# Multigrid for an HDG method

@article{Cockburn2013MultigridFA, title={Multigrid for an HDG method}, author={Bernardo Cockburn and Olivier Dubois and Jay Gopalakrishnan and S. Tan}, journal={IMA Journal of Numerical Analysis}, year={2013}, volume={34}, pages={1386-1425} }

We analyze the convergence of a multigrid algorithm for the Hybridizable Discontinuous Galerkin (HDG) method for diffusion problems. We prove that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it, and identify an abstract class of problems for which a nonnested two-level multigrid cycle with one smoothing step converges even when the prolongation…

## 70 Citations

### Homogeneous multigrid for embedded discontinuous Galerkin methods

- Computer ScienceBIT Numerical Mathematics
- 2021

A multigrid method is formed for an embedded discontinuous Galerkin (EDG) discretization scheme for Poisson’s equation that uses the injection operator developed in Lu et al. ( 2021) for HDG and shows optimal convergence rates under the assumption of elliptic regularity.

### Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

- Computer ScienceArXiv
- 2021

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes with Elliptic regularity used in the proofs.

### Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods

- Computer ScienceSIAM J. Sci. Comput.
- 2021

A local Fourier analysis (LFA) of the two-grid error-propagation operator of the Laplacian is presented and it is shown that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization is almost as efficient as when applied to a continuousGalerkin discretized.

### Analysis of a two-level algorithm for HDG methods for diffusion problems

- Computer Science
- 2015

An extended version of the Xu-Zikatanov (X-Z) identity is used to derive a sharp estimate of the convergence rate of the algorithm, and it is shown that the theoretical results also apply to weak Galerkin (WG) methods.

### An H-Multigrid Method for Hybrid High-Order Discretizations

- Computer Science, MathematicsSIAM Journal on Scientific Computing
- 2021

This work considers a second order elliptic PDE discretized by the Hybrid High-Order method, for which globally coupled unknowns are located at faces, and proposes a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level.

### HMG - Homogeneous multigrid for HDG

- Mathematics, Computer ScienceArXiv
- 2020

A stable injection operator is constructed and optimal convergence of the method is proved under the assumption of elliptic regularity to introduce a homogeneous multigrid method for Poisson's equation on all levels.

### A two-level algorithm for the weak Galerkin discretization of diffusion problems

- Computer ScienceJ. Comput. Appl. Math.
- 2015

### Analysis of a family of HDG methods for second order elliptic problems

- MathematicsJ. Comput. Appl. Math.
- 2016

### P-MULTIGRID PRECONDITIONERS APPLIED TO HIGH-ORDER DG AND HDG DISCRETIZATIONS

- Computer Science
- 2018

In this work the use of a p-multigrid preconditioned flexible GMRES solver to deal with the solution of stiff linear systems arising from high order time discretization is explored in the context of…

### Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

- Computer Science, MathematicsMath. Comput.
- 2014

This article develops and analyzes two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coecients, based on a decomposition of the DG nite element space that inherently hinges on the diusion coecient of the problem.

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