## Multigrid preconditioning of the non-regularized augmented Bingham fluid problem

Alexis Aposporidis, Panayot S. Vassilevski, and Alessandro Veneziani

### Abstract

In the numerical solution of visco-plastic fluids, one of the hard problems is the effective detection of rigid or plug regions. These occur when the strain-rate tensor vanishes and consequently the equations for the fluid region become singular. In order to manage this lack of regularity, different approaches are possible. Regularization procedures replace the plug regions with high-viscosity fluid regions, featuring a regularization parameter $\varepsilon>0$. In Aposporidis et al. [Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2434–2446], an augmented formulation for Bingham fluids was introduced to improve the regularity properties of the problem. Results presented there show that the augmented formulation is more effective for numerical purposes and it works also in the non-regularized case ($\varepsilon=0$) without a significant degradation of the non-linear solver's performance. However, when solving high-dimensional Bingham problems, the augmented formulation leads to more challenging linear systems. In this paper we develop a strategy for preconditioning large non-regularized augmented Bingham systems. We use the regularized problem as a preconditioner for the non-regularized case. Then, we resort to a nonlinear geometric multilevel preconditioner to accelerate the convergence of the flexible Krylov linear solver for the regularized Bingham preconditioner. Results presented here demonstrate the effectiveness of the strategy also in realistic (non-academic) test cases.

Full Text (PDF) [686 KB], BibTeX

### Key words

multigrid, multilevel flexible GMRES, Bingham flow, mixed finite elements

### AMS subject classifications

65F10, 65N30, 65N55

### Links to the cited ETNA articles

 [22] Vol. 35 (2009), pp. 257-280 Howard C. Elman and Ray S. Tuminaro: Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations

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