@Article{CiCP-33-5,
author = {Waixiang, Cao and Lueling, Jia and Zhimin, Zhang},
title = {Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case},
journal = {Communications in Computational Physics},
year = {2023},
volume = {33},
number = {5},
pages = {1466--1508},
abstract = {<p style="text-align: justify;">In this paper, we study three families of $C^m (m=0,1,2)$ finite element methods for one dimensional fourth-order equations. They include $C^0$ and $C^1$ Galerkin
methods and a $C^2-C^0$ 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 $k$th-order elements, the $C^0$ and $C^1$ finite element solutions and their derivative are superconvergent with rate $h^{2k−2} (k≥3)$ at all mesh nodes; while the solution of the $C^2-C^0$ Petrov-Galerkin method and its first- and second-order derivatives are superconvergent with rate $h^{2k−4} (k≥5)$ at all mesh nodes. Furthermore, interior superconvergence
points for the $l$-${\rm th} (0≤l≤m+1)$ 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 $C^m$ finite element solution is superconvergent towards
a particular Jacobi projection of the exact solution in the $H^l
(0≤ l ≤ m+1)$ norms. All
theoretical findings are confirmed by numerical experiments.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2022-0311},
url = {https://global-sci.com/article/79456/superconvergence-analysis-of-cm-finite-element-methods-for-fourth-order-elliptic-equations-i-one-dimensional-case}
}