Two-Level Coupled and Decoupled Parallel Correction Methods for Stationary Incompressible Magnetohydrodynamics

Abstract

In this paper, we propose two-level coupled correction and decoupled parallel correction finite element methods for solving the stationary magnetohydrodynamics (MHD) equations. We prove the error estimates for the methods which show that if coarse mesh size $(H)$ and fine mesh size $(h)$ satisfy the relation $H=O(\sqrt{h})$, the methods provide optimal convergence rates. Further, we study the dependence of the errors of the methods on parameters. Numerically, investigations for 2D/3D Hartmann flows with different physical parameters are conducted to validate theoretical analyses, which show the efficiency of the methods to solve the MHD problems.

Appendix

Proof of Theorem 1

We define a function space

$$\begin{aligned} Z = \left\{ ({\mathbf {u}}, {\mathbf {b}}) \in {\mathbf {H}}^{1}_{0}(\varOmega ) \times {\mathbf {H}}^{1}_{n}(\varOmega ): B(q;{\mathbf {u}}, {\mathbf {b}})=0, \forall q \in L^{2}_{0}(\varOmega )\right\} . \end{aligned}$$

For $\forall ({\mathbf {w}},{\mathbf {d}}) \in Z,$ the linear equations: solving ${\mathbf {u}} \in {\mathbf {H}}^{1}_{0}(\varOmega ), {\mathbf {b}} \in {\mathbf {H}}^{1}_{n}(\varOmega )$ and $p \in L^{2}_{0}(\varOmega )$ from

$$\begin{aligned}&A({\mathbf {u}}, {\mathbf {b}}; {\mathbf {v}}, {\mathbf {c}})+C({\mathbf {w}}, {\mathbf {d}}; {\mathbf {u}}, {\mathbf {b}}; {\mathbf {v}}, {\mathbf {c}})+B(p; {\mathbf {v}}, {\mathbf {c}})-B(q; {\mathbf {u}},{\mathbf {b}})= \langle {\mathbf {F}}, ({\mathbf {v}}, {\mathbf {c}})\rangle , \end{aligned}$$

$\forall ({\mathbf {v}}, {\mathbf {c}}) \in \mathbf {H}_{0}^{1}(\varOmega ) \times \mathbf {H}_{n}^{1}{(\varOmega )}, q \in L_{0}^{2}(\varOmega ),$ admit a unique solution $({\mathbf {u}}, {\mathbf {b}} ) \in Z$ and $p \in L^{2}_{0}(\varOmega )$ satisfying

$$\begin{aligned} ||({\mathbf {u}}, {\mathbf {b}})||_{1}\le \frac{||{\mathbf {F}}||_{-1}}{\underline{\nu }}. \end{aligned}$$

For details, see Part I. Corollary 4.1 in [12]. We define a map

$$\begin{aligned} \Phi : Z\rightarrow Z, \quad ({\mathbf {w}}, {\mathbf {d}})\mapsto ({\mathbf {u}}, {\mathbf {b}}). \end{aligned}$$

Supposing $\Phi ({\mathbf {w}}_{i}, {\mathbf {d}}_{i})=({\mathbf {u}}_{i}, {\mathbf {b}}_{i}), i=1,2$ and using (9), (10), we have

$$\begin{aligned}&\underline{\nu }||\Phi ({\mathbf {w}}_{1}, {\mathbf {d}}_{1})- \Phi ({\mathbf {w}}_{2}, {\mathbf {d}}_{2})||_{1}^{2}=\underline{\nu }||({\mathbf {u}}_{1}- {\mathbf {u}}_{2}, {\mathbf {b}}_{1}- {\mathbf {b}}_{2})||_{1}^{2}\\&\quad \le A({\mathbf {u}}_{1}-{\mathbf {u}}_{2}, {\mathbf {b}}_{1}- {\mathbf {b}}_{2}; {\mathbf {u}}_{1}-{\mathbf {u}}_{2}, {\mathbf {b}}_{1}-{\mathbf {b}}_{2})\\&\quad =-C({\mathbf {w}}_{1}-{\mathbf {w}}_{2}, {\mathbf {d}}_{1}-{\mathbf {d}}_{2}; {\mathbf {u}}_{2}, {\mathbf {b}}_{2}; {\mathbf {u}}_{1}-{\mathbf {u}}_{2}, {\mathbf {b}}_{1}- {\mathbf {b}}_{2} ) \\&\quad \le \frac{N ||{\mathbf {F}}||_{-1} }{\underline{\nu }}||({\mathbf {w}}_{1}-{\mathbf {w}}_{2}, {\mathbf {d}}_{1}-{\mathbf {d}}_{2})||_{1} ||({\mathbf {u}}_{1}-{\mathbf {u}}_{2}, {\mathbf {b}}_{1}-{\mathbf {b}}_{2})||_{1}. \end{aligned}$$

