Divergence-free decoupled finite element methods for incompressible flow problems

TL;DR

This work develops divergence-free, H(div)-conforming finite element methods for incompressible Stokes flow by building a divergence-free velocity basis that decouples velocity from pressure. A modified enrichment yields a pressure-free velocity solve, with a rigorous error framework and practical strategies for non-homogeneous boundary conditions and pressure reconstruction. The methods demonstrate optimal convergence and notable efficiency gains compared to full and reduced saddle-point schemes, in both 2D and 3D tests, and are extensible to Navier–Stokes with appropriate nonlinear treatment. Overall, the approach offers robust, scalable velocity-pressure decoupling with accurate pressure recovery and effective boundary-condition handling.

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.

Figures (14)

• Figure 1: Example \ref{['ex:2d']}. The first three computational grids.
• Figure 2: Example \ref{['ex:2d']} with $\nu = 1$. $L^2(\Omega)$ velocity error (top) and pressure error (bottom) for the polynomial degree $k=2$ (left), $k=3$ (center), and $k=4$ (right), $\delta = 1$ (skew-symmetric case).
• Figure 3: Example \ref{['ex:2d']} with $\nu = 1$. $L^2(\Omega)$ velocity error (top) and pressure error (bottom) for the polynomial degree $k=1$ (left), $k=2$ (center), and $k=3$ (right), $\delta=-1$ (symmetric case).
• Figure 4: Example \ref{['ex:2d']} with $\nu = 10^{-6}$. $L^2(\Omega)$ velocity error (top) and pressure error (bottom) for the polynomial degree $k=1$ (left), $k=2$ (center), and $k=3$ (right), $\delta = 1$ (skew-symmetric case).
• Figure 5: Example \ref{['ex:2d']} with $\nu = 10^{-6}$. $L^2(\Omega)$ velocity error (top) and pressure error (bottom) for the polynomial degree $k=2$ (left), $k=3$ (center), and $k=4$ (right), $\delta=-1$ (symmetric case).

Theorems & Definitions (8)

• Remark 3.1: Basis for $\boldsymbol{RT}_0^0$
• Lemma 3.2: Characterization of the space of discretely divergence-free functions
• Proof 1
• Lemma 3.3: Properties of $a_h$
• Theorem 3.4
• Theorem 3.4
• Theorem 5.1
• Proof 2