解有邊界流形上雙調和算子的特徵值問題

Abstract

本文研究以數值方法，求在流形上具箝制邊界條件的雙調和算子的特徵值問題。我們採用「最近點法」，其精神為將流形嵌入至歐氏空間，並將其上之偏微分方程問題轉化成流形附近一等價的偏微分方程問題，再用歐氏空間中的差分方法求解之。本研究中主要的困難為如何處理箝制邊界條件。過去針對低次偏微分方程所常用的鏡射技巧在此均不適用。本文主要的貢獻是提出一個能搭配箝制邊界條件的外插法，並結合最近點法，而得一個解決此問題的二階方法。我們進行一維及二維具邊界流形的測試，數值結果顯示此方法收斂，但似乎只有一階。此原因是來自邊界高階外插造成的誤差。

The numerical eigenvalue problem for the biharmonic operator with clamped boundary condition on a curved manifold is considered in this article. The recently developed closest point method is adopted. The closest point method for solving PDEs on a manifold is to solve an equivalent problem in a neighborhood of the manifold in the Euclidean space the manifold embedded in, where finite difference method can be easily applied. In the present study, the main difficulty lies on the clamped boundary condition. A sophisticated but natural extrapolation method is introduced for the clamped boundary condition, which is a second order method theoretically; however, the present numerical tests show that the accuracy is less than second order. Our numerical investigation shows the coefficient of the truncation error from the boundary is too large due to the high order extrapolation, yielding the discrepancy between the theoretical and numerical results.