Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data

Abstract

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $L^{3+\delta }$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a $\ell ^2(W^{1,3+\delta })$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.

Appendix: Well-posedness of the PDE problem (1.6)–(1.11)

Theorem A.1

There exists a unique weak solution of (1.6)–(1.11) with the following regularity:

$$\begin{aligned}&\psi \in C([0,T];{\mathcal L}^2)\cap L^\infty (0,T;{\mathcal H}^1) , \quad \partial _t\psi \in L^2(0,T;{\mathcal L}^2),\\&|\psi |\le 1\quad \text{ a.e. }~\text{ in }~\,\Omega \times (0,T),\\&\mathbf{A}\in C([0,T];\mathbf{L}^2)\cap L^{\infty }(0,T;\mathbf{H}(\mathrm{curl},\mathrm{div})) , \quad \partial _t\mathbf{A}\in L^2(0,T;\mathbf{L}^2) . \end{aligned}$$

Proof

Global well-posedness of time-dependent Ginzburg–Landau equations in curved polyhedra was proved in [34]. The convergence of numerical solutions proved in this paper yields an alternative proof.

In fact, from (3.95) and (3.98) we see that there exists a weak solution $(\Psi ,{\varvec{\Lambda }})$ of (1.6)–(1.11) with the regularity above. It remains to prove the uniqueness of the weak solution.

Suppose that there are two weak solutions $(\psi ,\mathbf{A})$ and $(\Psi ,{\varvec{\Lambda }})$ for the system (1.6)–(1.11). Then we define $e=\psi -\Psi$ and $\mathbf{E}=\mathbf{A}-{\varvec{\Lambda }}$ and consider the difference equations

$$\begin{aligned}&\int _0^T\Big [\big (\eta \partial _t e ,\varphi \big ) + \frac{1}{\kappa ^2}\big (\nabla e, \nabla \varphi \big ) + \big (|\mathbf{A}|^2 e, \varphi \big ) \Big ]\mathrm{d}t\nonumber \\&\quad =\int _0^T\Big [-\frac{i}{\kappa }\big (\mathbf{A}\cdot \nabla e ,\varphi \big ) -\frac{i}{\kappa }\big (\mathbf{E}\cdot \nabla \Psi ,\varphi \big ) +\frac{i}{\kappa }\big (e \mathbf{A},\nabla \varphi \big ) +\frac{i}{\kappa }\big (\Psi \mathbf{E},\nabla \varphi \big ) \nonumber \\&\qquad - \big ((|\mathbf{A}|^2 -|{\varvec{\Lambda }}|^2) \Psi , \varphi \big ) -\big ( (|\psi |^{2}-1) \psi -(|\Psi |^{2}-1) \Psi ,\varphi \big )\Big ]\mathrm{d}t\nonumber \\&\qquad -\int _0^T\big (i\eta \kappa \psi \nabla \cdot \mathbf{E} +i\eta \kappa e\nabla \cdot {\varvec{\Lambda }},\varphi \big )\mathrm{d}t , \end{aligned}$$

(A.1)

and

$$\begin{aligned}&\int _0^T\Big [\big (\partial _t\mathbf{E} ,\mathbf{a}\big ) + \big (\nabla \times \mathbf{E},\nabla \times \mathbf{a}\big ) +\big (\nabla \cdot \mathbf{E},\nabla \cdot \mathbf{a}\big ) \Big ]\mathrm{d}t\nonumber \\&\quad =-\int _0^T\mathrm{Re} \bigg ( \frac{i}{\kappa }( \overline{\psi }\nabla \psi - \overline{\Psi }\nabla \Psi ) + \mathbf{A}(|\psi |^2-|\Psi |^2)+|\Psi |^2 \mathbf{E},\, \mathbf{a}\bigg ) \mathrm{d}t , \end{aligned}$$

(A.2)

which hold for any $\varphi \in L^2(0,T;{\mathcal H}^1)$ and $\mathbf{a}\in L^2(0,T;\mathbf{H}(\mathrm{curl},\mathrm{div}))$. Choosing $\varphi (x,t)=e(x,t)1_{(0,t')}(t)$ in (A.1) and considering the real part, we obtain

