(untitled) (CSIAM Transactions on Applied Mathematics, 2024)
This paper is concerned with a C^1-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 C^1 piecewise polynomials of degree not more than k, k≥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.