### Title

Solution verification for the Navier-Stokes equation over 3D domains

### Description

Demonstration of the solution verification for the Naiver-Stokes equation.

### Properties

Public

• DIRECTORY
• DEMO

This project demonstrates a computer-assisted solution existence verification method for the stationary Navier–Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation.

## About Verified Finite Element Method package for MATLAB/Octave.

The Verified Finite Element Method (VFEM) package contains the well used FEM spaces for solving PDEs, especially the Stokes equation. Currently, only the 3D version is available here.

List of finite element function spaces: - $H^1$ Conforming Lagrange FEM space with arbitrary degree. - Scott-Vogelius FEM space with arbitray degree. - Raviart-Thomas FEM space with arbitrary degree. - Discontinuous FEM space with arbitrary degree. - The Crouzeix-Raviart FEM space (only degree 1)

The polynomials in constructing the FEM space are based on the Bernstein polynomial.

Also, a very simple 3D mesh generation for domain of blocks is provided. It can create the Shangyou Zhang's mesh, which is required by the Scott-Vogelius FEM space.

### Verified computing

The verified computation is based on the INTVAL library. The Verified PDF Toolbox provide an interface to freely switch between approximate computing and verified computing with intervals.

### Notice

Current implementation just focuses on "one reasonable implementation" of the codes needed by the solution verification for the Navier-Stokes equation. The code structure will have a big change in near future and the efficiency will/should be improved in the following development.

## Step-by-step instruction on the running the solution verification example.

The detailed file will updated before 2021/6/30.

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## 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.