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Volume 15, Issue 1-2
Superconvergence of a Quadratic Finite Element Method on Adaptively Refined Anisotropic Meshes

Weiming Cao

Int. J. Numer. Anal. Mod., 15 (2018), pp. 288-306.

Published online: 2018-01

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  • Abstract

We establish in this paper the supercloseness of the quadratic finite element solution of a two dimensional elliptic problem to the piecewise quadratic interpolation of its exact solution. The assumption is that the partition of the solution domain is quasi-uniform under a Riemannian metric and that each pair of the adjacent elements in the partition forms an approximate parallelogram. This result extends our previous one in [7] for the linear finite element approximations based on adaptively refined anisotropic meshes. It also generalizes the results by Huang and Xu in [13] for the supercloseness of the quadratic elements based on the mildly structured quasi-uniform meshes. A distinct feature of our analysis is that we transform the error estimates on each physical element to that on an equilateral standard element, and then focus on the algebraic properties of the Jacobians of the affine mappings from the standard element to the physical elements. We believe this idea is also useful for the superconvergence study of other types of elements on unstructured meshes.

  • AMS Subject Headings

65N30, 65N15, 65N50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

weiming.cao@utsa.edu (Weiming Cao)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-288, author = {Cao , Weiming}, title = {Superconvergence of a Quadratic Finite Element Method on Adaptively Refined Anisotropic Meshes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {1-2}, pages = {288--306}, abstract = {

We establish in this paper the supercloseness of the quadratic finite element solution of a two dimensional elliptic problem to the piecewise quadratic interpolation of its exact solution. The assumption is that the partition of the solution domain is quasi-uniform under a Riemannian metric and that each pair of the adjacent elements in the partition forms an approximate parallelogram. This result extends our previous one in [7] for the linear finite element approximations based on adaptively refined anisotropic meshes. It also generalizes the results by Huang and Xu in [13] for the supercloseness of the quadratic elements based on the mildly structured quasi-uniform meshes. A distinct feature of our analysis is that we transform the error estimates on each physical element to that on an equilateral standard element, and then focus on the algebraic properties of the Jacobians of the affine mappings from the standard element to the physical elements. We believe this idea is also useful for the superconvergence study of other types of elements on unstructured meshes.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10569.html} }
TY - JOUR T1 - Superconvergence of a Quadratic Finite Element Method on Adaptively Refined Anisotropic Meshes AU - Cao , Weiming JO - International Journal of Numerical Analysis and Modeling VL - 1-2 SP - 288 EP - 306 PY - 2018 DA - 2018/01 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10569.html KW - Quadratic elements, superconvergence, anisotropic meshes. AB -

We establish in this paper the supercloseness of the quadratic finite element solution of a two dimensional elliptic problem to the piecewise quadratic interpolation of its exact solution. The assumption is that the partition of the solution domain is quasi-uniform under a Riemannian metric and that each pair of the adjacent elements in the partition forms an approximate parallelogram. This result extends our previous one in [7] for the linear finite element approximations based on adaptively refined anisotropic meshes. It also generalizes the results by Huang and Xu in [13] for the supercloseness of the quadratic elements based on the mildly structured quasi-uniform meshes. A distinct feature of our analysis is that we transform the error estimates on each physical element to that on an equilateral standard element, and then focus on the algebraic properties of the Jacobians of the affine mappings from the standard element to the physical elements. We believe this idea is also useful for the superconvergence study of other types of elements on unstructured meshes.

Weiming Cao. (2020). Superconvergence of a Quadratic Finite Element Method on Adaptively Refined Anisotropic Meshes. International Journal of Numerical Analysis and Modeling. 15 (1-2). 288-306. doi:
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