A low-degree strictly conservative finite element method for incompressible flows

TL;DR

This paper introduces a new finite element method for incompressible flows that ensures divergence-free velocity approximations, verified through theoretical analysis and numerical tests, improving stability and accuracy.

Contribution

A novel $P_{2}-P_{1}$ finite element pair that satisfies key stability conditions and guarantees divergence-free velocity fields for incompressible flow simulations.

Findings

Satisfies discrete inf-sup condition on general triangulations

Ensures exactly divergence-free velocity approximations

Demonstrates robustness through theoretical and numerical validation

Abstract

In this paper, a new $P_{2}-P_{1}$ finite element pair is proposed for incompressible fluid. For this pair, the discrete inf-sup condition and the discrete Korn's inequality hold on general triangulations. It yields exactly divergence-free velocity approximations when applied to models of incompressible flows. The robust capacity of the pair for incompressible flows are verified theoretically and numerically.