A C1-Conforming Gauss Collocation Method for Elliptic Equations and Superconvergence Analysis Over Rectangular Meshes
Year: 2024
Author: Waixiang Cao, Lueling Jia, Zhimin Zhang
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 320–349
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≥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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2022-0018
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 320–349
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Hermite interpolation C1-conforming superconvergence Gauss collocation methods Jacobi polynomials.