An all‐at‐once algebraic multigrid method for finite element discretizations of Stokes problem

• Published in: International journal for numerical methods in fluids 2023-02, Vol.95 (2), p.193-214
• Main Authors: Bacq, Pierre‐Loïc, Gounand, Stéphane, Napov, Artem, Notay, Yvan
• Format: Article
• Language: English
• Subjects: Algebra; algebraic multigrid for Stokes; Discretization; Engineering Sciences; Finite difference method; Finite element method; Genetic transformation; Mathematical analysis; Methods; Relaxation method (mathematics); Robustness (mathematics); solution of discrete Stokes problem; Time dependence; Transformations (mathematics); transform‐then‐solve approach; variable viscosity; Viscosity
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5145

We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform‐then‐solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over‐relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform‐then‐solve approach is also more robust. In particular, for problems with variable viscosity, the transform‐then‐solve approach demonstrates significant speed‐up with respect to the block diagonal preconditioner. The method is also particularly robust for time‐dependent problems whatever the time step size. The transform‐then‐solve approach is extended to standard finite element discretizations of the Stokes problem. The extension exploits two complementary strategies to limit the impact of the complexity increase due to the transformation. The comparison with a state‐of‐the‐art block diagonal preconditioner demonstrates that the transform‐then‐solve approach is always competitive while being significantly more robust for problems with open flow boundary condition and‐or variable viscosity.