四阶特征值问题基于降阶格式的一种有效的Legendre-Galerkin逼近

An Efficient Legendre-Galerkin Approximation Based on Reduced Order Scheme for Fourth Order Eigenvalue Problems

摘要: 本文提出了四阶特征值问题基于降阶格式的一种有效的Legendre-Galerkin逼近。首先，我们引入了一个辅助函数，将原问题转化为一个二阶混合格式。通过引入一些适当的Sobolev空间，其相应的变分形式被建立，并在解足够光滑条件下证明了其等价性。其次，基于Legendre多项式的正交性质，两组紧凑的基函数被构造，并导出具有稀疏系数矩阵的线性特征系统。最后，我们给出了两个数值例子，数值结果表明了算法的收敛性与高精度。

Abstract: In this paper, an efficient Legendre-Galerkin approximation based on reduced order scheme for fourth order eigenvalue problems is presented. First, we introduce an auxiliary function to trans-form the original problem into a second order mixed format. By introducing some suitable Sobolev Spaces, the corresponding variational form is established, and its equivalence is proved if the solu-tion is sufficiently smooth. Secondly, based on the orthogonal property of Legendre polynomials, two groups of compact basis functions are constructed, and a linear characteristic system with sparse coefficients matrix is derived. Finally, we give two numerical examples, and the numerical results show the convergence and high precision of the algorithm.