### 2006-08

#### Using rectangula

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