arXiv subjects

Wenzheng Kuang

Publications and source records attributed to Wenzheng Kuang.

4 recordsLinked to original sources

arXiv2023

$hp$-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 equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix-Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput. Methods Appl. Mech. Engrg., 311 (2016)), which allows for the direct application of geometric multigrid theory on the Crouzeix-Raviart discretization. The numerical experiments demonstrate the robustness of the proposed $hp$-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi & Olshanskii (SIAM J. Sci. Comput., 28(6) (2006)) and Farrell et al. (SIAM J. Sci. Comput., 41(5) (2019)), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier-Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to $10^3$.

math.NA↗

arXiv2023

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 proposed {\sf HDG-P0} schemes, which hasn't appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix-Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings.

math.NA↗

arXiv2022

Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming HDG schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique \cite{Xu96}. A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

math.NA↗

arXiv2020

A monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction model

We present a novel monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction (FSI) problem with a thick structure. A pressure-robust optimal energy-norm estimate is obtained for the semidiscrete scheme. When combined with a Crank-Nicolson time discretization, our fully discrete scheme is energy stable and produces an exactly divergence-free fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid (AMG) preconditioner.

math.NA↗