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The Streamline-Diffusion Method for Nonconforming $Q_1^{rot}$ Elements on Rectangular Tensor--product Meshes

by Martin Stynes, Lutz Tobiska.

Series: 1999-06, Preprints

65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N15 Error bounds
65N12 Stability and convergence of numerical methods

When the streamline-diffusion finite element method is applied to convection-diffusion problems
using nonconforming trial spaces, it has previously been observed that
stability and convergence problems may occur. It has consequently been
proposed that certain jump terms should be added to the bilinear form to obtain the
same stability and convergence
behaviour as in the conforming case. The analysis in this paper shows that
for the $Q_1^{rot}$ element on rectangular shape-regular tensor-product meshes, no jump terms
are needed to stabilize the method.
In this case moreover, for smooth solutions we derive in the
streamline-diffusion norm convergence of order
$h^{3/2}$ (uniformly in the diffusion coefficient of the problem), where $h$ is
the mesh diameter. (This estimate is already known for the conforming case.)
Our analysis also shows that similar stability and convergence results fail to
hold true for analogous piecewise linear nonconforming elements.

streamline-diffusion, nonconforming finite element method,

This paper was published in:
IMA Journal of Numerical Analysis 21, S. 123 - 142, 2001