於此篇論文中，我們延伸了配置譜方法(SCM：即假性譜方法，正方形Dirichlet邊界半線性參數相依問題(PDP：parameter-dependent problem))可參考[27]。於此文中，我們並且得到H1 模以及L2 模的最佳誤差界，倘若解足夠平滑那麼可得到非常快的收斂速度。由於配置譜方法的演算法編程簡單容易，僅需些基底函數求解PD 問題不但可以有高精度表現之外還可有效節省CPU 時間。更進一步來說，我們已完成PD 配置譜方法問題的穩定性分析，並從分析過程中得到與Poisson方程有相同成長率的病態數。最後，從數值實驗與理論分析中觀察到相互呼應驗證彼此的例證。

In this thesis, we extend the spectral collocation methods(SCM) (i.e., pseudo-spectral method) in Quarteroni and Valli [27] for the semilinear, parameter-dependentproblems(PDP) in the square with the Dirichlet boundary condition. The optimal error bounds are derived in this thesis for both H1 and L2 norms. For the solutions sufficiently smooth, the very high convergence rates can be obtained. The algorithms of the SCM are simple and easy to carry out. Only a few of basis functions are needed so that not only can the high accuracy of the PDP solutions be achieved, but also a great deal of CPU time may be saved. Moreover, for PDP the stability analysis of SCM is also made, to have the same growth rates of condition number as those for Poisson’s equation. Numerical experiments are carried out to verify the theoretical analysis made.

1 Introduction 5
2 Description of Methods 6
3 Parameter-dependent semilinear problems 16
4 Error Bounds of Spectral Collocation Method 18
5 Stability Analysis 28
6 A continuation algorithm 33
7 Numerical Experiments 35
8 Conclusions 40
References 41
Appendix 44

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