Superconvergence Analysis of Cm Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case
Year: 2023
Author: Waixiang Cao, Lueling Jia, Zhimin Zhang
Communications in Computational Physics, Vol. 33 (2023), Iss. 5 : pp. 1466–1508
Abstract
In this paper, we study three families of Cm(m=0,1,2) finite element methods for one dimensional fourth-order equations. They include C0 and C1 Galerkin methods and a C2−C0 Petrov-Galerkin method. Existence, uniqueness and optimal error estimates of the numerical solution are established. A unified approach is proposed to study the superconvergence property of these methods. We prove that, for kth-order elements, the C0 and C1 finite element solutions and their derivative are superconvergent with rate h2k−2(k≥3) at all mesh nodes; while the solution of the C2−C0 Petrov-Galerkin method and its first- and second-order derivatives are superconvergent with rate h2k−4(k≥5) at all mesh nodes. Furthermore, interior superconvergence points for the l-th(0≤l≤m+1) derivate approximations are also discovered, which are identified as roots of special Jacobi polynomials, Lobatto points, and Gauss points. As a by-product, we prove that the Cm finite element solution is superconvergent towards a particular Jacobi projection of the exact solution in the Hl(0≤l≤m+1) norms. All theoretical findings are confirmed by numerical experiments.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2022-0311
Communications in Computational Physics, Vol. 33 (2023), Iss. 5 : pp. 1466–1508
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 43
Keywords: Cm finite element methods superconvergence fourth-order elliptic equations.
Author Details
Waixiang Cao Email
Lueling Jia Email
Zhimin Zhang Email
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