Thus,

$$\begin{aligned} ||\Phi ({\mathbf {w}}_{1}, {\mathbf {d}}_{1})- \Phi ({\mathbf {w}}_{2}, {\mathbf {d}}_{2})||_{1}\le \frac{N ||{\mathbf {F}}||_{-1}}{\underline{\nu }^{2}}||({\mathbf {w}}_{1}-{\mathbf {w}}_{2}, {\mathbf {d}}_{1}-{\mathbf {d}}_{2})||_{1}. \end{aligned}$$

By the assumption in the Theorem, we know the map $\Phi$ is a contraction map from $Z$ to $Z.$ Using Banach contraction mapping principle, the map $\Phi$ admits one unique fixed point in $Z,$ which is the unique solution of (4). $\square$

Proof of Theorem 3

The existence, uniqueness and $H^{1}$ stability of discrete solution can be proved similar to Theorem 1. We only prove the error estimate (24).

Let ${\mathbf {e}}_{u}= R({\mathbf {u}},p)- {\mathbf {u}}_{h}, {\mathbf {e}}_{b}= \varLambda {\mathbf {b}}-{\mathbf {b}}_{h}, e_{p}= Q({\mathbf {u}},p)-p.$ The problem (4) subtracting problem (18) gives the error equation:

$$\begin{aligned}&A({\mathbf {u}}-{\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h}; {\mathbf {v}}_{h},{\mathbf {c}}_{h})+C({\mathbf {u}}-{\mathbf {u}}_{h}, {\mathbf {b}}-{\mathbf {b}}_{h}; {\mathbf {u}}, {\mathbf {b}}; {\mathbf {v}}_{h},{\mathbf {c}}_{h})\nonumber \\&\quad +C({\mathbf {u}}_{h},{\mathbf {b}}_{h}; {\mathbf {u}}-{\mathbf {u}}_{h},{\mathbf {b}}-{\mathbf {b}}_{h}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h}) \nonumber \\&\quad +B(p- p_{h};{\mathbf {v}}_{h}, {\mathbf {c}}_{h})- B(q_{h}; {\mathbf {u}}- {\mathbf {u}}_{h},{\mathbf {b}}-{\mathbf {b}}_{h})=0, \end{aligned}$$

(48)

for all $({\mathbf {v}}_{h}, q_{h}, {\mathbf {c}}_{h}) \in V_{h}\times Q_{h} \times C_{h}.$ By this error equation, we have for all $({\mathbf {v}}_{h}, q_{h}, {\mathbf {c}}_{h}) \in V_{h}\times Q_{h} \times C_{h},$

$$\begin{aligned}&A({\mathbf {e}}_{u}, {\mathbf {e}}_{b}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h}) +C({\mathbf {e}}_{u}, {\mathbf {e}}_{b}; {\mathbf {u}},{\mathbf {b}}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h}) +C({\mathbf {u}}_{h}, {\mathbf {b}}_{h}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h})+B(e_{p}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h})\\&\quad - B(q_{h}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b})\\&\quad = C(R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {u}}, {\mathbf {b}}; {\mathbf {v}}_{h},{\mathbf {c}}_{h})+C({\mathbf {u}}_{h}, {\mathbf {b}}_{h};R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {v}}_{h}, {\mathbf {c}}_{h}). \end{aligned}$$

Taking ${\mathbf {v}}_{h}= {\mathbf {e}}_{u}, {\mathbf {c}}_{h}={\mathbf {e}}_{b}, q_{h}=e_{p},$ by (11) we obtain

$$\begin{aligned}&A({\mathbf {e}}_{u}, {\mathbf {e}}_{b}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b}) + C({\mathbf {e}}_{u}, {\mathbf {e}}_{b}; {\mathbf {u}}, {\mathbf {b}};{\mathbf {e}}_{u},{\mathbf {e}}_{b}) \nonumber \\&\quad = C(R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {u}}, {\mathbf {b}}; {\mathbf {e}}_{u},{\mathbf {e}}_{b})+C({\mathbf {u}}_{h}, {\mathbf {b}}_{h};R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}};{\mathbf {e}}_{u}, {\mathbf {e}}_{b}).\nonumber \\ \end{aligned}$$

(49)

