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

by Knobloch, P., Tobiska, L..

Series: 1999-28, Preprints

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

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.