Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains

TL;DR

This paper introduces a computer-assisted method to rigorously verify the existence of stationary solutions to the Navier-Stokes equations in 3D domains, combining finite element error estimation and fixed point theory.

Contribution

It presents a novel verification approach that uses rigorous computation and error bounds to confirm solution existence for the 3D stationary Navier-Stokes equations.

Findings

Successfully verified solution existence in complex 3D domains.

Developed new quantitative error estimation techniques.

Applied fixed point theorem with explicit bounds.

Abstract

This paper proposes 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. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.