Published

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

Long Chen and Jinchao Xu.

Numerische Mathematik, 109:167--191, 2008

Pdf   Bibtex

ABSTRACT:

  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
  $\|u-u_{N}\|_{\infty}\leq C\inf_{v_{N}\in V^{N}}
                   \|u-v_{N}\|_{\infty},$ 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.