Using H $\ddot{o}$ lder inequality, Sobolev inequalities and inverse inequality, the right hand side of (49) can be estimated as follows:

$$\begin{aligned} RHS&\le c ||{\mathbf {u}}- R({\mathbf {u}}, p) ||_{0}\left( ||{\mathbf {u}}||_{2} ||\nabla {\mathbf {e}}_{u}||_{0} + ||{\mathbf {u}}_{h}||_{L^{\infty }} ||\nabla {\mathbf {e}}_{u}||_{0} + ||\nabla {\mathbf {u}}_{h}||_{0} ||{\mathbf {e}}_{u}||_{L^{\infty }}\right. \nonumber \\&\quad \left. +S||{\mathbf {b}}_{h}||_{L^{\infty }} ||{\mathbf {e}}_{b}||_{{\mathbf {H}}^{1}_{n}} \right) + c ||{\mathbf {b}}- \varLambda {\mathbf {b}} ||_{0}\left( S||{\mathbf {b}}||_{2} ||\nabla {\mathbf {e}}_{u}||_{0} + S||{\mathbf {e}}_{b}||_{{\mathbf {H}}^{1}_{n}} || {\mathbf {u}}||_{2}\right. \nonumber \\&\left. \quad + S ||{\mathbf {b}}_{h}||_{L^{\infty }}||\nabla {\mathbf {e}}_{u}||_{0} +S||{\mathbf {e}}_{u}||_{L^{\infty }} ||{\mathbf {b}}_{h}||_{{\mathbf {H}}^{1}_{n}} \right) \nonumber \\&\le c ||{\mathbf {u}}\!-\! R({\mathbf {u}}, p) ||_{0}\left( ||{\mathbf {u}}||_{2} ||\nabla {\mathbf {e}}_{u}||_{0} \!+\! h^{-1/2}||\nabla {\mathbf {u}}_{h}||_{0} ||\nabla {\mathbf {e}}_{u}||_{0} \!+\!Sh^{-1/2}||{\mathbf {b}}_{h}||_{{\mathbf {H}}^{1}_{n}} ||{\mathbf {e}}_{b}||_{{\mathbf {H}}^{1}_{n}} \!\right) \nonumber \\&\quad + c ||{\mathbf {b}}- \varLambda {\mathbf {b}} ||_{0}\left( S||{\mathbf {b}}||_{2} ||\nabla {\mathbf {e}}_{u}||_{0} + S||{\mathbf {e}}_{b}||_{{\mathbf {H}}^{1}_{n}} || {\mathbf {u}}||_{2} + Sh^{-1/2} ||{\mathbf {b}}_{h}||_{{\mathbf {H}}^{1}_{n}}||\nabla {\mathbf {e}}_{u}||_{0} \right) \nonumber \\&\le ch^{-1/2} || ({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1} ||({\mathbf {u}}- R({\mathbf {u}}, p), {\mathbf {b}}- \varLambda {\mathbf {b}}) ||_{0} \left( ||({\mathbf {u}}, {\mathbf {b}})||_{2}+||({\mathbf {u}}_{h}, {\mathbf {b}}_{h})||_{1} \right) . \end{aligned}$$

(50)

By (9), (10) and (14), the left hand side can be estimated as follows:

$$\begin{aligned} LHS\ge \underline{\nu } ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}^{2}-N ||({\mathbf {u}}, {\mathbf {b}})||_{1}||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}^{2} \ge \underline{\nu } (1-\sigma ) ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}^{2}. \end{aligned}$$

(51)

Combing (50) with (51), and using (20), (22), (23), (15), we get

$$\begin{aligned} \underline{\nu } (1\!-\!\sigma ) ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}\le c_{S}h^{3/2}||{\mathbf {F}}||_{0}\left( \underline{\nu }^{-2}||{\mathbf {F}}||_{0} \!+\! \underline{\nu }^{-2}||{\mathbf {F}}||_{-1}\right) \le c_{S}h^{3/2}||{\mathbf {F}}||_{0}\left( 1 \!+\! \frac{||{\mathbf {F}}||_{0}}{||{\mathbf {F}}||_{-1}}\right) . \end{aligned}$$

Thus, we deduce that

$$\begin{aligned} \underline{\nu } ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}\le c_{S}h ||{\mathbf {F}}||_{0}\left( 1 + \frac{||{\mathbf {F}}||_{0}}{||{\mathbf {F}}||_{-1}}\right) . \end{aligned}$$

(52)

Then, by triangle inequality and (22), (20), we obtain

$$\begin{aligned}&\underline{\nu } ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1} \le \underline{\nu } ||({\mathbf {u}}- R ({\mathbf {u}},p), {\mathbf {b}}- \varLambda {\mathbf {b}})||_{1} + \underline{\nu } ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1} \le C h. \end{aligned}$$

