### 2002-09

by Stynes,M., Tobiska,L..

**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