Coupled Iterative Analysis for Stationary Inductionless Magnetohydrodynamic System Based on Charge-Conservative Finite Element Method

Abstract

This paper considers charge-conservative finite element approximation and three coupled iterations of stationary inductionless magnetohydrodynamics equations in Lipschitz domain. Using a mixed finite element method, we discretize the hydrodynamic unknowns by stable velocity-pressure finite element pairs, discretize the current density and electric potential by $\varvec{H}(\mathrm{div},\varOmega )\times L^{2}(\varOmega )$-comforming finite element pairs. The well-posedness of this formula and the optimal error estimate are provided. In particular, we show that the error estimates for the velocity, the current density and the pressure are independent of the electric potential. With this, we propose three coupled iterative methods: Stokes, Newton and Oseen iterations. Rigorous analysis of convergence and stability for different iterative schemes are provided, in which we improve the stability conditions for both Stokes and Newton iterative method. Numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed methods.

Appendix

In this appendix, we compare the result obtained in Theorem 16 and the one by application of the Newton–Kantorovich theorem. The following theorem is cited from [29].

Theorem 23

Let X and Y be Banach spaces and $F:D\subset$ $X\rightarrow Y.$ Suppose that on an open convex set $D_{0}\subset D,F$ is Fréchet differentiable and that $\left\| F^{\prime }(x)-F^{\prime }(y)\right\| \le K\Vert x-y\Vert$ for $x,y\in D_{0}.$ For some $x_{0}\in D_{0}$ assume that $\varGamma _{0}=$ $\left[ F^{\prime }\left( x_{0}\right) \right] ^{-1}$ is defined on Y and $\left\| \varGamma _{0}\right\| \le \beta .$ Set $\left\| \varGamma _{0}Fx_{0}\right\| \le \eta$ and suppose that $\tau =\beta K\eta \le 1/2$. Set

$$\begin{aligned} t^{*}=(\beta K)^{-1}(1-\sqrt{1-2\tau }) \end{aligned}$$

and

$$\begin{aligned} t^{**}=(\beta K)^{-1}(1+\sqrt{1-2\tau }) \end{aligned}$$

and suppose that

$$\begin{aligned} S=\left\{ x:\left\| x-x_{0}\right\| \le t^{*}\right\} \subset D_{0}. \end{aligned}$$

Then the Newton iterates

$$\begin{aligned} x_{k+1}=x_{k}-\left[ F^{\prime }\left( x_{k}\right) \right] ^{-1}Fx_{k},\quad k=0,1,2,\cdots , \end{aligned}$$

(77)

are well-defined, lie in S,  and converge to a solution $x^{*}$ of $Fx=0$ which is unique in $D_{0}\cap \left\{ x:\left\| x_{0}-x\right\| <t^{**}\right\} .$ Moreover, if $\tau <\frac{1}{2}$ the order of convergence is at least quadratic.

To make the presentation clear, we consider the reduced formulation of Method II in the discrete kernel space $\varUpsilon _h$. Given $\varXi _{h}^{n-1}\in \varUpsilon _{h}$, find $\varXi _{h}^{n}\in \varUpsilon _{h}$ satisfying for all $\varTheta _{h}\in \varUpsilon _{h}$,

$$\begin{aligned} \mathcal {A}(\varXi _{h}^{n},\varTheta _{h})+\mathcal {C}(\varXi _{h}^{n},\varXi _{h}^{n-1},\varTheta _{h})+\mathcal {C}(\varXi _{h}^{n-1},\varXi _{h}^{n},\varTheta _{h})-\mathcal {C}(\varXi _{h}^{n-1},\varXi _{h}^{n-1},\varTheta _{h})=\langle L,\varTheta _{h}\rangle . \end{aligned}$$

(78)

The reduced formulation of the finite element approximation (21) in $\varUpsilon _h$ reads: Find $\varXi _{h}\in \varUpsilon _{h}$ satisfying for all $\varTheta _{h}\in \varUpsilon _{h}$,

$$\begin{aligned} \mathcal {A}(\varXi _{h},\varTheta _{h})+\mathcal {C}(\varXi _{h},\varXi _{h},\varTheta _{h})= \langle L,\varTheta _{h}\rangle . \end{aligned}$$

The equivalence to original formulation follows from the inf-sup condition (27). In the operator level, it can be rewritten as

$$\begin{aligned} F\varXi _{h}=0, \end{aligned}$$

(79)

where $F:\varUpsilon _h\rightarrow \varUpsilon _h^{\prime }$ is defined by

$$\begin{aligned} \left\langle F\varXi _{h},\varTheta _{h}\right\rangle :=\mathcal {A}(\varXi _{h},\varTheta _{h})+\mathcal {C}(\varXi _{h},\varXi _{h},\varTheta _{h})-\langle L,\varTheta _{h}\rangle ,\forall \varXi _h,\varTheta _h\in \varUpsilon _h. \end{aligned}$$

Then the Fréchet derivative of F at $\varXi _h$, $F^{\prime }(\varXi _{h})\in \mathcal {L}\left( \varUpsilon _h,\varUpsilon _h^{\prime }\right)$, is given by

$$\begin{aligned} \left\langle \left( F^{\prime }(\varXi _{h})\right) \varPhi _{h},\varTheta _{h}\right\rangle :=\mathcal {A}(\varPhi _{h},\varTheta _{h})+\mathcal {C}(\varPhi _{h},\varXi _{h},\varTheta _{h})+\mathcal {C}(\varXi _{h},\varPhi _{h},\varTheta _{h})\quad \forall \varPhi _{h},\varTheta _h\in \varUpsilon _h. \end{aligned}$$

Note that (77) corresponds to (78).

