains all polynomials up to a given degree with many fewer degrees of freedom. However, some speedup techniques possible in spectral methods due to their tensor-product character are no longer available. The name p-version means that accuracy is increased by increasing the order of the approximating polynomials (thus, p) rather than decreasing the mesh size, h.
• The hp finite element method (hp-FEM) combines the advantages of the h and p refinements to obtain extremely fast, exponential convergence rates.