Papers › Low-order preconditioning of the Stokes equations
Low-order preconditioning of the Stokes equations
Alexey Voronin, Yunhui He, Scott MacLachlan, Luke N. Olson, Ray Tuminaro
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A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor-Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the ℚ₁iso ℚ₂/ ℚ₁ discretization of the Stokes operator as a preconditioner for the ℚ₂/ℚ₁ discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the ℚ₂/ℚ₁ system, our ultimate motivation is to apply algebraic multigrid within solvers for ℚ₂/ℚ₁ systems via the ℚ₁iso ℚ₂/ ℚ₁ discretization, which will be considered in a companion paper.
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