### 2006-08

by Stynes, M., Tobiska, L..

**Series:** 2006-08, Preprints

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

**Abstract:**

The streamline diffusion finite element method (SDFEM; the method

is also known as SUPG) is applied to a convection-diffusion

problem posed on the unit square whose solution has exponential

boundary layers. A rectangular Shishkin mesh is used. The trial

functions in the SDFEM are piecewise polynomials that lie in the

space $Q_p$, i.e., are tensor products of polynomials of degree

$p$ in one variable, where $p>1$. The error bound $\|I_N

u-u^N\|_{SD}\le C N^{-(p+1/2)}$ is proved; here $u^N$ is the

computed SDFEM solution, $I_N u$ is chosen in the finite element

space to be a special approximant of the true solution $u$, and

$\|\cdot\|_{SD}$ is the streamline-diffusion norm. This result is

compared with previously known results for the case $p=1$. The

error bound is a superclose result; $u^N$ can be enhanced using

local postprocessing to yield a modified solution $\tilde u^N$ for

which $\|u-\tilde u^N\|_{SD}\le C N^{-(p+1/2)}$.

**Keywords:**

convection-diffusion problems, streamline-diffusion method, finite elements, error estimates