A Stabilized Semi-Implicit Euler Gauge-Invariant Method for the Time-Dependent Ginzburg–Landau Equations

Abstract

In this paper, we propose and analyze a stabilized semi-implicit Euler gauge-invariant method for numerical solution of the time-dependent Ginzburg–Landau (TDGL) equations in the two-dimensional space. The proposed method uses the well-known gauge-invariant finite difference approximations with staggered variables in a rectangular mesh, and a stabilized semi-implicit Euler discretization for time integration. The resulted fully discrete system leads to two decoupled linear systems at each time step, thus can be efficiently solved. We prove that the proposed method unconditionally preserves the point-wise boundedness of the solution and is also energy-stable. Moreover, the proposed method under the zero-electric potential gauge is shown to be equivalent to a mass-lumped version of the lowest order rectangular Nédélec edge element approximation and the Lorentz gauge scheme to a mass-lumped mixed finite element method. These indicate the method is also effective in solving the TDGL problems in non-convex domains although the solutions are often of low-regularity in such situation. Various numerical experiments are also presented to demonstrate effectiveness and robustness of the proposed method.

Appendices

Appendix

An Example with Explicit Coefficient Matrices

In this appendix, we provide an example on a $2\times 2$ mesh to show the equivalence of the gauge invariant finite difference approximation and the mass-lumped mixed finite element method. To this end, we shall only consider the Lorentz gauge scheme for ${\mathbf {A}}$ (2.34) and the mixed FEM (4.8)–(4.9). For simplicity, we take $\Omega = (0,1)^2$ and set $h= \frac{1}{2}$. There are 9 degrees of freedom (DOFs) for $\Phi$ and 12 DOFs for ${\mathbf {A}}$, see Fig. 23.

Fig. 23

An illustration of labeled global DOFs for ${\mathbf {A}}$ and $\Phi$ on a $2 \times 2$ mesh

The Lorentz gauge scheme (2.33)–(2.34) can be written as

$$\begin{aligned} \frac{1}{\Delta t} \overrightarrow{A}^{n} - \nabla _h \, D_h \overrightarrow{A}^{n} + {\mathbf {K}}_1 \overrightarrow{A}^{n} = \text {RHS}, \end{aligned}$$

(A.1)

where $\nabla _h$ is a $12 \times 9$ gradient matrix and $D_h$ is a $9 \times 12$ divergence matrix defined respectively by

$$\begin{aligned} {\nabla }_h&= \frac{1}{h} \left[ \begin{array}{*{9}{r}} -1&{} \quad 1&{} \quad 0&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0&{} \quad -1&{} \quad 1&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -1&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0&{} \quad &{} \quad &{} \quad \\ 0&{} \quad -1&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad &{} \quad &{} \quad \\ 0&{} \quad 0&{} \quad -1&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad -1&{} \quad 1&{} \quad 0&{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad 0&{} \quad -1&{} \quad 1&{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad -1&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0&{} \quad 0\\ &{} \quad &{} \quad &{} \quad 0&{} \quad -1&{} \quad 0&{} \quad 0&{} \quad 1&{} \quad 0\\ &{} \quad &{} \quad &{} \quad 0&{} \quad 0&{} \quad -1&{} \quad 0&{} \quad 0&{} \quad 1\\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -1&{} \quad 1&{} \quad 0\\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0&{} \quad -1&{} \quad 1\\ \end{array} \right] ,\\ D_h&= \frac{1}{h} \left[ \begin{array}{rrrrrrrrrrrr} 2 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -1 &{} \quad 1 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad -2 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad -1 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad &{} \quad \\ &{} \quad &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad -1 &{} \quad 1 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad &{} \quad \\ &{} \quad &{} \quad 0 &{} \quad 0 &{} \quad -1&{} \quad 0 &{} \quad -2 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -2 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad -2 &{} \quad 0 &{} \quad -1 &{} \quad 1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad 0&{} \quad -2 &{} \quad 0 &{} \quad -2 \\ \end{array} \right] , \end{aligned}$$

