Divergence-free decoupled finite element methods for incompressible flow 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.