Stability and accuracy of adapted finite element methods for singularly perturbed problems

Long Chen, Jinchao Xu

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that ∥u - uN∥∞ ≤ C infvN ∈ VN ∥ u - vN∥∞, where u N is the SDFEM approximation in linear finite element space V N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. © 2008 Springer-Verlag.
Original languageEnglish (US)
Pages (from-to)167-191
Number of pages25
JournalNumerische Mathematik
Issue number2
StatePublished - Apr 1 2008
Externally publishedYes

Bibliographical note

Generated from Scopus record by KAUST IRTS on 2023-02-15

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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