and ${\mathbf {K}}_1$ is the coefficient matrix defined by

$$\begin{aligned} {\mathbf {K}}_1 = \frac{1}{h^2} \left[ { \begin{array}{*{12}{r}} 2&{} \quad 0&{} \quad -2&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad 0&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -2&{} \quad 0&{} \quad 2&{} \quad -2&{} \quad 0&{} \quad 2&{} \quad 0&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 1&{} \quad -1&{} \quad -1&{} \quad 2&{} \quad -1&{} \quad -1&{} \quad 1&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -1&{} \quad 0&{} \quad 1&{} \quad -1&{} \quad 0&{} \quad 2&{} \quad 0&{} \quad -1&{} \quad 1&{} \quad 0&{} \quad -1&{} \quad 0 \\ 0&{} \quad -1&{} \quad 0&{} \quad 1&{} \quad -1&{} \quad 0&{} \quad 2&{} \quad 0&{} \quad -1&{} \quad 1&{} \quad 0&{} \quad -1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -2&{} \quad 0&{} \quad 2&{} \quad -2&{} \quad 0&{} \quad 2&{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 1&{} \quad -1&{} \quad -1&{} \quad 2&{} \quad -1&{} \quad -1&{} \quad 1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0&{} \quad 2&{} \quad 0&{} \quad -2&{} \quad 2&{} \quad 0&{} \quad -2 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -2&{} \quad 0&{} \quad 2&{} \quad -2&{} \quad 0&{} \quad 2&{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0&{} \quad -2&{} \quad 0&{} \quad 2&{} \quad -2&{} \quad 0&{} \quad 2 \end{array} } \right] . \end{aligned}$$

Let us recall that the mixed FEM introduces $\Phi = -\mathrm {div}\, {\mathbf {A}}$ as an extra variable. Based on the lowest order bilinear element space $S_h$ and lowest order rectangular Nédélec edge element space $V_h$, we look for $(\Phi _h^n, {\mathbf {A}}_h^n) \, \in \, S_h \times V_h$ such that

$$\begin{aligned}&(\Phi _h^n, \omega _h) - ({\mathbf {A}}_h^n, \nabla \omega _h) = 0, \quad \forall \,\omega _h \in S_h , \end{aligned}$$

(A.2)

$$\begin{aligned}&\frac{1}{\Delta t}({\mathbf {A}}_h^{n}, {\mathbf {v}}_h) + (\nabla \Phi _h^n, {\mathbf {v}}_h) + (\mathrm {curl}{\mathbf {A}}_h^n, \mathrm {curl}{\mathbf {v}}_h ) = \text {RHS}\,, \quad \forall \, {\mathbf {v}}_h \in V_h, \end{aligned}$$

(A.3)

which further can be represented by the following matrix equations

$$\begin{aligned}&{\mathbf {M}}_{\text {bilinear}} \overrightarrow{\Phi }^{n} - {\mathbf {D}}_h\overrightarrow{A}^{n} = 0, \end{aligned}$$

(A.4)

$$\begin{aligned}&\frac{1}{\Delta t} \mathbf {\varvec{M}} \overrightarrow{A}^{n} + {\mathbf {D}}_h^{T} \overrightarrow{\Phi }^{n} + {\mathbf {K}}_2\overrightarrow{A}^{n} = \text {RHS}\,. \end{aligned}$$

(A.5)

On the $2 \times 2$ mesh, the above mass matrix ${\mathbf {M}}_{\text {bilinear}}$ generated by the lowest order bilinear Lagrange element space and the matrix $\mathbf {\varvec{D}}_h$ are defined by

