p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

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 $$L^2$$ 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.

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

“…We state here the design properties of each of the bilinear forms a h and d h , and the trilinear form t h . Most of these properties have already been obtained on the assumption of zero Dirichlet boundary conditions in [21,Section 8,9]. However, we require several of these results boundary conditions that either only impose vanishing normal components, and/or vanishing tangential curl of vector fields (see (1.1g), (1.1f)).…”

“…Remark 3 (Hybrid pressure spaces). A variation of this scheme can be devised which considers hybrid scalar spaces for the Lagrange multipliers q, r. Such an approach is outlined for the Stokes problem in [9]. The use of hybrid pressure spaces automatically achieves continuity of the normal components of the velocity and magnetic fields [52].…”

“…A thorough review of the analysis and application of HHO methods is given in the monograph [21]. There have been numerous studies of hybrid high-order discretisations of the Stokes [24,9] and Navier-Stokes [10,25] equations. By considering a hybrid pressure space, an HHO discretisation of the Navier-Stokes problem can be devised which locally preserves the conservation of mass of the fluid, and even exhibit robustness in the incompressible Euler limit [11].…”

Towards robust, fast solutions of elliptic equations on complex domains through hybrid high‐order discretizations and non‐nested multigrid methods

“…Finally, also conserving face-defined polynomials at every level, the latest multigrid algorithm, Reference 12 developed by the present authors and which this work follows up on, is based on a prolongation operator that internally reverses the static condensation to recover coarse element-defined polynomials, before computing their traces on the fine faces. Prior to detailing this solver, we also point out the efforts made to design p-multigrid preconditioners for nonelliptic equations, [13][14][15] as well as other techniques such as domain decomposition 16,17 and nested dissection. 18 In this article, we focus on HHO discretizations, which support general polytopal meshes and arbitrary degrees of approximation.…”

HHO methods for the incompressible Navier-Stokes and the incompressible Euler equations

“…From the matrix assembly viewpoint HHO methods are generally more expensive than DG methods, also due to the computational cost of static condensation, nevertheless matrix assembly and static condensation are intrinsically parallel tasks that are expected to show optimal scalability on multicore and manycore architectures. Besides computational efficiency considerations, HHO formulations have demonstrated to be robust with respect to mesh distortion and grading [5,17]. These features are of crucial importance in the context of CFD applications, where boundary layers are commonly employed to improve the resolution near wall.…”