Article history: Received 2 January 2014 Received in revised form 23 March 2014 Accepted 27 March 2014 Available online 2 April 2014

We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and L norm error bounds. These bound...

A finite element method for the 1-periodic Korteweg-de Vries equation "t + 2uux + "xxx = ° is analyzed. We consider first a semidiscrete method (i.e., discretization only in the space variable), and then we analyze some unconditionally stable fully discrete methods. In a special case, the fully discrete methods reduce to twelve point finite difference schemes (three time levels) which have seco...

A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimen...

is finite, and that K(1) = 1, K(2) = 2. For k > 2, Favard gives no quantitative information about K(k). An estimate for the supremum under the additional restriction that only uniform t be considered can be found in Jerome and Schumaker [5]. Their argument was extended by Golomb [4] as far as it will go, viz., to include nonuniform t’s whose global mesh ratio Rt := maxi ∆ti/mini ∆ti is bounded....

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, n...

where x = (x1, . . . , xd) ∈ IR , t > 0. HJ equations arise in many practical areas such as differential games, mathematical finance, image enhancement and front propagation. It is well known that solutions of (1) are Lipschitz continuous but derivatives can become discontinuous even if the initial data is smooth. There is a close relation between HJ equations and hyperbolic conservation laws. ...