© 2001 by Institute of Mathematics and its Applications
The streamline–diffusion method for nonconforming Qrot1 elements on rectangular tensor–product meshes
1 Department of Mathematics, National University of Ireland, Cork, Ireland, e-mail: stynes{at}ucc.ie 2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, PF 4120, D-39016 Magdeburg, Germany, e-mail: tobiska{at}mathematik.uni-magdeburg.de
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 Qrot1 1 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 h3/2 (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.
Key Words: streamline–diffusion, nonconforming finite element method, stability, convergence, error estimate
Received 22 March 1999. Accepted 21 December 1999.