HMG -- Homogeneous multigrid for HDG

Homogeneous multigrid method for HDG applied to the Stokes equation

We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart--Thomas, or a hybrid Brezzi--Douglas--Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Numerical experiments underline our analytical findings.

Optimal geometric multigrid preconditioners for HDG-P0 schemes for the reaction-diffusion equation and the generalized Stokes equations

We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0, for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the propo…

$hp$-Multigrid preconditioner for a divergence-conforming HDG scheme for the incompressible flow problems

In this study, we present an $hp$-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The $hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equ…

Manycore parallel computing for a hybridizable discontinuous Galerkin nested multigrid method

We present a parallel computing strategy for a hybridizable discontinuous Galerkin (HDG) nested geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse-grain and fine-grain parallelism to improve time to solution performance. In this work we focus on fine-grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) many-core processor. The GMG method achieves ideal convergence rates of $0.2$ or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensit…

Homogeneous multigrid for embedded discontinuous Galerkin methods

We introduce a homogeneous multigrid method in the sense that it uses the same embedded discontinuous Galerkin (EDG) discretization scheme for Poisson's equation on all levels. In particular, we use the injection operator developed in [LuRK2020] for HDG and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

用于混合离散化的齐次多重网格：在 HHO 方法中的应用

Homogeneous multigrid for hybrid discretizations: application to HHO methods

本文研究 HHO 等非连续骨架方法的几何多重网格 V-cycle，证明其一致收敛性，并通过数值实验验证理论结论。