arrow
Volume 33, Issue 5
Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case

Waixiang Cao, Lueling Jia & Zhimin Zhang

Commun. Comput. Phys., 33 (2023), pp. 1466-1508.

Published online: 2023-06

Export citation
  • Abstract

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.

  • AMS Subject Headings

65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-33-1466, author = {Cao , WaixiangJia , Lueling and Zhang , Zhimin}, 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 = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0311}, url = {http://global-sci.org/intro/article_detail/cicp/21768.html} }
TY - JOUR T1 - Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case AU - Cao , Waixiang AU - Jia , Lueling AU - Zhang , Zhimin JO - Communications in Computational Physics VL - 5 SP - 1466 EP - 1508 PY - 2023 DA - 2023/06 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0311 UR - https://global-sci.org/intro/article_detail/cicp/21768.html KW - $C^m$ finite element methods, superconvergence, fourth-order elliptic equations. AB -

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.

Cao , WaixiangJia , Lueling and Zhang , Zhimin. (2023). Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case. Communications in Computational Physics. 33 (5). 1466-1508. doi:10.4208/cicp.OA-2022-0311
Copy to clipboard
The citation has been copied to your clipboard