Hermite 有限元素法於解Monge-Ampere方程與重建光學反射面之應用

Abstract

Monge-Amp`ere (MA) 方程式為一完全非線性的偏微分方程式，此一方程式在許多的研究領域，如：微分幾何、質量傳輸、地球自轉流體和光學設計，都被使用。數值方法像是有限元素法被Feng , Neilan and Awanou等人所提出成功地用於解出標準型的MA方程式。在這一份工作裡，由CaRarelli、Oliker 、 Mader 和Wang等人 所提出來被大家所熟知的理論及上述的數值方法將會被介紹。與他人不同的是，我們使用vanished-moment method解出經由自由型態表面 (free form surface , FFS) 設計幾合所建構出來的MA方程式來處理幾合光學上面的問題。為了確保從數值計算中所得到的FFS結果為連續且可以簡單地計算出曲率，我們使用BCIZ有限元素法來處理線性化過後的MA問題。關於這個方法的精確性及穩定性我們透過一些典型的例子呈現在這個工作中。

The Monge-Amp`ere (MA) equation is a fully nonlinear PDE arising in various research fields including differential geometry, mass transportation, geostrophic fluid and optical design. Numerical methods such as finite difference methods proposed by Oberman etc [1], Dean and Glowinshi [2,3] and finite element methods proposed by Feng and Neilan [4, 5] and Awanou [6] have been successfully applied to solve the standard MA equation. In this work, known theoretical are introduced results obtained by CaRarelli [7], Oliker [8], Mader [9] and Wang [10] etc., and the above mentioned numerical methods. Particularly, we employee the vanished-moment method to solve the MA equation arising from the free form surface (FFS) design problem in geometric optics [11, 12]. To ensure the FFS obtained from numerical computation is continuous and to be able to compute the curvature of the FFS easily, we solve the linearized MA problem using BCIZ finite element method [13]. Accuracy and robustness of our approach are demonstrated on several benchmark examples.