Efficient and Robust Solvers for Incompressible Flow

Classical Stokes solvers often struggle with two key issues: the need to satisfy the inf-sup stability condition and the lack of pressure robustness, where velocity errors worsen as viscosity decreases. These challenges cause fluid simulations to become inefficient in high dimensions and unreliable in low-viscosity regimes, precisely the conditions that arise in realistic fluid-porous media applications.

My research develops uniform and pressure-robust finite element methods that overcome these limitations, enabling accurate and efficient simulations of incompressible flows across porous and fluid domains.

Pressure-Robust Enriched Galerkin (EG) Methods for the Stokes Equations

To address these challenges, my collaborators and I developed enriched Galerkin (EG) methods with velocity reconstruction that achieve pressure robustness while retaining computational efficiency. The key idea is to reconstruct discrete velocity fields into a divergence-conforming space, so that velocity errors remain independent of the pressure variable. This guarantees accurate velocity approximations even in low-viscosity regimes, where traditional methods often fail. At the same time, the EG framework preserves minimal degrees of freedom and supports efficient parallel matrix assembly, making it well-suited for large-scale simulations in high dimensions. These features provide stable, efficient, and physically reliable discretizations for incompressible fluid flow.

We consider an L-shaped cylinder and a rotational vector field (an exact solution) with viscosity 10^-6. The figure above shows streamlines and the colored magnitude of the numerical velocity solutions.

 A Uniformly Robust EG Framework for Brinkman Flow

Extending these ideas, we developed a uniformly robust enriched Galerkin (EG) framework for the Brinkman equations, which interpolate between Stokes flow and Darcy flow through a viscosity parameter. In the Stokes regime, velocity fields require H1-conformity, while in the Darcy regime they require H(div)-conformity, a distinction that makes standard finite element methods unreliable across both limits. Our framework overcomes this challenge by reconstructing velocities into an H(div)-conforming space, ensuring stability and optimal convergence uniformly across the entire range of viscosity values. This unification enables a single method to handle both free-flow and porous media regimes, providing a reliable numerical foundation for coupled Stokes–Darcy systems in porous media applications.

The Brinkman equations in the cube domain, with K(x) = 10^-6 within a ball and K(x) = 1 elsewhere. The other conditions are: viscosity 10^-6, boundary condition u = [1, 0, 0], and body force f = [1, 1, 1]. The streamlines and colored magnitude of the numerical velocity solutions are shown in the figure.

Related Publications

• S. Lee and L. Mu. Stable lowest-order finite element schemes for the time-dependent Stokes equations. (manuscript in preparation).

• S. Lee and L. Mu, A uniform and pressure-robust enriched Galerkin method for the Brinkman equations, Journal of Scientific Computing, 99, 39 (2024).

• S. Lee and L. Mu, A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, 166 (2024) 51-64.

• X. Hu, S. Lee, L. Mu, and S.-Y. Yi, Pressure-robust enriched Galerkin methods for the Stokes equations, Journal of Computational and Applied Mathematics, 436 (2024) 115449.