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Low-order preconditioning of the Stokes equations. (English) Zbl 07511607

Summary: 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 \(\pmb{\mathbb{Q}}_1\) iso \(\pmb{\mathbb{Q}}_2/\mathbb{Q}_1\) discretization of the Stokes operator as a preconditioner for the \(\pmb{\mathbb{Q}}_2/\mathbb{Q}_1\) 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 \(\pmb{\mathbb{Q}}_2/\mathbb{Q}_1\) system, our ultimate motivation is to apply algebraic multigrid within solvers for \(\pmb{\mathbb{Q}}_2/\mathbb{Q}_1\) systems via the \(\pmb{\mathbb{Q}}_1\) iso \(\pmb{\mathbb{Q}}_2/\mathbb{Q}_1\) discretization, which will be considered in a companion paper.

MSC:

65F08 Preconditioners for iterative methods
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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