99-28

The $P_1^{mod}$ Element: A New Nonconforming Finite Element for Convection--Diffusion Problems

by Knobloch, P.; Tobiska, L.

 

Preprint series: 99-28, Preprints

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

 

Abstract: We consider a nonconforming streamline diffusion finite element method for solving convection--diffusion problems. The loss of the Galerkin orthogonality of the streamline diffusion method when applied to nonconforming finite element approximations results in an additional error term which cannot be estimated uniformly with respect to the perturbation parameter for the standard piecewise linear or rotated bilinear elements. Therefore, we construct a modified nonconforming first order finite element space on shape regular triangular meshes satisfying the patch test of order two and being related to the Crouzeix/Raviart element. A rigorous error analysis of this $P_1^{\mbox{\scriptsize\it mod}}$ element applied to a streamline diffusion discretization is given. The numerical tests show the robustness of the new method and the improved algebraic properties of the discrete problem.


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