Harmonic Analysis and Numerical Analysis on Functional Defferential Equations and Partial Differential Equations

1.关于解空间、稳定性、泛函、偏微分方程周期解或概周期微分方程解的存在性的分析研究结果：我们研究了线性偏微分方程解的有界解的Fourier-Carleman谱，主要结果是谱分离对应的有界解的分解，它在P-解或AP-解的存在性和函数空间的可容许性中的应用。差分方程用类似的方法处理。给出了Banach空间中泛函微分方程解的存在唯一性的基本结果。利用线性泛函微分方程解半群的谱理论研究了P-解的稳定性和存在性。在抽象相空间上建立了一种新的泛函微分方程常数变分公式；该公式对相空间的分解和P-解或AP解的存在是有效的。具体应用实例的数值分析结果和数值计算技术的发展：关于机翼周围二维理想流体的有限元方法，我们将有限元结果应用于圆盘内部的拉普拉斯问题，研究了机翼保角函数的数值构造。对于无界区域的散射问题，我们研究了用区域分解方法和虚拟区域方法进行数值求解的方法，其中我们得到了Dirichlet-Neumann问题数值处理的新观点，包括对Poisson方程的混合型有限元逼近，观察了本质谱在有限元研究中的重要性。

1. The results on the analytic investigation of solution spaces, stability, existence of peridic(P-) or almost periodic(AP-) solutions in functional, partial or stochastic differential equations(DE's) : We studied the Fourier-Carleman spectrum of bounded solutions of linear DE's. Main result are decompositions of bounded solutions corresponding to the separation of spectrum, its application to the existence of P- or AP-solutions and admissiblility of function spaces. Difference equations are treated similary. Fundamental results are given on the existence and uniqueness of solutions to functional differential equations (FDE's) in Banach spaces. The spectral theory of solution semigroups of linear FDE's are applied to stability and existence of P-solutions. A new variation-of-constants formula for FDE's are established on the abstract phase space; the formula are shown to be effective on the decomposition of the phase space and the existence of P- or AP solutions.2. The results on the numerical analysis on the example of concrete applications and the development of the technique of numerical computation: About the finite element method(FEM) on 2 dimensional perfect fluid around a wing, we studied the numerical construction of conformal function of a wing by using the results on FEM to the discrite version of Laplace problem in the interior of a disk. About the scattering problem in unbounded region, we studied the numerical solving manner by using 'the domain decompositon method as well as the fictitious domain method. Among them we obtained a new view of numerical treatments of Dirichlet-Neumann problem. Including the FEM approximation of mixed type to Poisson equation, we observed the importance of the essential spectrum in the study of FEM.