In this paper, using the Newton's formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way. It is known that the polynomial interpolations are the foundations of construction the finite elements and the interpolation error estimates play a key role in derivi...

We give characterizations of radial Fourier multipliers as acting on radial L functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L −L bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces....

Topics Direct Approach for Discrete Systems Strong and Weak Forms for One-Dimensional Problems Second order boundary value problems (1D) Scalar field problems: heat conduction, advection-diffusion Triangular elements Computer implementation Isoparametric elements Numerical integration Vector field problems: elasticity equations Beam elements Eigenvalue problems Time dependent problems Special t...

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonome...

In this paper we introduce new techniques for the eecient computation of a Fourier transform on a nite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the Cooley-Tukey FFT to arbitrary nite groups. We apply our general results to special linear groups and low rank symmetric groups, and obtain new eecient al...

The Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator o...

Abstract. We present and analyze a subgrid viscosity Lagrange-Galerkin method that combines the subgrid eddy viscosity method proposed in W. Layton, A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comp., 133: 147-157, 2002, and a conventional Lagrange-Galerkin method in the framework of P1⊕ cubic bubble finite elements. This results in an efficient and easy to i...

For simplex finite elements, the native Lagrange family with arbitrarily placed nodes is presented in hierarchy-ranking expressions. It includes the well-known complete Lagrange family as well as the mid-edge Lagrange family to he proposed in this paper. This new family enables us to utilize harmonious combinations of interpolations of different orders in finite element analysis. As an applicat...