### Title

Lehmann-Goerisch Method for high-precision eigenvalue bounds

### Description

The project demonstrates the high-precision eigenvalue bounds via Lehmann-Goerisch method

Public

• DIRECTORY
• DEMO

# 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

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

#### About the directory

Folders or files beginning with a dot are not displayed by default.

## Virtual Machine Setting

#### Warning!

You are starting the virtual machine as a visitor to current project. As a visitor, you can change files in the booted virtual machine, but the changed files will be aborted when the server is shut down.

(Please login first to start the virtual machine.)

#### About Machine Type

The machine with type as "n1-standard-1" has 1 CPU Core and 4GB memory. The Google app compute engine provides a detailed guide of the machine type. For more detailed information, please refer to More detail.
If you need a high-spec machine type, please contact the site manager.