Spectral element methods : theory and applications

Some mathematical aspects of finite and spectral element discretizations for partial differential equations are presented. The weighted residual method is introduced and several kinds of collocation (finite difference and finite volume) and Galerkin (spectral and finite element) methods are derived as particular cases to that method. The concept of the spectral methods is described and an example of the application of the spectral element method to a second-order elliptic equation provides the reader practical information about it. Some direct and iterative methods to solve the resulting linear algebraic systems are described and some stabilization methods are introduced. An overview of the most commonly used time integration methods for unsteady problems is given in the context of the spectral space discretization. Different approaches for solution of the steady and unsteady Navier-Stokes are introduced in the context of the spectral and finite element methods. Some results of the practical implementation of SEM to 2-D problems are presented.