### 2002-09

#### The SDFEM for a convection-dif

Series: 2002-09, Preprints

MSC:
65N15 Error bounds
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Abstract:
The streamline diffusion finite element method (SDFEM) is applied to a
convection-diffusion problem posed on the unit square, using a
Shishkin rectangular mesh with piecewise bilinear trial functions.
The hypotheses of the problem exclude interior layers but allow
exponential boundary layers. An error bound is proved for
$\|u^I-u^N\|_{SD}$, where $u^I$ is the interpolant of the solution
$u$, $u^N$ is the SDFEM solution, and $\|\cdot\|_{SD}$ is the
streamline-diffusion norm. This bound implies that
$\|u-u^N\|_{L^2}$ is of optimal order, thereby settling an open
question regarding the $L^2$-accuracy of the SDFEM on rectangular
meshes. Furthermore, the bound shows that $u^N$ is superclose to
$u^I$, which allows the construction of a simple postprocessing
that yields a more accurate solution. Enhancement of the rate of
convergence by using a discrete streamline-diffusion norm is also
discussed. Finally, the verification of these rates of convergence
by numerical experiments is examined, and it is shown that this
practice is less reliable than was previously believed.

Keywords:
streamline diffusion, finite element method, singular perturbation, convection-diffusion, Shishkin mesh