Next, we identify the key parameters in Newton-Kantorovich theorem successively. For the estimation for K, by the definition of $F^{\prime }$, it yields that for any $\varXi _{h}^{\star },\varXi _{h}^{\star \star }\in \varUpsilon _{h}$

$$\begin{aligned}&\left\| F^{\prime }(\varXi _{h}^{\star })-F^{\prime }(\varXi _{h}^{\star \star })\right\| _{\mathcal {L}\left( \varUpsilon _h,\varUpsilon _h^{\prime }\right) }\\&\quad =\sup _{\varPhi _{h},\varTheta _h\in \varUpsilon _h}\frac{\left\langle \left( F^{\prime }(\varXi _{h}^{\star })-F^{\prime }(\varXi _{h}^{\star \star })\right) \varPhi _{h},\varTheta _{h}\right\rangle }{\left\| \varPhi _{h}\right\| _{1}\left\| \varTheta _{h}\right\| _{1}}\\&\quad =\sup _{\varPhi _{h},\varTheta _h\in \varUpsilon _h}\frac{\left| \mathcal {C}(\varPhi _{h},\varXi _{h}^{\star }-\varXi _{h}^{\star \star },\varTheta _{h})+\mathcal {C}(\varXi _{h}^{\star }-\varXi _{h}^{\star \star },\varPhi _{h},\varTheta _{h})\right| }{\left\| \varPhi _{h}\right\| _{1}\left\| \varTheta _{h}\right\| _{1}}\\&\quad \le 2\lambda _{2}\left\| \varXi _{h}^{\star }-\varXi _{h}^{\star \star }\right\| _{1}. \end{aligned}$$

It mean that $K=2\lambda _{2}$.

Then, we estimate $\beta$. From (65) and the definition of $F^{\prime }$, we have that for any $\varPhi _h \in \varUpsilon _h$,

$$\begin{aligned} \left\langle F^{\prime }\left( \varXi _{h}^{0}\right) \varPhi _{h}, \varPhi _{h}\right\rangle \ge C_{\min }(1-\sigma )\left\| \varPhi _{h}\right\| _{1}^{2}. \end{aligned}$$

(80)

Using (28), we have $C_{\min }(1-\sigma )>0$. This implies that

$$\begin{aligned} \left\| F^{\prime }(\varXi _{h}^{0})^{-1}\right\| _{\mathcal {L}\left( \varUpsilon _h^{\prime },\varUpsilon _h\right) }\le \frac{1}{C_{\min }\left( 1-\sigma \right) }. \end{aligned}$$

Thus, we have

$$\begin{aligned} \beta =\frac{1}{C_{\min }\left( 1-\sigma \right) }. \end{aligned}$$

Now we are in a position to estimate $\eta =\left\| \varGamma _{0}F\varXi _{h}^{0}\right\| _1$. Note that $\varPhi _{h}:=\varGamma _{0} F \varXi _{h}^{0}$ can be obtained by solving the following problem: Find $\varPhi _{h}\in \varUpsilon _h$ such that

$$\begin{aligned} \left\langle F^{\prime }\left( \varXi _{h}^{0}\right) \varPhi _{h}, \varTheta _{h}\right\rangle =\left\langle F \varXi _{h}^{0}, \varTheta _{h}\right\rangle ,\quad \varTheta _{h}\in \varUpsilon _{h}. \end{aligned}$$

By (34) and the definitions of F and $F^{\prime }$, the problem can be simplified as: Find $\varPhi _{h}\in \varUpsilon _h$ such that for all $\varTheta _{h}\in \varUpsilon _{h}$,

$$\begin{aligned} \left\langle F^{\prime }\left( \varXi _{h}^{0}\right) \varPhi _{h}, \varTheta _{h}\right\rangle =\mathcal {C}(\varXi _{h}^0,\varXi _{h}^{0},\varTheta _{h}). \end{aligned}$$

Taking $\varTheta _{h}=\varPhi _{h}$, using (80) and (26), we deduce that

$$\begin{aligned} C_{\min }(1-\sigma )\left\| \varPhi _{h}\right\| _{1}\le \lambda _{2}\left\| \varXi _{h}^{0}\right\| _{1}^{2}. \end{aligned}$$

Invoking with (38), it gives that

$$\begin{aligned} \left\| \varPhi _{h}\right\| _{1}\le \frac{\lambda _2\left\| \varXi _{h}^{0}\right\| _{1}^2}{C_{\min }\left( 1-\sigma \right) } \le \frac{\sigma \Vert L\Vert _{*}}{C_{\min }\left( 1-\sigma \right) }. \end{aligned}$$

This yields

$$\begin{aligned} \eta =\frac{\sigma \Vert L\Vert _{*}}{C_{\min }\left( 1-\sigma \right) }. \end{aligned}$$

Finally, we apply Theorem 23 to the operator equation (79). Taking $X=D=D_0=\varUpsilon _h$, $Y=\varUpsilon _h^{\prime }$ and $x=\varTheta _h$ in Theorem 23, and then using (40) and the fact that $\tau =\frac{2\sigma ^2}{\left( 1-\sigma \right) ^{2}},$ we get the global quadratic-convergent condition,

$$\begin{aligned} \frac{2\sigma _N^2}{\left( 1-\sigma _N\right) ^{2}}<1/2. \end{aligned}$$

Direct calculation shows that $0<\sigma _N<1/3$. Thus, our result is a little better than one by application of the Newton-Kantorovich theorem straightway.