High-precision eigenvalue bounds via Lehamnn-Goerisch method.

This project domonstrate the two-stage algorithm to provide high-precision guaranteed eigenvalue bounds for differential operators.

In the first stage, the non-conforming finite element methods are used to obtain lower eigenvalue bounds, which is regarded a generalization of the method proposed by the first author to handle the positive semi-definite bilinear forms in the eigenvalue problem formulation.

In the second stage, the Lehmann--Goerisch method along with high-order finite element methods and raw eigenvalue bounds are used to obtain high-precision eigenvalue bounds.

Two eigenvalue problems will be considered:

• The Laplacian eigenvalue problem

$$ -\Delta u = \lambda u \quad \text{ in } \Omega; \quad u=0 \text{ on } \partial \Omega~. $$

• The Steklov eigenvalue problem

$$ -\Delta u + u = 0 \quad \text{ in } \Omega; \quad \frac{\partial{u}}{\partial{n}}=\lambda u \text{ on } \partial \Omega~. $$