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I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element.

The continuous Stokes Equation in the weak formulation is

Find $u \in H_0^1(\Omega, \mathbb{R}^d)$ and $p \in L_0^2(\Omega)$ with

\begin{align} \int_\Omega \nabla u : \nabla v dx - \int_\Omega p \text{ div} (v) dx &= \int_{\Omega} fv dx \qquad \ \forall v \in H_0^1(\Omega, \mathbb{R}^d)\\ \int_\Omega q \text{ div} (u) dx&= 0 \qquad \qquad \quad \forall q \in L_0^2(\Omega) \end{align}

It has a unique solution in the spaces $V = H_0^1(\Omega, \mathbb{R}^d)$ and $Q= L_0^2(\Omega)$.

For the discrete spaces I have

\begin{align} V_h &= \{ v_h \in H_0^1(\Omega, \mathbb{R}^d) : v_h|_T\in P^3(T),v_h|_E\in P^2(E),\forall T \in T_h\forall E\subset \partial T\}\\ Q_h &= \{q_h\in L_0^1(\Omega):q_h|_T\in P^1(T),\forall T \in T_h \} \end{align}

To prove solvability, I want to prove the discrete LBB condition

\begin{align} \sup\limits_{v_h\in V_h} \frac{\int_\Omega \text{div}(v_h)q_h dx}{\|v_h\|_{V}} \geq \beta_1 \|q_h\|_{Q} \qquad \forall q_h \in Q_h \end{align}

For the continuous spaces $V,Q$ the LBB condition holds.

Assume there exists a Fortin operator $\Pi_h:V \rightarrow V_h$ with \begin{align} \|\Pi_h v \|_V &\preccurlyeq \|v\|_V \qquad \quad \forall v\in V \\ b(\Pi_h v, q_h) &= b(v, q_h) \qquad \ \forall q_h \in Q_h \end{align} Then the continuous LBB implies the discrete LBB.

Can you please tell me how I can construct such an Fortin Operator and then prove its two properties.

Especially it would be very helpful to see the full scaling argument that proves its continuity.

I would be thankful for any help.

I have also posted this question on Math-Stack exchange, but so far without success: https://math.stackexchange.com/questions/3710994/fortin-operator-for-crouzeix-raviart-element

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  • $\begingroup$ If someone mentions the Crouzeix-Raviart element, I think about something else, i.e., the nonconforming $P_1$ element for velocity and $P_0$ for the pressure. See, e.g., dx.doi.org/10.1002/num.21892 $\endgroup$
    – knl
    Jun 9, 2020 at 13:16
  • $\begingroup$ Have you looked at the proofs of the discrete LBB condition for other kinds of elements? $\endgroup$ Jun 9, 2020 at 15:27
  • $\begingroup$ I found the name in the book "Mixed and HybridFinite Element Methods" by Brezzi and Fortin. Actually they do a prove of the discrete LBB condition on page 222 for this element (they construct a Fortin operator). But I don't understand the prove completely. The Fortin operator is made up from two operators F1 and F2 were F=F1+F2-F2F1 is the Fortin operator for my element. F1 is the Fortin operator from the P2-P0 element. Now what is unclear is: How do I fix the coefficients of the bubble element by this two operators? How do I prove continuity of the operators F1 and F2(scaling argument)? $\endgroup$
    – Pepe
    Jun 10, 2020 at 8:22

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