A numerical test case demonstrates that Lobatto and Gauss points are not natural superconvergent points for cubic and quartic finite elements under equilateral triangular mesh. 2000 Mathematics Subject Classification. Primary 65N30, Secondary 65N15, 41A10, 41A25, 41A27, 41A63.

In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are presented to evaluate the rational integrals of the element matrix for a general quadrilateral. These integrals arise in finite element formulation for second order Partial Differential Equation via Galerkin weighted residual method in closed form. Convergence to the analytical solutions and efficiency are depicted by numerical ex...

We address quadrature-based approximations of the bilinear inverse form u>A−1u, where A is a real symmetric positive definite matrix, and analyze properties of the Gauss, Gauss-Radau, and Gauss-Lobatto quadrature. In particular, we establish monotonicity of the bounds given by these quadrature rules, compare the tightness of these bounds, and derive associated convergence rates. To our knowledg...

In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundar...

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The m...

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...

We formulate collocation Runge–Kutta time-stepping schemes applied to linear parabolic evolution equations as space-time Petrov–Galerkin discretizations, and investigate their a priori stability for the parabolic space-time norms, that is the continuity constant of the discrete solution mapping. We focus on collocation based on A-stable Gauss–Legendre and L-stable right-Radau nodes, addressing ...