$$\begin{aligned} {\mathbf {M}}_{\text {bilinear}}&=h^2 \left[ { \begin{array}{*{9}{r}} \frac{1}{9} &{} \quad \frac{1}{18}&{} \quad 0 &{} \quad \frac{1}{18}&{} \quad \frac{1}{36}&{} \quad 0 &{} \quad &{} \quad &{} \quad \\ \frac{1}{18}&{} \quad \frac{2}{9} &{} \quad \frac{1}{18}&{} \quad \frac{1}{36}&{} \quad \frac{1}{9}&{} \quad \frac{1}{36} &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad \frac{1}{18}&{} \quad \frac{1}{9} &{} \quad 0 &{} \quad \frac{1}{36}&{} \quad \frac{1}{18}&{} \quad &{} \quad &{} \quad \\ \frac{1}{18} &{} \quad \frac{1}{36} &{} \quad 0 &{} \quad \frac{2}{9} &{} \quad \frac{1}{9} &{} \quad 0 &{} \quad \frac{1}{18}&{} \quad \frac{1}{36} &{} \quad 0 \\ \frac{1}{36} &{} \quad \frac{1}{9} &{} \quad \frac{1}{36} &{} \quad \frac{1}{9} &{} \quad \frac{4}{9} &{} \quad \frac{1}{9} &{} \quad \frac{1}{36}&{} \quad \frac{1}{9} &{} \quad \frac{1}{36} \\ 0 &{} \quad \frac{1}{36} &{} \quad \frac{1}{18} &{} \quad 0 &{} \quad \frac{1}{9} &{} \quad \frac{2}{9} &{} \quad 0 &{} \quad \frac{1}{36} &{} \quad \frac{1}{18} \\ &{} \quad &{} \quad &{} \quad \frac{1}{18}&{} \quad \frac{1}{36}&{} \quad 0 &{} \quad \frac{1}{9} &{} \quad \frac{1}{18}&{} \quad 0 \\ &{} \quad &{} \quad &{} \quad \frac{1}{36}&{} \quad \frac{1}{9}&{} \quad \frac{1}{36}&{} \quad \frac{1}{18} &{} \quad \frac{2}{9} &{} \quad \frac{1}{18} \\ &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{36}&{} \quad \frac{1}{18}&{} \quad 0 &{} \quad \frac{1}{18}&{} \quad \frac{1}{9} \end{array} } \right] ,\\ \mathbf {\varvec{D}}_h&= \left[ { \begin{array}{*{12}{r}} -\frac{1}{3} &{} \quad 0 &{} \quad -\frac{1}{3} &{} \quad -\frac{1}{6} &{} \quad 0 &{} \quad -\frac{1}{6}&{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ \frac{1}{3} &{} \quad -\frac{1}{3} &{} \quad -\frac{1}{6} &{} \quad -\frac{2}{3} &{} \quad -\frac{1}{6}&{} \quad \frac{1}{6}&{} \quad -\frac{1}{6}&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad \frac{1}{3} &{} \quad 0 &{} \quad -\frac{1}{6} &{} \quad -\frac{1}{3}&{} \quad 0 &{} \quad \frac{1}{6} &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -\frac{1}{6} &{} \quad 0 &{} \quad \frac{1}{3} &{} \quad \frac{1}{6} &{} \quad 0 &{} \quad -\frac{2}{3}&{} \quad 0 &{} \quad -\frac{1}{3}&{} \quad -\frac{1}{6} &{} \quad 0 &{} \quad -\frac{1}{6} &{} \quad 0 \\ \frac{1}{6} &{} \quad -\frac{1}{6} &{} \quad \frac{1}{6} &{} \quad \frac{2}{3} &{} \quad \frac{1}{6}&{} \quad \frac{2}{3}&{} \quad -\frac{2}{3}&{} \quad -\frac{1}{6}&{} \quad -\frac{2}{3} &{} \quad -\frac{1}{6} &{} \quad \frac{1}{6} &{} \quad -\frac{1}{6} \\ 0 &{} \quad \frac{1}{6} &{} \quad 0 &{} \quad \frac{1}{6} &{} \quad \frac{1}{3}&{} \quad 0 &{} \quad \frac{2}{3}&{} \quad 0 &{} \quad -\frac{1}{6} &{} \quad -\frac{1}{3} &{} \quad 0 &{} \quad \frac{1}{6} \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -\frac{1}{6}&{} \quad 0 &{} \quad \frac{1}{3}&{} \quad \frac{1}{6} &{} \quad 0 &{} \quad -\frac{1}{3} &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{6}&{} \quad -\frac{1}{6}&{} \quad \frac{1}{6}&{} \quad \frac{2}{3} &{} \quad \frac{1}{6} &{} \quad \frac{1}{3} &{} \quad -\frac{1}{3} \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{6}&{} \quad 0 &{} \quad \frac{1}{6} &{} \quad \frac{1}{3} &{} \quad 0 &{} \quad \frac{1}{3} \\ \end{array} } \right] , \end{aligned}$$

