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A $C^1$-Conforming Gauss Collocation Method for Elliptic Equations and Superconvergence Analysis Over Rectangular Meshes

A $C^1$-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 $C^1$-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.

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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 $C^1$-conforming superconvergence Gauss collocation methods Jacobi polynomials.

Author Details

Waixiang Cao

Lueling Jia

Zhimin Zhang