Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

TL;DR

This paper introduces explicit a posteriori and a priori error estimations for finite element solutions of the Stokes equations, enabling rigorous and efficient error bounds in 2D and 3D domains.

Contribution

It develops novel explicit error estimations for Stokes equations using the extended hypercircle method and Scott-Vogelius elements, with applications to eigenvalue bounds.

Findings

Explicit error bounds for Stokes solutions in 2D and 3D.

Rigorous eigenvalue bounds for the Stokes operator.

Validated error estimates demonstrated on complex domains.

Abstract

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.