Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

Pazner, Will; Kolev, Tzanio

doi:10.1007/s42967-021-00136-3

Cited by 10 publications

(8 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We adapted the unified framework for DG methods for elliptic problems presented in [2] to the VEF equations to derive analogues of the interior penalty (IP), second method of Bassi and Rebay (BR2), Minimal Dissipation Local Discontinuous Galerkin (MDLDG), and continuous finite element (CG) methods. The uniform subspace correction preconditioner developed by Pazner and Kolev [3] was shown to be effective leading to iteration counts independent of the mesh size, polynomial order, and penalty parameter. GMRES convergence estimates for the preconditioned system were derived for the nonsymmetric VEF system of equations under relatively mild assumptions.…”

Section: Discussionmentioning

confidence: 99%

“…When the mesh T is nonconforming (e.g. as the result of adaptive mesh refinement), or when the DG finite element space Y p has variable polynomial degrees, then a more sophisticated subspace decomposition is required [3]. In this case, the boundary subspace Y B is decomposed into a collection of smaller subspaces defined on each non-conforming edge.…”

Section: Decomposition Into Conforming and Interface Subspacesmentioning

confidence: 99%

“…In particular, we extend the unified analysis of DG methods developed for elliptic problems presented by Arnold et al [2] to the VEF equations to derive analogues of the interior penalty (IP), second method of Bassi and Rebay (BR2), Minimal Dissipation Local Discontinuous Galerkin (MDLDG), and continuous finite element (CG) techniques. We show that the IP and BR2 VEF methods are effectively preconditioned by the subspace correction method from Pazner and Kolev [3] and that Algebraic Multigrid (AMG) is effective for the CG and MDLDG discretizations. Anistratov and Warsa [33] also applied DG techniques to the VEF equations but they discretize the first-order form of the VEF equations and only consider lowest-order elements in one dimension.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Family of Independent Variable Eddington Factor Methods with Efficient Preconditioned Iterative Solvers

Olivier¹,

Pazner²,

Haut³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have highorder accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with the Discontinuous Galerkin transport discretization from [1] to form an effective highorder, linear transport method. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems presented by Arnold et al. [2] to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners [3], which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. In addition, a parallel weak scaling study of the interior penalty VEF discretization demonstrates the scalability of the method out to 1152 processors.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Decomposition Into Conforming and Interface Subspacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Family of Independent Variable Eddington Factor Methods with Efficient Preconditioned Iterative Solvers

Olivier¹,

Pazner²,

Haut³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 12 (CG preconditioning for DG methods). The use of continuous Galerkin discretizations (together with a smoothing operation, such as Jacobi or Gauss-Seidel) as preconditioners for discontinuous Galerkin methods has been studied extensively in the literature [3,31,66,67]. Theorem 4 gives an alternative, elementary, proof of the optimality of CG preconditioning for DG methods.…”

Section: Mass Matrix Preconditioningmentioning

confidence: 99%

“…Fischer and Lottes developed Schwarz solvers for the pressure solver of the incompressible Navier-Stokes equations using low-order preconditioning in [36,55]. Pazner and Kolev applied low-order preconditioning to high-order continuous and discontinuous Galerkin methods with (nonconforming) hp-refinement [66,67]. The aforementioned works use tensor-product elements (mapped quadrilaterals and hexahedra); Chalmers and Warbuton [25] and Olson [61] considered the extension to simplex elements.…”

Section: Introductionmentioning

confidence: 99%

Low-order preconditioning for the high-order finite element de Rham complex

Pazner¹,

Kolev²,

Dohrmann³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we present a unified framework for constructing spectrally equivalent low-orderrefined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div), and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for highorder interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, using the finite element library MFEM, with which the construction of such preconditioners requires only one or two lines of code. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.

show abstract

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

Wilson,

Eaton,

Kópházi

2024

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

Cited by 10 publications

References 57 publications

A Family of Independent Variable Eddington Factor Methods with Efficient Preconditioned Iterative Solvers

A Family of Independent Variable Eddington Factor Methods with Efficient Preconditioned Iterative Solvers

Low-order preconditioning for the high-order finite element de Rham complex

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

Contact Info

Product

Resources

About