Volume 33, Issue 1
Graded Meshes for Higher Order FEM

Hans-Görg Roos, L. Teofanov & Zorica Uzelac

J. Comp. Math., 33 (2015), pp. 1-16.

Published online: 2015-02

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  • Abstract

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an ‘optimal’ mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.

  • Keywords

Singular perturbation Boundary value problem Layer-adapted meshes Graded meshes Finite element method

  • AMS Subject Headings

65L11 65L50 65L60 65L70.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

Hans-Goerg.Roos@tu-dresden.de (Hans-Görg Roos)

ljiljap@uns.ac.rs (L. Teofanov)

zora@uns.ac.rs (Zorica Uzelac)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-1, author = {Roos , Hans-Görg and Teofanov , L. and Uzelac , Zorica }, title = {Graded Meshes for Higher Order FEM}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {1}, pages = {1--16}, abstract = {

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an ‘optimal’ mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1405-m4362}, url = {http://global-sci.org/intro/article_detail/jcm/9824.html} }
TY - JOUR T1 - Graded Meshes for Higher Order FEM AU - Roos , Hans-Görg AU - Teofanov , L. AU - Uzelac , Zorica JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 16 PY - 2015 DA - 2015/02 SN - 33 DO - http://doi.org/10.4208/jcm.1405-m4362 UR - https://global-sci.org/intro/article_detail/jcm/9824.html KW - Singular perturbation KW - Boundary value problem KW - Layer-adapted meshes KW - Graded meshes KW - Finite element method AB -

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an ‘optimal’ mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.

Hans-Görg Roos, L. Teofanov & Zorica Uzelac . (2019). Graded Meshes for Higher Order FEM. Journal of Computational Mathematics. 33 (1). 1-16. doi:10.4208/jcm.1405-m4362
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