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A two-level iterative method with Newton-type linearization for the stationary micropolar fluid equations

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Abstract

In this paper, a two-level Newton iterative method is proposed for the stationary micropolar fluid equations. Firstly, the original equations are solved on a coarse grid based on Newton-type linearization. Then, the simplified linearized equations are solved on a fine grid. The stability and error estimates of the method are given in the theoretical part. The results of the theoretical analysis show that when the coarse mesh size \(\varvec{H}\) and fine mesh size \(\varvec{h}\) satisfy the relation \(\varvec{h=O(H^{2}})\), the two-level Newton iterative method can achieve an optimal convergence rate. Finally, the effectiveness and applicability of the method are verified by some numerical experiments.

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Funding

National Natural Science Foundation of China (No. 12061075); Key Laboratory of Xinjiang Province (No. 2022D04014). National Key Research and Development Program of China (No. 2023YFB3001604)

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Xin Xing & Demin Liu

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Xin Xing

Authors
  1. Xin Xing
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  2. Demin Liu
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Contributions

Formal analysis: XX and DL. Methodology: XX and DL.

Corresponding author

Correspondence to Demin Liu.

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Appendix

Appendix

1.1 The Proof of Theorem 4

Proof

The first inequality (27) can be obtained by using Lemma 2.1.

Then, let us prove (28). Subtracting (25) from (3), we have

$$\begin{aligned}&A((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{v},\varvec{\uppsi })) +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u},\varvec{\upomega }),(\textbf{v},\varvec{\uppsi })) \nonumber \\&\quad +B((\textbf{u}_{\mu },\varvec{\upomega }_{\mu }),(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{v},\varvec{\uppsi })) -d((\textbf{v},\varvec{\uppsi }),p-p_{\mu }) \nonumber \\&\quad +d((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),q)=0, \quad \forall ((\textbf{v},\varvec{\uppsi }),q) \in \textbf{W}_{\mu } \times M_{\mu }. \end{aligned}$$
(53)

Taking \((\textbf{v},\varvec{\uppsi })=(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }), q=\rho _{\mu }p-p_{\mu }\) in (53), we can derive that

$$\begin{aligned}&\quad A((R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&\quad +B((R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u},\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&=A((R_{\mu }\textbf{u}-\textbf{u},Q_{\mu }\varvec{\upomega }-\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&\quad +B((R_{\mu }\textbf{u}-\textbf{u},Q_{\mu }\varvec{\upomega }-\varvec{\upomega }),(\textbf{u},\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \\&\quad +B((\textbf{u}_{\mu },\varvec{\upomega }_{\mu }),(R_{\mu }\textbf{u}-\textbf{u},Q_{\mu }\varvec{\upomega }-\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&\quad +d((R_{\mu }\textbf{u}\!-\!\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }),p\!-\!\rho _{\mu }p) \!-\!d((\textbf{u}\!-\!R_{\mu }\textbf{u},\varvec{\upomega }\!-\!Q_{\mu }\varvec{\upomega }),\rho _{\mu }p\!-\!p_{\mu }).\nonumber \end{aligned}$$
(54)

For the left-hand side of (54), by using the (11)–(13), we have

$$\begin{aligned} L.H.S&\ge C_{\min }\Vert (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1}\nonumber \\&\quad -\lambda \Vert (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1}\Vert (\textbf{u},\varvec{\upomega })\Vert _{1} \nonumber \\&\ge C_{\min }(1-\sigma )\Vert (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1}. \end{aligned}$$
(55)

Moreover, from (12), (13) and (18), we have

$$\begin{aligned}&A((R_{\mu }\textbf{u}-\textbf{u}, Q_{\mu }\varvec{\upomega }-\varvec{\upomega }), (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&\quad +B((R_{\mu }\textbf{u}-\textbf{u},Q_{\mu }\varvec{\upomega }-\varvec{\upomega }),(\textbf{u},\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })) \nonumber \\&\quad +B((\textbf{u}_{\mu },\varvec{\upomega }_{\mu }),(R_{\mu }\textbf{u}-\textbf{u}, Q_{\mu }\varvec{\upomega }-\varvec{\upomega }),(R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }))\nonumber \\&\le C\Vert (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} \Vert (R_{\mu }\textbf{u}-\textbf{u},Q_{\mu }\varvec{\upomega }-\varvec{\upomega }) \Vert _{1}. \end{aligned}$$
(56)

