Fundamental Research and Applied Numerical Analysis in Partial Differential Equations and Functional Partial Differential Equations

1.将半Fredholm型算子的性质有效地应用于由非紧算子描述的线性非齐次泛函微分方程周期解的存在性。在一般相空间中找到了无穷时滞发展方程的解半群的生成元公式。它被应用于半群的谱、特征函数的研究以及稳定性的研究。适当引入人工边界上的透明非局部边界条件，通过对速度场流函数的有限元计算，可以尽可能精确地计算出满足库塔条件的流动。研究了结构-声学耦合振动问题的摄动法及其合理性：提出了基于区域分解技术的Helmholtz方程的迭代方法并证明了其有效性；考虑了扰动条件与谱污染的关系。文中推导了弹性弦的振动方程。三维性，并对其进行了一定的论证。提出了一种求解对流方程或对流方程的收敛有限元格式。证明了最好可能的误差阶估计。将该格式成功地应用于密度相关的Stokes方程中密度的平流方程。当目标流形具有非正截曲率时，建立了与p-调和映射相关的发展问题解的存在性和正则性。

1. The property of Semi-Fredholm operators are effectively applied to the existence of periodic solutions of linear, nonhomogeneous functional differential equations described by the noncompact operators. A formula of generators is found for the solution semigroups of evolution equations with infinite delay on the general phase space. It is applied to the study of the spectrum, eigenfunctions of the semigroup, as well as to the study of stability.2. The flow satisfying Kutta condition can be computed numerically as precisely as possible through a finite element computation of stream function of the velocity field using the transpearent non local boundary conditions imposed on artificial boundaries suitably introduced.3. The vibration and wave propagation phenomena are studied for the following themes : Proposal of perturbation method for structural-acoustic coupled vibration problem and its justification : Proposal of the iteration method for the Helmholtz equation based on the domain decomposition technique and proof of its efficiency : Consideration of the relation between inf-sup condition and spectral pollution. The equation is derived for the vibration of a elastic string in. 3-dimensional, and its certain justification is shown.4. A convergent finite element scheme for advection equations of convection equations or convection equations is proposed. The error order estimates which is best possible is proved. The scheme is successfully applied to an advection equation for the densities in the density dependent Stokes equations.5. Existence and regularity for a solution of the evolution problem associated to p-harmonic maps is established if the target manifold has a non-positive sectional curvature.