$$\begin{aligned}&\frac{\eta }{2} \Vert e(\cdot ,t') \Vert _{\mathcal{L}^2}^2 + \int _0^{t'}\Big (\frac{1}{\kappa ^2}\Vert \nabla e\Vert _{\mathcal{L}^2}^2 + \Vert \mathbf{A} e\Vert _{\mathbf{L}^2}^2\Big )\mathrm{d}t \\&\quad \le \int _0^{t'}\Big (C\Vert \mathbf{A}\Vert _{\mathbf{L}^{3+\delta }}\Vert \nabla e\Vert _{\mathcal{L}^2} \Vert e\Vert _{\mathcal{L}^{6-4\delta /(1+\delta )}} +C\Vert \mathbf{E}\Vert _{\mathbf{L}^{3+\delta }}\Vert \nabla \Psi \Vert _{\mathcal{L}^2} \Vert e\Vert _{\mathcal{L}^{6-4\delta /(1+\delta )}}\\&\qquad +C\Vert e\Vert _{\mathcal{L}^{6-4\delta /(1+\delta )}} \Vert \mathbf{A}\Vert _{\mathbf{L}^{3+\delta }} \Vert \nabla e\Vert _{\mathcal{L}^2} +C\Vert \mathbf{E}\Vert _{\mathbf{L}^2}\Vert \nabla e\Vert _{\mathcal{L}^2} \\&\qquad +C(\Vert \mathbf{A}\Vert _{\mathbf{L}^{3+\delta }}+\Vert {\varvec{\Lambda }}\Vert _{\mathbf{L}^{3+\delta }}) \Vert \mathbf{E}\Vert _{\mathbf{L}^2} \Vert e\Vert _{\mathcal{L}^{6-4\delta /(1+\delta )}} +C\Vert e\Vert _{\mathcal{L}^2}^2 +C\Vert \nabla \cdot \mathbf{E}\Vert _{L^2}\Vert e\Vert _{\mathcal{L}^2}\Big )\mathrm{d}t \\&\quad \le \int _0^{t'}\Big (C\Vert \nabla e\Vert _{L^2} (C_\epsilon \Vert e\Vert _{\mathcal{L}^2}+\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2}) +C\Vert \mathbf{E}\Vert _{\mathbf{H}(\mathrm{curl},\mathrm{div})} (C_\epsilon \Vert e\Vert _{\mathcal{L}^2}+\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2})\\&\qquad +C\Vert \nabla e\Vert _{\mathcal{L}^2}(C_\epsilon \Vert e\Vert _{\mathcal{L}^2} +\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2}) +C\Vert \mathbf{E}\Vert _{\mathbf{L}^2}\Vert \nabla e\Vert _{\mathcal{L}^2} \\&\qquad +C\Vert \mathbf{E}\Vert _{\mathbf{L}^2}(C_\epsilon \Vert e\Vert _{\mathcal{L}^2} +\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2}) +C\Vert e\Vert _{\mathcal{L}^2}^2 +C\Vert \nabla \cdot \mathbf{E}\Vert _{L^2}\Vert e\Vert _{\mathcal{L}^2} \Big )\mathrm{d}t\\&\quad \le \int _0^{t'}\Big (\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2}^2+ \epsilon \Vert \nabla \times \mathbf{E}\Vert _{\mathbf{L}^2}^2 + \epsilon \Vert \nabla \cdot \mathbf{E}\Vert _{L^2}^2 +C_\epsilon \Vert e\Vert _{\mathcal{L}^2}^2 + C_\epsilon \Vert \mathbf{E}\Vert _{\mathbf{L}^2}^2\Big )\mathrm{d}t , \end{aligned}$$

where $\epsilon$ can be arbitrarily small. By choosing $\mathbf{a}(x,t)=\mathbf{E}(x,t)1_{(0,t')}(t)$ in (A.2), we get

$$\begin{aligned}&\frac{1}{2}\Vert \mathbf{E}(\cdot ,t')\Vert _{\mathbf{L}^2}^2 +\int _0^{t'}\Big (\Vert \nabla \times \mathbf{E} \Vert _{\mathbf{L}^2}^2 +\Vert \nabla \cdot \mathbf{E} \Vert _{\mathbf{L}^2}^2 \Big )\mathrm{d}t\\&\quad \le \int _0^{t'}\Big (C \Vert e\Vert _{\mathcal{L}^{6-4\delta /(1+\delta )}} \Vert \nabla \psi \Vert _{L^2}\Vert \mathbf{E}\Vert _{\mathbf{L}^{3+\delta }} +C\Vert \nabla e \Vert _{\mathcal{L}^2} \Vert \mathbf{E}\Vert _{\mathbf{L}^2} \\&\qquad + (\Vert e\Vert _{L^{6-4\delta /(1+\delta )}}\Vert \mathbf{A}\Vert _{\mathbf{L}^{3+\delta }} +\Vert \mathbf{E}\Vert _{\mathbf{L}^2})\Vert \mathbf{E}\Vert _{L^2}\Big )\mathrm{d}t\\&\quad \le \int _0^{t'}\Big (C(C_\epsilon \Vert e\Vert _{\mathcal{L}^2} +\epsilon \Vert \nabla e \Vert _{\mathcal{L}^2}) \Vert \mathbf{E}\Vert _{\mathbf{H}(\mathrm{curl},\mathrm{div})} +\Vert \nabla e \Vert _{\mathcal{L}^2} \Vert \mathbf{E}\Vert _{L^2} \\&\qquad + (\Vert e\Vert _{\mathcal{L}^2} +\Vert \nabla e\Vert _{\mathcal{L}^2} +\Vert \mathbf{E}\Vert _{\mathbf{L}^2})\Vert \mathbf{E}\Vert _{\mathbf{L}^2}\Big )\mathrm{d}t\\&\quad \le \int _0^{t'}\Big (\epsilon \Vert \nabla e\Vert _{\mathcal{L}^2}^2 +\epsilon \Vert \nabla \times \mathbf{E}\Vert _{\mathbf{L}^2} +\epsilon \Vert \nabla \cdot \mathbf{E}\Vert _{\mathbf{L}^2} +C_\epsilon \Vert e\Vert _{\mathcal{L}^2}^2 + C_\epsilon \Vert \mathbf{E}\Vert _{\mathbf{L}^2}^2\Big )\mathrm{d}t , \end{aligned}$$

where $\epsilon$ can be arbitrarily small. By choosing $\epsilon <\frac{1}{4} \min (1, \kappa ^{-2} )$ and summing up the two inequalities above, we have

$$\begin{aligned}&\frac{\eta }{2}\Vert e(\cdot ,t')\Vert _{L^2}^2 +\frac{1}{2}\Vert \mathbf{E}(\cdot ,t')\Vert _{L^2}^2 \le \int _0^{t'}\Big (C\Vert e\Vert _{L^2}^2 +C\Vert \mathbf{E}\Vert _{L^2}^2\Big )\mathrm{d}t , \end{aligned}$$

which implies

$$\begin{aligned}&\max _{t\in (0,T)} \bigg (\frac{\eta }{2}\Vert e\Vert _{L^2}^2 +\frac{1}{2}\Vert \mathbf{E}\Vert _{L^2}^2\bigg )=0 \end{aligned}$$

via Gronwall’s inequality. Uniqueness of the weak solution is proved. $\square$