金茲堡與朗道曾提出了金茲堡・朗道理論，宣稱超導體最穩定的狀態為其
對應的金茲堡・朗道能量達到最小值的時候，此時滿足最小值的變數函數即為
金茲堡・朗道方程的解。幾年後，戈爾柯夫與伊利埃伯格建構了時變金茲堡・
朗道方程來模擬第二型超導體的渦旋型態。
這篇論文裡我們介紹了兩種最低階的混合有限元法ＶＥ格式與ＣＦ格式，
金茲堡・朗道方程與時軸規範下的時變金茲堡・朗道方程均有。其中ＶＥ格式
與杜強於1998 年提出的協體積法完全相同，其論文著重於理論分析，而我們的
工作只是補足了其缺少的數值實作，因此我們的新方法只有ＣＦ格式。這兩種
方法針對金茲堡・朗道方程的格式均滿足數值上的規範不變性，而針對時軸規
範下的時變金茲堡・朗道方程的格式均能在一般情況下達到一階誤差精度，在
非凸域上亦是如此，因著我們使用的解為人工光滑解。
在論文的最末我們比較了這兩種方法在各項實驗下的結果，我們發現新提
出來的ＣＦ格式在解金茲堡・朗道方程時較有效率，而在解時軸規範下的時變
金茲堡・朗道方程時其中一個變數函數能在沒有擾動的規則網格上達到二階誤
差精度，這是兩個優於ＶＥ格式的結果。

Ginzburg and Landau proposed the Ginzburg-Landau theory which has the
postulate that the superconducting sample is in a state such that the Ginzburg-Landau
energy functional is a minimum and the minimizer serves as a solution of the Ginzburg-
Landau (GL) equations. Several years later, Gor’kov and Eliashberg established the
time-dependent Ginzburg-Landau (TDGL) equations which has a microscopic
description of the vortex state for the type II superconductors.
In this paper, we introduce two lowest-order mixed finite element methods, VEscheme
and CF-scheme, both for GL equations and TDGL equations under the temporal
gauge. The VE-scheme is identical to the covolume method proposed by Qiang Du in
1998 and the CF-scheme is our new method. Du focused on theoretical analysis of the
covolume method. We introduce the VE-scheme to serve it as the implementation of
Du’s method and compare its results with our CF-scheme. These two schemes for the
GL equations both achieve the gauge invariance property numerically. Furthermore,
these two schemes for the TDGL equations under the temporal gauge both achieve the
first error order in general case and simulate the vortex motion well not only on convex
domains but also on non-convex domains. The good results for the non-convex case
may be due to the artificial smooth solution we adopted in the experiments.
The comparisons of these two methods show that our CF-scheme is more
efficient than VE-scheme in solving the GL equations. What is more, one of the
numerical solutions in CF-scheme achieves the second error order on unperturbed
regular triangle mesh while solving the TDGL equations which is better than the VEscheme.
Accordingly, we believe our new method will have great potential for further
study in the future.

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