Jacobi Spectral Galerkin and Iterated Methods for Nonlinear Volterra Integral Equation

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Superconvergence of a New Nonconforming Mixed Finite Element Scheme for Elliptic Problem

A new nonconforming mixed finite element scheme for the second-order elliptic problem is proposed based on a new mixed variational form. It has the lowest degrees of freedom on rectangular meshes. The superclose property is proven by employing integral identity technique. Then global superconvergence result is derived through interpolation postprocessing operators. At last, some numerical experiments are carried out to verify the theoretical analysis.

Convergence and Stability in Collocation Methods of Equationu′(t)=au(t)+bu([t])

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods foru′(t)=au(t)+bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.