Numerical methods for spectral problems: theory and applications
Explicit lower eigenvalue bounds of differential operators
*Xuefeng LIU (Niigata University)
In the past 10 years, there has been a noticeable progress on the explicit eigenvalue bounds for differential operators. It is well known that the strict upper bounds for eigenvalues can be easily obtained by applying the Rayleigh--Ritz method and the min-max principle. However, it is difficult to give rigorous lower eigenvalue bounds of, e.g., the Laplace operator. The recently proposed methods based on non-conforming finite element method supply a very concise and efficient way to provide the lower eigenvalue bounds for several model eigenvalue problems. In this talk, we will review the methods that have been developed in bounding lower eigenvalue bounds. The application of rigorous eigenvalue bounds and some open problems will also be introduced.