By using non-conforming and conforming FEM, one can easily obtain rigorous eigenvalue bounds for various differential operators, such as the Laplacian, biharmoic, Steklov, Stokes, Maxwell. However, current approaches cannot take advantage of high-degree FEM schemes to provide sharp eigenvalue bounds. In this talk, we will review the ideas of Kato (1949), Lehmann (1963) and Goerisch (1986), and propose an efficient way to obtain high-precision eigenvalue bounds by using high-degree conforming FEM schemes. The newly proposed method is well stated in a newly published book: Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems (Springer) : https://link.springer.com/book/10.1007/978-981-97-3577-8
中文: 协调和非协调有限元法（FEM）可以用于获得各种微分算子的严格特征值界，例如拉普拉斯算子、双调和算子、Steklov算子、Stokes算子以及Maxwell算子。然而，此类方法无法利用高阶有限元方案来提供精确的特征值界。本次报告中，我们将回顾Kato（1949年）、Lehmann（1963年）和Goerisch（1986年）的方法，并提出一种利用低阶有限元的特征值粗糙估计以及高阶协调有限元来获得高精度特征值上下界的有效方法。 该新方法在报告者的新著中有详细说明：《Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems》（Springer）：https://link.springer.com/book/10.1007/978-981-97-3577-8