本文章探討解能量極小問題的一種新方法，傳統上使用歐拉-拉格朗日方程式後，再解偏微分方程式。本文中先將能量方乘離散化，再解其最小值。令計算域為有限維歐氏空間中的長方體，使用等距分割點將其離散後，再解出其極小函數。我們提出一種用於能量函數積分的高階精度方法。在線性情形下，將證明離散解與原解會符合相同的偏微分方程與自然邊界條件。此方法亦可應用到曲線上的極小問題與障礙問題。計算域為曲線時，則先將計算域拓展為帶子後，再做歐式空間上的有限差分，即使是複雜的曲線也可使用。文內會提出一些使用此方法的範例。

In this thesis, we propose a direct method for solving energy minimization problems on function spaces. Usually, people solve the related Euler-Lagrange equations to obtain the minimizers. In this thesis, we start with discretizing energy function and solve the minimizer for the discrete system. First, we consider computational domain to be a rectangle in finite dimensional Euclidean space. We use Cartesian grids and find the minimizer numerically. We proposed a higher order discretization method for integral of energy function. In the linear case, we show that the numerical approximation is also satisfied the Euler-Lagrange equation and the natural boundary condition. Moreover, this method can be applied to the minimization problems and obstacle problems on unit circle by extending the domain to a band area. Some numerical examples are presented.

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