The XFEM for high-gradient solutions in convection-dominated problems

Safdar Abbas, Alaskar Alizada, Thomas Peter Fries*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Convection-dominated problems typically involve solutions with high gradients near the domain boundaries (boundary layers) or inside the domain (shocks). The approximation of such solutions by means of the standard finite element method requires stabilization in order to avoid spurious oscillations. However, accurate results may still require a mesh refinement near the high gradients. Herein, we investigate the extended finite element method (XFEM) with a new enrichment scheme that enables highly accurate results without stabilization or mesh refinement. A set of regularized Heaviside functions is used for the enrichment in the vicinity of the high gradients. Different linear and non-linear problems in one and two dimensions are considered and show the ability of the proposed enrichment to capture arbitrary high gradients in the solutions.

Original languageEnglish
Pages (from-to)1044-1072
Number of pages29
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
Publication statusPublished - 21 May 2010


  • Boundary layers
  • Convection-diffusion
  • Convection-dominated
  • Extended finite element method (XFEM)
  • High-gradient solutions
  • Shocks

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics
  • Numerical Analysis


Dive into the research topics of 'The XFEM for high-gradient solutions in convection-dominated problems'. Together they form a unique fingerprint.

Cite this