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Volume 35, Issue 2
An $hp$-FEM for Singularly Perturbed Transmission Problems

Serge Nicaise & Christos Xenophontos

DOI: 10.4208/jcm.1607-m2014-0187

J. Comp. Math., 35 (2017), pp. 152-168.

Published online: 2017-04

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  • Abstract

We perform the analysis of the $hp$ finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree $p$ of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the $hp$-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.

  • Keywords

Singularly perturbed transmission problem, Boundary layers, Interface layers, $hp$-FEM, Balanced norm, Exponential convergence.

  • AMS Subject Headings

65N30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

serge.nicaise@univ-valenciennes.fr (Serge Nicaise)

xenophontos@ucy.ac.cy (Christos Xenophontos)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-152, author = {Nicaise , Serge and Xenophontos , Christos}, title = {An $hp$-FEM for Singularly Perturbed Transmission Problems}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {2}, pages = {152--168}, abstract = {

We perform the analysis of the $hp$ finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree $p$ of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the $hp$-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1607-m2014-0187}, url = {http://global-sci.org/intro/article_detail/jcm/10425.html} }
TY - JOUR T1 - An $hp$-FEM for Singularly Perturbed Transmission Problems AU - Nicaise , Serge AU - Xenophontos , Christos JO - Journal of Computational Mathematics VL - 2 SP - 152 EP - 168 PY - 2017 DA - 2017/04 SN - 35 DO - http://doi.org/10.4208/jcm.1607-m2014-0187 UR - https://global-sci.org/intro/article_detail/jcm/10425.html KW - Singularly perturbed transmission problem, Boundary layers, Interface layers, $hp$-FEM, Balanced norm, Exponential convergence. AB -

We perform the analysis of the $hp$ finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree $p$ of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the $hp$-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.

Nicaise , Serge and Xenophontos , Christos. (2017). An $hp$-FEM for Singularly Perturbed Transmission Problems. Journal of Computational Mathematics. 35 (2). 152-168. doi:10.4208/jcm.1607-m2014-0187
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