论文标题
定期收敛和特征值问题的有限元方法
Regular Convergence and Finite Element Methods for Eigenvalue Problems
论文作者
论文摘要
自1970年代以来，已经研究了定期收敛以及各种其他类型的收敛性，以实现线性运算符的离散近似值。在本文中，我们考虑了紧凑型操作员的特征值近似，这些频谱问题可以写为某些Holomollic Fredholm操作员功能的特征值问题。为了关注有限元方法（符合元素，不连续的Galerkin等），我们表明，离散全态运算符函数的常规收敛性来自有限元空间的近似属性以及在某些合适的Sobolev空间中离散操作员的紧凑型收敛。然后，使用离散近似理论获得特征值的收敛性，以解决全体形态弗雷姆操作员函数的特征值问题。结果可用于显示各种有限元方法的融合，以解决特征值问题，例如dirhcilet特征值问题和Biharmonic eigenvalue问题。
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem can be written as the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the compact convergence of the discrete operators in some suitable Sobolev space. The convergence for eigenvalues is then obtained using the discrete approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to show the convergence of various finite element methods for eigenvalue problems such as the Dirhcilet eigenvalue problem and the biharmonic eigenvalue problem.
