擬譜補償微分運算子之積分預處理矩陣及於微分方程之應用

Abstract

對於解牽涉到純粹微分運算子d^m/dx^m的微分方程式，譜與擬譜積分預處理矩陣是有效的工具。在本論文的第一部分當中，對於在Gauss-Radau-Legendre網格點上的混和微分運算子d/dr a(r) d/dr，我們採用了一種可分離的建構框架，來明確地建構條件好的逆擬譜補償矩陣。對於在極座標系統中變係數二階微分方程式，這種逆矩陣可以作為解運算子，或是有效的預處理運算子。而在本論文的第二部分當中，對於在Gauss-Lobatto-Legendre網格點上的一階微分運算子d/dx，我們明確地建構條件好的多域逆擬譜補償矩陣。對於直角坐標系統上具有分段連續係數的一階微分方程式，這些逆矩陣可以作為解運算子，或是有效的預處理運算子。

Spectral and pseudospectral integration preconditioning matrices are effective tools for solving differential equations involving pure differential operators d^m/dx^m. In the first part of this thesis we adopt a separable construction framework to explicitly construct a well-conditioned inverse pseudospectral penalty matrix for the mixed differential operator d/dr a(r) d/dr on Gauss-Radau-Legendre grid points. This inverse matrix can be used either as a solution operator or an effective preconditioner for variable coefficient second order differential equations in polar coordinate system. In the second part of this thesis we explicitly construct well-conditioned multidomain inverse pseudospectral penalty matrices for the first order differential operator d/dx on Gauss-Lobatto-Legendre grid points. These inverse matrices can be used as either solution operators or effective preconditioners for first order differential equations with a piecewise continuous coefficient in Cartesian coordinate system.