多重網格法應用於高振盪係數的橢圓方程

Abstract

這篇碩士論文的主題是利用多重網格法計算諾伊曼邊界條件，並且帶有不連續或 者高振盪係數的橢圓方程，離散的方式是利用五點的有限差分法，首先我們討論一 個ill-posed的特殊案例，在這個案例中，當執行多重網格法時，粗網格校正問題是否可解是必須討論的議題，我們發展了一些理論來挑選限制算子，藉此避免不可解的情 形，以確保多重網格法的執行，由於擴散係數的不連續以及高振盪性，我們利用擴散 係數來定義一個新的插補算子來改善收斂速率，最後，我們展示一些數值結果並且比 較不同的迭代法，由這些數值結果可以看出多重網格法在計算諾伊曼邊界條件，並帶 有高振盪係數的橢圓方程是可實行而且有效率的。

The subject of this thesis is the application of the multi-grid method to solve Neumann boundary condition and the elliptic equations with discontinuous or highly oscillating coefficients. Numerical discretization is based on fi ve-point finite difference method. First, we discuss a special case which is ill-posed. When we execute the multi-grid method in this case, whether the coarse-grid correction problem is solvable or not is an issue which we need to discuss. To avoid a situation that the coarse-grid correction problem is not solvable and ensure that the multi-grid method works, we develop a theory to choose the restriction operator. Because of the discontinuous and highly oscillating coefficient, we employ a new interpolation operator which is dependent on the diffusion coefficient of the elliptic equation to improve the rate of convergence. Finally, we display some numerical results and compare it with other iterative methods. With these results, the multi-grid method appears to be feasible and efficient on solving Neumann boundary condition and the elliptic equations with highly oscillating coefficients.