拋物方程式之初始值與邊界值不相容的數值解法

Abstract

In this thesis, the corner singularities from space-time domains for parabolic equations are considered. Our work is to give an improved penalty method, with an initial condition approximated by a Helmholtz equation with a boundary layer, and a boundary condition approached by a second order ordinary differential equation. Basically, we extend the work in \cite{CQT11_1} and \cite{CQT11_2} and intend to numerically fix the incompatibility problems with the presence of both zeroth and first order incompatibilities. Through two examples on two dimensional domains (rectangle and disk), we demonstrate this method's practicality and usage. In short, from the point of view of comparative errors in $L^{\infty}$, $L^2$ and $H^1$ norm, this improved penalty method has better performance.

In this thesis, the corner singularities from space-time domains for parabolic equations are considered. Our work is to give an improved penalty method, with an initial condition approximated by a Helmholtz equation with a boundary layer, and a boundary condition approached by a second order ordinary differential equation. Basically, we extend the work in \cite{CQT11_1} and \cite{CQT11_2} and intend to numerically fix the incompatibility problems with the presence of both zeroth and first order incompatibilities. Through two examples on two dimensional domains (rectangle and disk), we demonstrate this method's practicality and usage. In short, from the point of view of comparative errors in $L^{\infty}$, $L^2$ and $H^1$ norm, this improved penalty method has better performance.