Local accuracy in finite element analysis using curved isoparametric elements

Pranjal Saxena; Chandra Shekhar Upadhyay

Applications of Mathematics (2025)

  • Issue: 2, page 257-292
  • ISSN: 0862-7940

Abstract

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The finite element method (FEM) is popularly used for numerically approximating PDE(s) over complicated domains due to its rich mathematical background, versatility, and ease of implementation. In this article, we investigate one of its important features, i.e., the approximation of PDE(s) over nonpolygonal Lipschitz domains by higher-order simplicial elements in 2D and 3D. This important issue is not well understood and often ignored by engineers due to its mathematical complexity, i.e., the FEM approximation of curved domains results in inexact boundary conditions, which is a variational crime. This article explores the role of approximation at curved boundaries. Further, the effect of incompleteness of the approximation space also contributes to the error induced in the curved elements. A simple benchmark test for errors is proposed. Tests are conducted for subparametric and isoparametric approximations. Comparison with isogeometric analysis (IGA) is also presented to highlight the basic differences and advantages of isoparametric elements.

How to cite

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Saxena, Pranjal, and Upadhyay, Chandra Shekhar. "Local accuracy in finite element analysis using curved isoparametric elements." Applications of Mathematics (2025): 257-292. <http://eudml.org/doc/299974>.

@article{Saxena2025,
abstract = {The finite element method (FEM) is popularly used for numerically approximating PDE(s) over complicated domains due to its rich mathematical background, versatility, and ease of implementation. In this article, we investigate one of its important features, i.e., the approximation of PDE(s) over nonpolygonal Lipschitz domains by higher-order simplicial elements in 2D and 3D. This important issue is not well understood and often ignored by engineers due to its mathematical complexity, i.e., the FEM approximation of curved domains results in inexact boundary conditions, which is a variational crime. This article explores the role of approximation at curved boundaries. Further, the effect of incompleteness of the approximation space also contributes to the error induced in the curved elements. A simple benchmark test for errors is proposed. Tests are conducted for subparametric and isoparametric approximations. Comparison with isogeometric analysis (IGA) is also presented to highlight the basic differences and advantages of isoparametric elements.},
author = {Saxena, Pranjal, Upadhyay, Chandra Shekhar},
journal = {Applications of Mathematics},
keywords = {curved boundary; error estimate; isoparametric FEM; isogeometric analysis; patch test; local convergence},
language = {eng},
number = {2},
pages = {257-292},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Local accuracy in finite element analysis using curved isoparametric elements},
url = {http://eudml.org/doc/299974},
year = {2025},
}

TY - JOUR
AU - Saxena, Pranjal
AU - Upadhyay, Chandra Shekhar
TI - Local accuracy in finite element analysis using curved isoparametric elements
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 2
SP - 257
EP - 292
AB - The finite element method (FEM) is popularly used for numerically approximating PDE(s) over complicated domains due to its rich mathematical background, versatility, and ease of implementation. In this article, we investigate one of its important features, i.e., the approximation of PDE(s) over nonpolygonal Lipschitz domains by higher-order simplicial elements in 2D and 3D. This important issue is not well understood and often ignored by engineers due to its mathematical complexity, i.e., the FEM approximation of curved domains results in inexact boundary conditions, which is a variational crime. This article explores the role of approximation at curved boundaries. Further, the effect of incompleteness of the approximation space also contributes to the error induced in the curved elements. A simple benchmark test for errors is proposed. Tests are conducted for subparametric and isoparametric approximations. Comparison with isogeometric analysis (IGA) is also presented to highlight the basic differences and advantages of isoparametric elements.
LA - eng
KW - curved boundary; error estimate; isoparametric FEM; isogeometric analysis; patch test; local convergence
UR - http://eudml.org/doc/299974
ER -

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