### Abstract

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation sqrt(λ) + tan sqrt(λ) = 0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.

Original language | English |
---|---|

Pages (from-to) | 3750-3758 |

Number of pages | 9 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 196 |

Issue number | 37-40 SPEC. ISS. |

DOIs | |

Publication status | Published - 2007 Aug 1 |

### Fingerprint

### Keywords

- Babuška-Aziz constant
- Error estimates
- FEM
- Interpolation error constants
- Triangular element

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mechanics

### Cite this

_{0}and P

_{1}triangular finite elements.

*Computer Methods in Applied Mechanics and Engineering*,

*196*(37-40 SPEC. ISS.), 3750-3758. https://doi.org/10.1016/j.cma.2006.10.029

**Estimation of interpolation error constants for the P _{0} and P_{1} triangular finite elements.** / Kikuchi, Fumio; Liu, Xuefeng.

Research output: Contribution to journal › Article

_{0}and P

_{1}triangular finite elements',

*Computer Methods in Applied Mechanics and Engineering*, vol. 196, no. 37-40 SPEC. ISS., pp. 3750-3758. https://doi.org/10.1016/j.cma.2006.10.029

_{0}and P

_{1}triangular finite elements. Computer Methods in Applied Mechanics and Engineering. 2007 Aug 1;196(37-40 SPEC. ISS.):3750-3758. https://doi.org/10.1016/j.cma.2006.10.029

}

TY - JOUR

T1 - Estimation of interpolation error constants for the P0 and P1 triangular finite elements

AU - Kikuchi, Fumio

AU - Liu, Xuefeng

PY - 2007/8/1

Y1 - 2007/8/1

N2 - We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation sqrt(λ) + tan sqrt(λ) = 0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.

AB - We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but some results are also given for the non-conforming case. We obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles. In particular, we explicitly determine the Babuška-Aziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The equation for determination is the transcendental equation sqrt(λ) + tan sqrt(λ) = 0, so that the solution can be numerically obtained with desired accuracy and verification. Such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions.

KW - Babuška-Aziz constant

KW - Error estimates

KW - FEM

KW - Interpolation error constants

KW - Triangular element

UR - http://www.scopus.com/inward/record.url?scp=34447096654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34447096654&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2006.10.029

DO - 10.1016/j.cma.2006.10.029

M3 - Article

VL - 196

SP - 3750

EP - 3758

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

IS - 37-40 SPEC. ISS.

ER -