Divergence-free decoupled finite element methods for incompressible flow problems

TL;DR

This paper introduces divergence-free, decoupled finite element methods for incompressible flow problems, improving computational efficiency by constructing divergence-free basis functions and addressing implementation challenges.

Contribution

It develops $oldsymbol{H}( ext{div})$-conforming finite element methods that decouple velocity and pressure, extending previous divergence-free approaches with practical algorithmic solutions.

Findings

Numerical results show improved efficiency over previous methods.

Decoupling simplifies the computational process.

Methods are validated in 2D and 3D Stokes problems.

Abstract

Incompressible flows are modeled by a coupled system of partial differential equations for velocity and pressure, Starting from a divergence-free mixed method proposed in [John, Li, Merdon and Rui, Math. Models Methods Appl. Sci. 34(05):919--949, 2024], this paper proposes $\vecb{H}(\mathrm{div})$-conforming finite element methods which decouple the velocity and pressure by constructing divergence-free basis functions. Algorithmic issues like the computation of this basis and the imposition of non-homogeneous Dirichlet boundary conditions are discussed. Numerical studies at two- and three-dimensional Stokes problems compare the efficiency of the proposed methods with methods from the above mentioned paper.