近年來，反問題（Inverse Problems）不論在理論探究或數值計算上皆成為相當重要的研究方向。本碩士論文針對定義於多邊形夾雜物（Polygonal Inclusion）之導電率方程式（Conductivity Equation），建立了一套有效的數值求解方法。本研究利用FreeFEM++ 程式語言建構有限元素法（Finite Element Method， FEM）以求解正問題，並定義相應的目標函數（Cost Function）；隨後，透過有限差分法（Finite Difference Method）計算梯度，並結合梯度下降法（Gradient Descent Method）迭代求得逆問題之近似解。數值實驗結果表明，當多邊形夾雜物具備銳角特徵、導電率數值較大、或多邊形面積較大時，本方法能展現出較高的反演準確度。;In recent years， inverse problems have attracted significant research interest in both theoretical exploration and numerical computation. This master’s thesis develops an efficient numerical method for solving the conductivity equation with a polygonal inclusion. Using the FreeFEM++ programming language， we first construct a finite element method (FEM) framework to solve the forward problem and formulate the corresponding cost function. Subsequently， the finite difference method is employed to compute the gradient， and the gradient descent algorithm is applied to iteratively obtain an approximate solution to the inverse problem. Numerical experiments demonstrate that the proposed method yields higher reconstruction accuracy and better convergence when the polygonal inclusion features sharp corners， higher conductivity contrast， or a larger polygon area.
