Galerkin特征向量逼近
Galerkin Eigenvector Approximations
AI总结：
本文研究Galerkin特征向量逼近的最优性，证明其渐近逼近精确特征向量在试验子空间的投影，并应用于非自伴问题、椭圆PDE离散及Krylov子空间方法，同时给出算子sep的新下界。
摘要：
Galerkin特征向量与试验子空间内可获得的最佳逼近有多接近？在各种条件下，Galerkin方法给出的近似特征向量渐近地趋近于精确特征向量在试验子空间上的投影——而且这比近似特征向量的基本收敛速度更快。本文同时考虑了正交Galerkin和Petrov-Galerkin方法，并特别关注非自伴问题。讨论了这些结果对用有限元方法或谱方法离散的椭圆型偏微分方程数值处理的影响，并给出了在大规模矩阵特征值问题的Krylov子空间方法中的应用。此外，还发展了一对算子的sep的新下界。
英文摘要：
How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed and an application to Krylov subspace methods for large scale matrix eigenvalue problems is presented. New lower bounds to the $sep$ of a pair of operators are developed as well.
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