在這篇論文中，我們以Cahn Hilliard方程的相場模型去數值驗證Modica Mortola定理的Γ-收斂性。首先，我們先整合[7]中Modica Mortola定理的證明細節。根據Modica Mortola定理，若Ω⊂R^n是一個有界開集且有利普希茨(Lipschitz)邊界∂Ω，則重新標度的Cahn Hilliard方程能量泛函F_ε在L^1 (Ω)中會Γ-收斂。Modica Mortola 定理可以推論出：若F_ε在L^1 (Ω)的最小值發生在u_ε且u_ε以L^1量度收斂到某個L^1 (Ω)中的 u ，則當ε遞減到0時，F_ε [u_ε ]會趨近於{u=1}在Ω中的周長再乘上某個常數倍，其中u的值域為{-1,1}，並且此u所對應的{u=-1}和{u=1}之間的界面在質量守恆的限制下是最小的周長。我們以數值方法解Cahn Hilliard方程足夠長的時間區間，試圖近似F_ε在L^1 (Ω)最小值發生的位置u_ε，去驗證上述事實。我們藉由[9]中的係數輔助變數法(scalar auxiliary variable method)去解Cahn Hilliard 方程，此方法在每一個時間節點只需要解較小型的常係數的線性系統。事實上，我們的簡單的數值收斂測試結果驗證了F_ε滿足Γ-收斂的部分定義。

In this paper, we numerically verify the convergence of Modica Mortola Theorem by using the phase field model of Cahn Hilliard equation. First, we go through the detials of the proof for Modica Mortola Theorem in [7], that is, rescaling energy functional F_ε of Cahn Hilliard equation Γ-converges on L^1 (Ω) if Ω⊂R^n is a bounded open set with Lipschitz boundary ∂Ω. Modica Mortola Theorem implies that if the minimizer u_ε of F_ε in L^1 (Ω) converges to a function u∈L^1 (Ω) in L^1 sense, then F_ε [u_ε ] tends to the perimeter of {u=1} in Ω as ε→0^+ with a constant and u only takes values -1 and 1 with the minimal perimeter interface between the sets {u=-1} and {u=1} confined to the mass conservation. To verify this fact numerically, we attempt to find the minimizer u_ε of F_ε in L^1 (Ω) by solving the standard Cahn Hilliard equation numerically for a time interval large enough. Thanks for scalar auxiliary variable (SAV) method in [9], we may solve the standard Cahn Hilliard equation numerically with smaller size linear systems with constant coefficients in each time step. Actually, our simple numerical test of convergence verify a part of definition of Γ-convergence numerically.

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