and the mass matrix ${\mathbf {M}}$ and stiffness matrix ${\mathbf {K}}_2$ generated by the lowest order rectangular Nédélec edge element space are defined respectively by

$$\begin{aligned} {\mathbf {M}}&= h^2 \left[ { \begin{array}{*{12}{r}} \frac{1}{3}&{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{6} &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad \frac{1}{3}&{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{6} &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad \frac{1}{3}&{} \quad \frac{1}{6}&{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad \frac{1}{6}&{} \quad \frac{2}{3}&{} \quad \frac{1}{6}&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad 0 &{} \quad \frac{1}{6}&{} \quad \frac{1}{3}&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ \frac{1}{6}&{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad \frac{2}{3} &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{6}&{} \quad 0 \\ 0 &{} \quad \frac{1}{6}&{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{2}{3} &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{6} \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{3} &{} \quad \frac{1}{6}&{} \quad 0 &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{6} &{} \quad \frac{2}{3}&{} \quad \frac{1}{6} &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{6}&{} \quad \frac{1}{3} &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{6} &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{3} &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{6} &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad \frac{1}{3} \end{array} } \right] ,\\ {\mathbf {K}}_2&= \left[ { \begin{array}{*{12}{r}} 1 &{} \quad 0 &{} \quad -1 &{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad -1&{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -1 &{} \quad 0 &{} \quad 1 &{} \quad -1&{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 1 &{} \quad -1 &{} \quad -1 &{} \quad 2&{} \quad -1&{} \quad -1 &{} \quad 1 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad -1&{} \quad 1&{} \quad 0 &{} \quad -1 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ -1 &{} \quad 0 &{} \quad 1 &{} \quad -1&{} \quad 0&{} \quad 2 &{} \quad 0 &{} \quad -1 &{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 1&{} \quad -1&{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad -1&{} \quad 1 &{} \quad 0 &{} \quad -1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -1 &{} \quad 0 &{} \quad 1 &{} \quad -1 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 1 &{} \quad -1 &{} \quad -1 &{} \quad 2 &{} \quad -1 &{} \quad -1 &{} \quad 1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad -1 &{} \quad 1 &{} \quad 0 &{} \quad -1 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad -1 &{} \quad 0 &{} \quad 1 &{} \quad -1 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad 1 &{} \quad -1 &{} \quad 0 &{} \quad 1 \end{array} } \right] . \end{aligned}$$

If the mass-lumping is used, then it holds

$$\begin{aligned}&(\Phi _h^n, \omega _h)_h - ({\mathbf {A}}_h^n, \nabla \omega _h)_h = 0, \end{aligned}$$

(A.6)

$$\begin{aligned}&\frac{1}{\Delta t}({\mathbf {A}}_h^{n}, {\mathbf {v}}_h)_h + (\nabla \Phi _h^n, {\mathbf {v}}_h)_h + (\mathrm {curl}{\mathbf {A}}_h^n, \mathrm {curl}{\mathbf {v}}_h) = \text {RHS}\, , \end{aligned}$$

(A.7)

and we can obtain the following matrix equations

$$\begin{aligned}&\widetilde{{\mathbf {M}}}_{\text {bilinear}} \overrightarrow{\Phi }^{n} -\widetilde{{\mathbf {D}}}_h \overrightarrow{A}^{n} = 0 \,, \end{aligned}$$

