### Title

Steklov Eigenvalue

### Description

By using the hypercircle method, this project describes how to obtain the guareenteed lower eigenvalue bound of Steklov operator with FEM.

Public

• DIRECTORY
• DEMO

# Guaranteed lower eigenvalue bound of Steklov operator with FEM

Qin LI (Last updated: 2020/11/07)

### Problem description

We consider the Steklov eigenvalue problem over a bounded 2D domain.

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

Let $\lambda_{k,h}$ be the linear conforming FEM solution to the above problem. We provide a guaranteed lower bound of $\lambda_{k}$ with an explicit value of $M_h$ such that $$\lambda_{k}\geq \frac{\lambda_{k, h}}{1+M^{2}_{h}\lambda_{k, h}}.$$

The hypercircle method is adopted here. See detail in https://doi.org/10.1137/120878446 .

### Codes

• create_mat_Kappa.py: The python code to create the matrices.
• load_matrix_Kappa.m: The matlab code to calculate the quantity $\kappa_h$.
• load_matrix_CG_Square.m,load_matrix_CG_Lshaped.m: Calculate the explicit value (upper bound) of $C_h$ and the quantity $M_{h}:=\sqrt{C_h^2+\kappa_{h}^2}$ over the square and L-shaped domains, respectively .

### Environment.

• Python3 + FEniCS
• Matlab

### Setting of computing

In create_mat_kappa.py, set the mesh size and degree of finite element method.

Here is a sample setting. It defines an uniform mesh with subdivision number 16 for the unit square domain. The FEM is selected as the linear conforming one.

N = 16
mesh = UnitSquareMesh(N,N)
degree = 1


#### 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.