A finite element wave-packet procedure is presented to solve problems of wave propagation in multiscale behavior. The proposed scheme combines the advantages of the finite element and spectral methods. The basic formulation is presented, and the capabilities of the procedure are demonstrated through the solution of some illustrative problems, including a problem that characterizes the mode-conv...

Recent work on octree-based finite-element systems has developed a multigrid solver for Poisson equations on meshes. While the idea of defining a regularly indexed function space has been successfully used in a number of applications, it has also been noted that the richness of the function space is limited because the function values can be coupled across locally disconnected regions. In this ...

Spectral and pseudo spectral methods for advection equations are investigated. A basic framework is given which allows the application of techniques used in finite element analysis to spectral methods with trigonometric polynomials. Error estimates for semidiscrete spectral and pseudo spectral as well as fully discrete explicit pseudo spectral methods are given. The approximation schemes are sh...

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares; a finite element approximation is used on the first square and a spectral discretization is used on the second. Two kinds of matching conditions on the interface are presented and compared; in both cases, error estimates are proved.

Abstract The Spectral Finite Element Technique (SFEM) has Several Applications in the Sciences, Engineering, and Mathematics, which will be Covered this Review Article. Method is a Variant of Traditional FEM that Makes use Higher Order Basis Functions (FEM). One most Fundamental Numerical Techniques Employed Simulation SFEM, Outperforms Other Terms Faster Convergence, Reduced Diffusion Dispersi...

In the past few years we have been involved in the development and implementation of parallel finite element and pseudo-spectral methods for porous media flow. In this paper we present a brief review of our methods and show several examples of highly nonlinear simulations in two and three-dimensions, on different current high performance computers. New issues on finite element techniques and ps...

An orthogonal system of rational functions is introduced. Some results on rational approximations based on various orthogonal projections and interpolations are established. These results form the mathematical foundation of the related spectral method and pseudospectral method for solving differential equations on the half line. The error estimates of the rational spectral method and rational p...

Standard spectral methods are capable of providing very accurate approximations to well-behaved smooth functions with significantly less degrees of freedom when compared with finite difference or finite element methods (cf. [6,7,11]). However, if a function exhibits localized behaviors such as sharp interfaces, very thin internal or boundary layers, using a standard Gauss-type grid usually fail...

Polynomial extensions play a vital role in the analysis of the p and h-p FEM and the spectral element method. We construct explicitly polynomial extensions on standard elements: cubes, triangular prisms and pyramids, which together with the extension on tetrahedrons are used by the p and h-p FEM in three dimensions. These extensions are proved to be stable and compatible with FEM subspaces on t...