99-06

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

by Martin Stynes; Lutz Tobiska

 

Preprint series: 99-06, Preprints

The paper is published: IMA Journal of Numerical Analysis 21, S. 123 - 142, 2001

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

 

Abstract: 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.

Keywords: streamline-diffusion, nonconforming finite element method,


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