According to (54), we can get the equation of pressure

$$\begin{aligned}&\vert d((R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }),p-\rho _{\mu }p) \vert -\vert d((\textbf{u}-R_{\mu }\textbf{u},\varvec{\upomega }-Q_{\mu }\varvec{\upomega }),\rho _{\mu }p-p_{\mu }) \vert \nonumber \\&\quad =\vert d((R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu }),p-\rho _{\mu }p)\vert \nonumber \\&\quad \le \dfrac{C_{\min }(1-\sigma )}{4}\Vert (R_{\mu }\textbf{u}-\textbf{u}_{\mu },Q_{\mu }\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1} +\dfrac{2}{C_{\min }(1-\sigma )}\Vert p-\rho _{\mu }p\Vert ^{2}_{0}. \end{aligned}$$
(57)

Combining the three equations above, we have

$$\begin{aligned} {\left| \hspace{-0.63747pt}\left| \hspace{-0.63747pt}\left| (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }) \right| \hspace{-0.63747pt}\right| \hspace{-0.63747pt}\right| } _{1} \le C\mu \Vert \textbf{F}\Vert _{0}. \end{aligned}$$
(58)

Then, by using (11)–(13) and discrete inf-sup condition (21), we can obtain

$$\begin{aligned} \beta _{0}\Vert \rho _{\mu }p-p_{\mu }\Vert _{0}&\le C_{\max }\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} + \lambda \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} \nonumber \\&\quad \times \big (\Vert (\textbf{u},\varvec{\upomega })\Vert _{1} + \Vert (\textbf{u}_{\mu },\varvec{\upomega }_{\mu })\Vert _{1}\big ) +\Vert p-\rho _{\mu }p\Vert _{0} \nonumber \\&\le C\big ({\left| \hspace{-0.63747pt}\left| \hspace{-0.63747pt}\left| (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }) \right| \hspace{-0.63747pt}\right| \hspace{-0.63747pt}\right| }_{1}+\Vert p-\rho _{\mu }p\Vert _{0}\big ) \nonumber \\&\le C\mu \Vert \textbf{F}\Vert _{0}. \end{aligned}$$
(59)

The \(L^2(\Omega )^d\) norm error estimate can be obtained by the dual technique. Firstly, we construct the dual problem. The error equation (53) is rewritten as

$$\begin{aligned}&A((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{v}_{\mu },\varvec{\uppsi }_{\mu })) + B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u},\varvec{\upomega }),(\textbf{v}_{\mu },\varvec{\uppsi }_{\mu })) \nonumber \\&\quad + B((\textbf{u},\varvec{\upomega }),(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{v}_{\mu },\varvec{\uppsi }_{\mu })) + d((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),q_{\mu }) \nonumber \\&\quad - d((\textbf{v},\varvec{\uppsi }),p-p_{\mu }) = B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{h}),(\textbf{v}_{\mu },\varvec{\uppsi }_{\mu })). \end{aligned}$$
(60)

If we denote \((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })=(\textbf{v},\varvec{\uppsi })\), \(p-p_{\mu }=q\), \((\textbf{v}_{\mu },\varvec{\uppsi }_{\mu })=(\textbf{w},\varvec{\upphi })\), and \(q_{\mu }=s\), the dual problem can be given: find \(((\textbf{w},\varvec{\upphi }),s) \in \textbf{W} \times M\) such that

$$\begin{aligned}&A((\textbf{v},\varvec{\uppsi }),(\textbf{w},\varvec{\upphi })) +B((\textbf{v},\varvec{\uppsi }),(\textbf{u},\varvec{\upomega }),(\textbf{w},\varvec{\upphi })) +B((\textbf{u},\varvec{\upomega }),(\textbf{v},\varvec{\uppsi }),(\textbf{w},\varvec{\upphi })) \nonumber \\&\quad -d((\textbf{w},\varvec{\upphi }),q) +d((\textbf{v},\varvec{\uppsi }),s) =((\textbf{v},\varvec{\uppsi }),\textbf{G}), \quad \forall ((\textbf{v},\varvec{\uppsi }),q) \in \textbf{W} \times M, \end{aligned}$$
(61)

where \(((\textbf{v},\varvec{\uppsi }),\textbf{G})=B((\textbf{v},\varvec{\uppsi }),(\textbf{v},\varvec{\uppsi }),(\textbf{w},\varvec{\upphi }))\), \(\textbf{G}:= (G_{1},G_{2}) \in L^{2}(\Omega )^{d}\times L^{2}(\Omega )^{d}\).

