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Accessible Unlicensed Requires Authentication Published by De Gruyter June 21, 2018

Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem

Rahul Biswas, Asha K. Dond and Thirupathi Gudi

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 32 in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings.

Funding statement: The second author gratefully acknowledges financial support from the National Board for Higher Mathematics (NBHM), Government of India.

A Appendix

The H1- and L2-stability of the L2-orthogonal projection Jh:L2(Ω)CR1(𝒯) lead to the following approximation properties for locally quasi-uniform plus shape-regular meshes.

Lemma A.1 (L2-Orthogonal Projections).

The L2-projection Jh:L2(Ω)CR1(T) satisfies

h𝒯-1(w-Jhw)+h(w-Jhw)Ch𝒯w2for all wH2(Ω).

Further, the trace inequality (2.5) over each edge implies

(ehhe-1w-JhwL2(e)2)12Ch𝒯32wfor all wH2(Ω).

Proof.

The L2-orthogonal projection Jh:L2(Ω)CR1(𝒯) exhibits the following H1- and L2-stability properties [2, Theorem 4.1,4.2], [12, Theorem 4]:

  1. (a)

    Jhvv,

  2. (b)

    h𝒯-1JhvCh𝒯-1v,

  3. (c)

    hJhvChv.

Let Πh be any local quasi-interpolation. Then

(A.1)v-Πhv+h𝒯(v-Πhv)1Ch𝒯2v2.

Since w-Jhw is L2-orthogonal to CR1(𝒯), we have

w-Jhw2=(w-Jhw,w-Jhw)=(w-Jhw,w-Πhw).

Then the Cauchy–Schwarz inequality and (A.1) imply w-Jhww-ΠhwCh𝒯2w2. Using the quasi-interpolation, the triangle inequality and the H1 stability property (c), we find

h(w-Jhw)=h(w-Πhw)+hJh(Πhw-w)
(A.2)h(w-Πhw)+Ch(Πhw-w)Ch𝒯w2.

The weighted L2-stability as defined in (b) gives

h𝒯-1(w-Jhw)h𝒯-1(w-Πhw)+h𝒯-1Jh(w-Πhw)
(A.3)h𝒯-1(w-Πhw)+Ch𝒯-1(w-Πhw)Ch𝒯w2.

The trace inequality over each edge and (A.2)–(A.3) imply

ehhe-1(w-Jhw)L2(e)C(h𝒯-1(w-Jhw)+((w-Jhw)))Ch𝒯32w2.

This concludes the proof. ∎

Remark A.2.

For 𝐰[CR1(𝒯)]2, the trace inequality implies

|ehehe|(𝐰-𝐉h𝐰)𝐧|2ds|ehhe(𝐰-𝐉h𝐰)L2(e)2
(A.4)C(𝐰-𝐉h𝐰12+hT2𝐰-𝐉h𝐰22).

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 H1-stability of the L2-projection as defined in (c) result in

hT(𝐰-𝐉h𝐰)2hT(𝐰-Πh𝐰)2+hT𝐉h(𝐰-Πh𝐰)2
hT(𝐰-Πh𝐰)2+𝐉h(𝐰-Πh𝐰)1Ch𝒯𝐰2.

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Received: 2017-10-13
Revised: 2018-04-24
Accepted: 2018-06-08
Published Online: 2018-06-21
Published in Print: 2019-04-01

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