In this note, we make a few comments concerning the paper of Hughes and Akin [4]. Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the finite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements.

In this paper the quadratic spline difference scheme for a convection-diffusion problem is derived. With the suitable choice of collocation points we provide the discrete minimum principle. The numerical results implies the uniform convergence of order O(n−2 ln n).

In this paper, the aim is to solve the differential-difference equations using a radial basis collocation method. We present the advantages and improvement of using the proposed method for solving differential-difference equations. Some experiments are also employed to illustrate the validity and flexibility of the proposed method even where the data points are scattered. 2010 mathematics subje...

In this paper, we consider two interpolations of Birkhoff-type with integer-order derivative. The Birkhoff interpolation is related with collocation method for the corresponding initial or boundary value problems of differential equations. The solvability of the interpolation problems is proved. For Gauss-type interpolating points, error of interpolation approximation is deduced. Also, we give ...

It is shown that simple eigenvalues of an mth order ordinary differential 2k equation are approximated within 0(1 AI ) by collocation at Gauss points with piecewise polynomial functions of degree < m + k on a mesh A. The same rate is achieved by certain averages in case the eigenvalue is not simple. The argument relies on an extension and simplification of Osborn's recent results concerning the...

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ ∈ (0, 1) to compute that of any order k + μ with in...

[1] An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen-Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen-Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic co...

In this paper, we present a boundary integral based approach to isogeometric analysis and shape optimization. For analysis, it uses the same basis, Non-Uniform Rational B-Spline (NURBS) basis, for both representing object boundary and for approximating physical fields in analysis via a Boundary-Integral-Equation Method (BIEM). We propose the use of boundary points corresponding to Greville absc...

A simple and explicit expression is given for the inner product of (higher order) derivatives of multivariate box splines and their translates. We also show that the energy inner product related to a linear partial differential equation discretized with a set of shifted box splines can be interpreted as a collocation of the problem with a higher order box spline at certain points.

In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...