A C1-Conforming Gauss Collocation Method for Elliptic Equations and Superconvergence Analysis Over RectangularMeshes

Abstract

This paper is concerned with a C¹-conforming Gauss collocation approximation to the solution of a model two-dimensional elliptic boundary problem. Superconvergence phenomena for the numerical solution at mesh nodes, at roots of a special Jacobi polynomial, and at the Lobatto and Gauss lines are identified with rigorous mathematical proof, when tensor products of C¹ piecewise polynomials of degree not more than $k,k\ge 3$ are used. This method is shown to be superconvergent with (2k-2)-th order accuracy in both the function value and its gradient at mesh nodes, ( k+2 )-th order accuracy at all interior roots of a special Jacobi polynomial, ( k+1 )-th order accuracy in the gradient along the Lobatto lines, and k-th order accuracy in the second-order derivative along the Gauss lines. Numerical experiments are presented to indicate that all the superconvergence rates are sharp.

Keywords

Hermite interpolation / C¹-conforming / superconvergence / Gauss collocation methods / Jacobi polynomials