(53)

In the below, we deduce the $L^{2}$ error estimate through superconvergence property of the Stokes projection and Maxwell projection, which allow us to avoid using duality technique. The right hand side of (49) can be rewritten as

$$\begin{aligned}&RHS= C(R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {u}}, {\mathbf {b}}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b}) +C(-{\mathbf {e}}_{u}, -{\mathbf {e}}_{b}; R({\mathbf {u}}, p)\!-\!{\mathbf {u}}, \varLambda {\mathbf {b}}\!-\!{\mathbf {b}}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b})\\&\quad +C(R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b})\\&\quad +C({\mathbf {u}}, {\mathbf {b}}; R({\mathbf {u}}, p)-{\mathbf {u}}, \varLambda {\mathbf {b}}-{\mathbf {b}}; {\mathbf {e}}_{u}, {\mathbf {e}}_{b}) \end{aligned}$$

Using (10), (12), (13), we have

$$\begin{aligned}&RHS \le c||({\mathbf {u}}- R({\mathbf {u}}, p), {\mathbf {b}}- \varLambda {\mathbf {b}}) ||_{0} ||({\mathbf {u}}, {\mathbf {b}})||_{2}|| ({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}\nonumber \\&\quad +c ||({\mathbf {u}}- R({\mathbf {u}}, p), {\mathbf {b}}- \varLambda {\mathbf {b}}) ||_{1}|| ({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}^{2} \nonumber \\&\quad +c ||({\mathbf {u}}- R({\mathbf {u}}, p), {\mathbf {b}}- \varLambda {\mathbf {b}}) ||_{1}^{2}|| ({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1}. \end{aligned}$$

(54)

Combing (51) with (54), and by (22), (20), (15) and (52), we derive that

$$\begin{aligned}&\underline{\nu } (1-\sigma ) ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1} \le c_{S}\left( \frac{3 ||{\mathbf {F}} ||_{0}}{||{\mathbf {F}} ||_{-1}} + \left( \frac{ ||{\mathbf {F}} ||_{0}}{||{\mathbf {F}} ||_{-1}}\right) ^{2}\right) ||{\mathbf {F}} ||_{0} h^{2} \le C h^{2}. \end{aligned}$$

Thus, by triangle inequality, (22), (20) and (15), we have

$$\begin{aligned}&\underline{\nu } (1-\sigma ) ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{0} \le \underline{\nu } ||({\mathbf {u}}- R({\mathbf {u}}, p), {\mathbf {b}}- \varLambda {\mathbf {b}}) ||_{0}\nonumber \\&\quad +c \underline{\nu } (1-\sigma ) ||({\mathbf {e}}_{u}, {\mathbf {e}}_{b})||_{1} \le C h^{2}. \end{aligned}$$

(55)

Finally, through (48), (8) and (10), we drive that

$$\begin{aligned}&B(p_{h}-\pi _{2}p; {\mathbf {v}}_{h}, {\mathbf {c}}_{h}) \le \overline{\nu } ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1} ||({\mathbf {v}}_{h}, {\mathbf {c}}_{h})||_{1}\\&\quad +c ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1} ||({\mathbf {u}}, {\mathbf {b}})||_{1} ||({\mathbf {v}}_{h}, {\mathbf {c}}_{h})||_{1}\\&\quad + c ||({\mathbf {u}}_{h}, {\mathbf {b}}_{h})||_{1} ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1} ||({\mathbf {v}}_{h}, {\mathbf {c}}_{h})||_{1} +c ||p-\pi _{2}p||_{0} ||\nabla {\mathbf {v}}_{h}||_{0}. \end{aligned}$$

Then, using (16), (14), (23) and (53), we have

$$\begin{aligned} \hat{\beta } ||p_{h} - \pi _{2}p||_{0}&\le \overline{\nu } ||({\mathbf {u}} - {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1} + c \underline{\nu }^{-2}||{\mathbf {F}}||_{-1}\underline{\nu } ||({\mathbf {u}}- {\mathbf {u}}_{h}, {\mathbf {b}}- {\mathbf {b}}_{h})||_{1}\\&\qquad +c ||p-\pi _{2}p||_{0}\\&\le \frac{\overline{\nu }}{\underline{\nu }} C h + C h+c ||p-\pi _{2}p||_{0}. \end{aligned}$$

Next, by triangle inequality and approximation property (17), we arrive at

$$\begin{aligned} \underline{\nu } \overline{\nu }^{-1}||p-p_{h}||_{0} \le C h. \end{aligned}$$

(56)

Finally, combining (53), (55) and (56) gives the error estimate (24). $\square$