Efficient and Robust Numerical Solvers for Fluid Dynamics

Pressure-robust numerical methods for the Stokes equations

The enriched Galerkin (EG) velocity and pressure spaces [Yi et al., 2022] have been proposed for solving the Stokes equations with minimal degrees of freedom. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. These velocity and pressure spaces satisfy the inf-sup condition for the Stokes equations, so they are stable Stokes elements.

Pressure-robustness is an important property of numerical methods for the Stokes equations in the case of small viscosity. Though the inf-sup condition is crucial for the well-posedness of the discrete problem, many inf-sup stable pairs may not guarantee accurate numerical velocity solutions. More precisely, such pairs produce the velocity solution whose error bound depends on a pressure term and is inversely proportional to the viscosity. In contrast, pressure-robust schemes can eliminate the pressure term from the velocity error bounds in the error estimates, so they guarantee accurate numerical velocity and pressure simultaneously. We develop efficient pressure-robust schemes by applying a velocity reconstruction operator, first introduced in [Linke, 2012], mapping Stokes elements into an H(div)-conforming space.

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

 A pressure-robust numerical method for the Brinkman equations

The Brinkman equations describe fluid flow in porous media characterized by interconnected pores that allow for the flow of fluids, considering both the viscous forces within the fluid and the resistance from the porous media. The Brinkman equations provide a mathematical framework for studying and modeling complex phenomena such as groundwater flow, multiphase flow in oil reservoirs, blood flow in biological tissues, and pollutant transport in porous media. Mathematically, the Brinkman equations can be seen as a combination of the Stokes and Darcy equations. With a nonzero viscous parameter, the Brinkman equations are in a Stokes regime affected by the viscous forces, so standard mixed formulations require the H1-conformity for velocity. On the other hand, since the Darcy model becomes more prominent as the viscous parameter approaches zero, finite-dimensional spaces for velocity are forced to satisfy the H(div)-conformity. This compatibility in velocity spaces makes it challenging to construct robust numerical solvers for the Brinkman equations in both the Stokes and Darcy regimes. Our research focuses on developing an efficient and robust numerical method for the Brinkman equations, showing uniform performance from the Stokes to Darcy regimes.

We apply piecewise constant permeability to the Brinkman equations in the cube domain, K(x) = 10^-6 in a ball and K(x) = 1 otherwise. The other conditions are given as; 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 given in the figure.

• 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.