(A.8)

$$\begin{aligned}&\frac{1}{\Delta t} \widetilde{{\mathbf {M}}} \overrightarrow{A}^{n} +\widetilde{{\mathbf {D}}}_h^{T}\overrightarrow{\Phi }^{n} +{\mathbf {K}}_2 \overrightarrow{A}^{n} = \text {RHS}\,, \end{aligned}$$

(A.9)

where $\widetilde{{\mathbf {M}}}_{\text {bilinear}}$ and $\widetilde{\mathbf {\varvec{M}}}$ are diagonal matrices as

$$\begin{aligned}&\widetilde{{\mathbf {M}}}_{\text {bilinear}} = \text {diag}\left( \left[ \begin{array}{*{9}{r}} \frac{1}{4}&\frac{1}{2}&\frac{1}{4}&\frac{1}{2}&1&\frac{1}{2}&\frac{1}{4}&\frac{1}{2}&\frac{1}{4} \end{array} \right] ^T\right) , \nonumber \\&\widetilde{\mathbf {\varvec{M}}} = \text {diag}\left( \left[ \begin{array}{*{12}{r}} \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&1&\frac{1}{2}&1&1&\frac{1}{2}&1&\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \end{array} \right] ^T\right) , \end{aligned}$$

and

$$\begin{aligned} {\widetilde{{\mathbf {D}}}_h = h \left[ { \begin{array}{*{12}{r}} -\frac{1}{2} &{} \quad 0 &{} \quad -\frac{1}{2} &{} \quad 0 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ \frac{1}{2} &{} \quad -\frac{1}{2} &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ 0 &{} \quad \frac{1}{2} &{} \quad 0 &{} \quad 0 &{} \quad -\frac{1}{2}&{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \\ &{} \quad &{} \quad \frac{1}{2} &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad -\frac{1}{2}&{} \quad 0 &{} \quad 0 &{} \quad &{} \quad \\ &{} \quad &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 1 &{} \quad -1 &{} \quad 0 &{} \quad -1 &{} \quad 0 &{} \quad &{} \quad \\ &{} \quad &{} \quad 0 &{} \quad 0 &{} \quad \frac{1}{2}&{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad -\frac{1}{2} &{} \quad &{} \quad \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad \frac{1}{2}&{} \quad 0 &{} \quad 0 &{} \quad -\frac{1}{2} &{} \quad 0 \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad \frac{1}{2} &{} \quad -\frac{1}{2} \\ &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 0 &{} \quad 0&{} \quad \frac{1}{2} &{} \quad 0 &{} \quad \frac{1}{2} \\ \end{array} } \right] .} \end{aligned}$$

By taking matrices $\widetilde{{\mathbf {M}}}_{\text {bilinear}}$, $\widetilde{\mathbf {\varvec{M}}}$ and $\widetilde{{\mathbf {D}}}_h$ into (A.9), we can deduce that

$$\begin{aligned}&\frac{1}{\Delta t} \overrightarrow{A}^{n} + \widetilde{{\mathbf {M}}}^{-1} \widetilde{{\mathbf {D}}}_h^{T} \widetilde{{\mathbf {M}}}_{\text {bilinear}}^{-1} \widetilde{{\mathbf {D}}}_h \overrightarrow{A}^{n} + \widetilde{{\mathbf {M}}}^{-1}{\mathbf {K}}_2\overrightarrow{A}^{n} = \text {RHS}\,. \end{aligned}$$

It is easy to verify the following matrix qualities

$$\begin{aligned} \nabla _h = \widetilde{{\mathbf {M}}}^{-1} \widetilde{{\mathbf {D}}}_h^{T}, \quad -D_h = \widetilde{{\mathbf {M}}}_{\text {bilinear}}^{-1}\widetilde{{\mathbf {D}}}_h, \quad {\mathbf {K}}_1 = \widetilde{{\mathbf {M}}}^{-1}{\mathbf {K}}_2, \end{aligned}$$

which confirm our observation. Finally, we shall point out that similar equivalences also hold for L-shape and multi-connected domains.