The MINI mixed finite element for the Stokes problem: An experimental investigation

TL;DR

This paper experimentally investigates super-convergence phenomena in the MINI mixed finite element method for the 2D Stokes problem, revealing potential for higher accuracy beyond classical theory on various mesh types.

Contribution

It demonstrates super-convergence of order 1.5 in pressure and velocity for the MINI element on structured and unstructured meshes, extending theoretical understanding.

Findings

Super-convergence of order 1.5 observed in pressure and velocity.

Complete velocity approximation is closer to the exact velocity.

Piecewise-linear velocity component better conserves mass.

Abstract

Super-convergence of order 1.5 in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretised with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that super-convergence of order 1.5 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of super-convergence of order 1.5, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better.