沉浸有限元素方法對於橢圓界面問題的數值研究

Abstract

沉浸有限元素方法(immersed finite element method)被運用在解具有界面的Laplace 和biharmonic 問題上。對於二維的biharmonic 界面問題，我們根據該方法建立了qubic 界面元素(interface element)。該界面元素能夠滿足界面上的jump 條件並且和其他非界面HCT 元素銜接在一起。在這篇論文中，我們和採用interface-fitted 網格策略的標準HCT 有限元素方法作比較。由我們建立的界面元素得到的數值解在收斂的準確度上並沒有比較好，但是依然有穩定的收斂準確度。此外，該數值解能夠確實 的滿足界面上的jump 條件。

The immersed finite element method is employed to solve the Laplace and biharmonic problem with interfaces. For solving two-dimensional biharmonic interface problem, we implement the interface elements so that the natural jump conditions can be satisfied and assembling with Hsieh-Clough-Tocher(HCT) element in the non-interface region is seamlessly. Although the efficiency and the accuracy of the convergent solutions are not better than those obtained by standard HCT finite element method with an interface-fitted mesh, we obtain a stable numerical solutions. Furthermore, the modified interface elements generate numerical solutions that exactly satisfying the natural jump conditions.