Multigrid for an HDG method

  title={Multigrid for an HDG method},
  author={Bernardo Cockburn and Olivier Dubois and Jay Gopalakrishnan and S. Tan},
  journal={IMA Journal of Numerical Analysis},
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… 

Figures and Tables from this paper

Homogeneous multigrid for embedded discontinuous Galerkin methods

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

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

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

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

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

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

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


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

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.



A convergent multigrid cycle for the hybridized mixed method

This work designs and proves mesh‐independent convergence of the variable V‐cycle algorithm for the hybridized mixed method for second‐order elliptic boundary‐value problems and designs a suitable intergrid transfer operator.

Convergence of nonconforming multigrid methods without full elliptic regularity

  • S. Brenner
  • Computer Science, Mathematics
    Math. Comput.
  • 1999
It is proved that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large.

A multilevel discontinuous Galerkin method

A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.

The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms

A theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms is provided and various numerical approximations of second-order elliptic boundary value problems are applied.

Convergence of Multigrid Algorithms for Interior Penalty Methods

V-cycle, F-cycle and W-cycle multigrid algorithms for interior penalty methods for second order elliptic boundary value problems are studied in this paper. It is shown that these algorithms converge

Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations

The iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems are presented and the convergence properties of these algorithms are analyzed.

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

It is shown that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively.

The analysis of multigrid methods

Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods

Numerical results support the theoretical analysis and demonstrate the potential of the proposed specific assembling process for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems.

Application of unified DG analysis to preconditioning DG methods