The computation of optical modes inside axisymmetric cavity resonators with a general spatial permittivity profile is a formidable computational task. In order to avoid spurious modes the vector Helmholtz equations are discretised by a mixed finite element approach. We formulate the method for first and second order Nédélec edge and Lagrange nodal elements. We discuss how to accurately compute ...

A connection is made between a recently introduced Lagrange-distributed approximating-functional and the Paley-Wiener sampling theorem. The Lagrange-distributed approximating-functional sampling is found to provide much superior results to that of Paley-Wiener sampling. The relations between discrete variable representation and Lagrange-distributed approximating functionals are discussed. The l...

The smoothing properties of variational methods for the design of fair surfaces are considered using a PDE (partial differential equation) model. The PDE is the Euler-Lagrange equation that is obtained from the variational formulation of the surface design problem. The smoothing properties of the design scheme are considered by looking at the scheme's efficiency in reducing the amplitude of the...

The problem of semi-cardinal spline interpolation was solved by Schoenberg exploiting the piecewise polynomial form of the splines. In the present paper, we propose a new construction for the Lagrange functions of semi-cardinal spline interpolation , based on a radial basis and Fourier transform approach. This approach suggests a way of extending semi-cardinal interpolation to polyharmonic spli...

We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by the n roots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, ...

The Lagrange-mesh method is a very accurate procedure to compute eigenvalues and eigenfunctions of a two-body quantum equation. The method requires only the evaluation of the potential at some mesh points in the configuration space. It is shown that the eigenfunctions can be easily computed in the momentum space by a Fourier transform using the properties of the basis functions. Observables in ...

In relation to Fourier-Pade approximation, Ed Sa¤ observed that Taylor and Lagrange interpolation projections satisfy the following property: P (f) P (g) 2 n =) P (f g) = P (f) P (g). We classify all projections that satisfy this property, thus answering a question of Sa¤. Some error formulas for approximation with the above mentioned projections are also produced. AMS Subject Classi…cation: 41...

In this article we investigate the application of high order approximation techniques to one-dimensional boundary layer problems. In particular, we use second order differential equations and coupled second order differential equations as case studies. The accuracy and convergence rate of numerical solution obtained with Lagrange, Hermite, B-spline finite elements and C∞ generalized finite elem...