Abstract
We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using L^2-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.
Highlights
In this work we develop and numerically validate p-multigrid solution strategies for nonconforming polytopal discretizations of the Stokes equations, governing the creeping flow of incompressible fluids.1 3 Vol.:(0123456789)Communications on Applied Mathematics and ComputationFor the sake of simplicity, we focus on a Newtonian fluid with uniform density and unit kinematic viscosity
The second scheme, inspired by the hybridizable discontinuous Galerkin (HDG) method of [52], see [47], hinges on hybrid approximations of both the velocity and the pressure and includes, with respect to the above reference, a different treatment of viscous terms that results in improved orders of convergence
The multilevel V-cycle iteration based on p-coarsened operators and incomplete lower-upper (ILU) preconditioned Krylov smoothers is an effective solution strategy for high-order HHO discretizations of the Stokes equations
Summary
In this work we develop and numerically validate p-multigrid solution strategies for nonconforming polytopal discretizations of the Stokes equations, governing the creeping flow of incompressible fluids. The second scheme, inspired by the hybridizable discontinuous Galerkin (HDG) method of [52], see [47], hinges on hybrid approximations of both the velocity and the pressure and includes, with respect to the above reference, a different treatment of viscous terms that results in improved orders of convergence. In both cases, the Dirichlet condition on the velocity is enforced weakly in the spirit of [20].
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