Steklov特征值问题的高效数值计算方法

Steklov eigenvalue problems, in which the eigenvalue parameter appears in the boundary condition, arise widely in physical mechanics, especially in computational fluid dynamics. Thus, Steklov eigenvalue problems have received increasing attention in physical and mathematical fields. This project attemps to make a systematic and deep research on high effective numerical methods for Steklov eigenvalue problems from three aspects: (1) To propose a multi-grid discretization scheme of finite element based on shifted-inverse power method, and combine this multi-grid discretization scheme with a posteriori error estimate to establish a new adaptive algorithm.(2) To establish a parallel algorithm based on local computation for Steklov eigenvalue problem, and prove the efficiency of the algorithm by theory and numerical experiments. (3) To study the Galerkin spectral method for Steklov eigenvalue problems. The issues mentioned above cannot be found in the existed literatures. We will analyze the convergence and stability of these new methods theoretically and carry out numerical experiments. The algorithms which will be studied in this project will improve the computational efficiency significantly and reduce the computational costs on the premise that numerical solutions have high accuracy. The achievements of this project are of important for enriching and developing the numerical computation theory and application of differential operator eigenvalue problems.

Steklov特征值问题广泛地出现在物理力学尤其是流体力学中，求解Steklov特征值问题的高效数值计算方法是数学物理界关注的课题。本项目拟从三个方面对Steklov特征值问题的高效数值计算方法作系统深入研究：（1）建立Steklov特征值问题基于移位反幂法的有限元多网格离散方案，并把多网格离散方案和后验误差估计相结合建立新的自适应算法；（2）建立Steklov特征值问题的基于局部计算的并行算法，并从理论和数值实验两个方面证明算法的高效性；（3）研究Steklov特征值问题的Galerkin谱方法。上述研究内容在现有文献中未见报导，我们将对这些新的方法的收敛性、稳定性进行数值分析和实验。本项目研究的数值方法将在保证数值解具有足够高精度的前提下，大大减少内存要求和计算时间，提高计算效率。本项目的研究成果对丰富和发展微分算子特征值问题数值计算的理论和实际应用具有非常重要的意义。

Steklov特征值问题广泛地出现在物理力学尤其是流体力学中，求解Steklov特征值问题的高效数值计算方法是数学物理界关注的课题。本项目从三个方面对Steklov特征值问题的高效数值计算方法进行了系统深入研究：（1）建立了Steklov特征值问题基于移位反幂法的有限元多网格离散方案，并把多网格离散方案和后验误差估计相结合建立了一种新的自适应算法；（2）建立了Steklov特征值问题的基于移位反迭代的自适应算法和基于局部计算的并行算法，并从理论和数值实验两个方面证明了算法的高效性；（3）研究了二阶和四阶Steklov特征值问题的Galerkin谱方法。我们对这些新的方法的收敛性、稳定性进行了数值分析和实验。本项目的研究成果对丰富和发展微分算子特征值问题数值计算的理论和实际应用具有非常重要的意义。