Abstract
In finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix–Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche’s technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and
Funding statement: The second author gratefully acknowledges financial support from the National Board for Higher Mathematics (NBHM), Government of India.
A Appendix
The
Lemma A.1 (L 2
The
Further, the trace inequality (2.5) over each edge implies
Proof.
The
Let
Since
Then the Cauchy–Schwarz inequality and (A.1) imply
The weighted
The trace inequality over each edge and (A.2)–(A.3) imply
This concludes the proof. ∎
Remark A.2.
For
The estimate for the first part on the right-hand side of
(A.4) follows from (A.2). In the second part, a use of
the inverse inequality and the
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