Spectral methods are essentially discretization methods
for the approximate solution of partial-differential equations
expressed in a weak form, based on high-order
Lagrangian interpolants used in conjunction with particular quadrature rules.
For spectral methods it is assumed that the solution
can be expressed as a series of polynomial basis functions,
which can approximate the solution well
in some norm as the polynomial degree tends to infinity.
These smooth basis-functions usually form an L2-complete basis.
For the sake of computational efficiency this basis is typically chosen
to be orthogonal in a weighted inner-product.
The spectral element method is a high-order finite element technique that
combines the geometric flexibility of finite elements with the high accuracy of
This method was pioneered in the mid 1980's by Anthony Patera
at MIT and Yvon Maday at Paris-VI.
It exhibits several favourable computational properties,
such as the use of tensor products,
naturally diagonal mass matrices, adequacy to implementations in a parallel computer system.
Due to these advantages, the spectral element method is a viable alternative to
currently popular methods such as finite volumes and finite elements,
if accurate solutions of regular problems are sought.
Main Features of the Spectral Element Method