期後誤差估計之通用架構

Abstract

各種邊界值問題在得到數值解之後，對於誤差估計之方法在此篇論文中給 予一個一般性的統一架構。 本文針對有限元素法、邊界元素法應用在 線性橢圓偏微分方程式、 Stokes 方程式、雙曲及混合型偏微分方程式及 邊界積分方程式等問題誤差估計之研究、探討。基於區域化弱餘式問題上 所求得之解，我們發展並分析其協調性與非協調性的誤差估計方法。 此法起始於決定一個餘式問題的變分公式，接著對此式在一適當建立的 區域化基底函數空間中求解。 我們先建立一個一般性的統一架構，然後 對於所要探討的兩類方法與幾種邊界值問題逐一驗證架構中所應具的 形式與條件。儘管誤差估計的公式能夠相當簡單與一致化，理論之確立 卻相對的顯得困難。此篇論文主要部份即在完成其誤差估計方法在上述 所提有限元素、邊界元素兩種方法於四個邊界值問題上所做的理論探究。

This thesis presents a general and unified framework for obtaining numerical estimates of the accuracy of approximations to solutions of various boundary value problems. The study focuses on finite element and boundary element methods for problems including linear elliptic partial differential equations, the Stokes equations, hyperbolic and mixed-type partial differential equations and boundary integral equations. Based on the solutions of local weak residual problems, conforming and nonconforming error estimators are developed and analyzed. The approach begins with the variational formulation of residual problems which are then solved element-by-element by proper construction of local shape functions on each element. A general setting is first given and then specifically verified for each individual method or each one of BVPs considered herein. Although the implementation of the error estimation can be fairly simple and unified, theoretical justification of the resulting error estimators is relatively problematic. This thus constitutes a major part of theoretical presentation of the thesis.