According to Theorem 1, there exists a unique solution \(((\textbf{w},\varvec{\upphi }),s) \in \textbf{W} \times M\) of the dual problem (61), and it satisfies

$$\begin{aligned} \Vert (\textbf{w},\varvec{\upphi })\Vert _{1} \le C\Vert \textbf{G}\Vert _{-1}, \end{aligned}$$
(62)
$$\begin{aligned} \Vert (\textbf{w},\varvec{\upphi })\Vert _{2}+\Vert s\Vert _{1} \le C\Vert \textbf{G}\Vert _{0}. \end{aligned}$$
(63)

Further, the projection operator \(((R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi }),\rho _{\mu }s)\) satisfies

$$\begin{aligned} \Vert (\textbf{w}-R_{\mu }\textbf{w},\varvec{\upphi }-Q_{\mu }\varvec{\upphi })\Vert _{2}+\Vert s-\rho _{\mu }s\Vert _{1} \le C\mu \Vert \textbf{G}\Vert _{0}. \end{aligned}$$
(64)

We set \(\textbf{G}=(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), \ (\textbf{v},\varvec{\uppsi })=(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\), and \(q=p-p_{\mu }\) in (61), it yields that

$$\begin{aligned}&A((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{w},\varvec{\upphi })) +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u},\varvec{\upomega }),(\textbf{w},\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u},\varvec{\upomega }),(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{w},\varvec{\upphi })) -d((\textbf{w},\varvec{\upphi }),p-p_{\mu }) \nonumber \\&\quad -d((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),s) =((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })). \end{aligned}$$
(65)

Let \((\textbf{v},\varvec{\uppsi })=(R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })\), \(q=\rho _{\mu }s\) in (3) and (25), we have

$$\begin{aligned}&A((\textbf{u}\!-\!\textbf{u}_{\mu },\varvec{\upomega }\!-\!\varvec{\upomega }_{\mu }), (R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })) \!+\!B((\textbf{u},\varvec{\upomega }),(\textbf{u}\!-\!\textbf{u}_{\mu },\varvec{\upomega }\!-\!\varvec{\upomega }_{\mu }), (R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{u},\varvec{\upomega }),(R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +d((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),\rho _{\mu }s) -d((R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi }),p-p_{\mu }) = 0. \end{aligned}$$
(66)

Making a difference between the above two formulas, we get

$$\begin{aligned}&\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{0} =A((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{w}-R_{\mu }\textbf{w}, \varvec{\upphi }-Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),(\textbf{u},\varvec{\upomega }), (\textbf{w}-R_{\mu }\textbf{w},\varvec{\upphi }-Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u},\varvec{\upomega }),(\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{w}-R_{\mu }\textbf{w},\varvec{\upphi }-Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +B((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }), (R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })) \nonumber \\&\quad +d((\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu }),s-\rho _{\mu }s) -d((\textbf{w}-R_{\mu }\textbf{w}, \varvec{\upphi }-Q_{\mu }\varvec{\upphi }),p-p_{\mu }). \end{aligned}$$
(67)

By using (11), (13), (57), and (64) to simplify the right-hand side of (67), we have

$$\begin{aligned} R.H.S&\le C_{\max }\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} \Vert (\textbf{w}-R_{\mu }\textbf{w}, \varvec{\upphi }-Q_{\mu }\varvec{\upphi })\Vert _{1} \nonumber \\&\quad +2\lambda \Vert (\textbf{u},\varvec{\upomega })\Vert _{1}\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} \Vert (\textbf{w}-R_{\mu }\textbf{w},\varvec{\upphi }-Q_{\mu }\varvec{\upphi })\Vert _{1} \nonumber \\&\quad +\lambda \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1} \Vert (R_{\mu }\textbf{w}, Q_{\mu }\varvec{\upphi })\Vert _{1} \nonumber \\&\quad +C\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1}\Vert s-\rho _{\mu }s\Vert _{0} \nonumber \\&\quad +C\Vert (\textbf{w}-R_{\mu }\textbf{w},\varvec{\upphi }-Q_{\mu }\varvec{\upphi })\Vert _{1}\Vert p-p_{\mu }\Vert _{0} \nonumber \\&\le C\mu \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{0} \nonumber \\&\quad +C\mu \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert ^{2}_{1} \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{0} \nonumber \\&\quad +C\mu \big (\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{1} +\Vert p-p_{\mu }\Vert _{0}\big )\Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{0}. \end{aligned}$$
(68)

Finally, we can obtain

$$\begin{aligned} \Vert (\textbf{u}-\textbf{u}_{\mu },\varvec{\upomega }-\varvec{\upomega }_{\mu })\Vert _{0} \le C\mu ^{2}\Vert \textbf{F}\Vert _{0}. \end{aligned}$$
(69)

The proof is completed. \(\square \)

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Xing, X., Liu, D. A two-level iterative method with Newton-type linearization for the stationary micropolar fluid equations. Numer Algor 97, 475–501 (2024). https://doi.org/10.1007/s11075-023-01711-w

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