RECORD
Unified geometric multigrid algorithm for hybridized high-order finite element methods
Numerical Analysis · Submitted 24 Nov 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. | contribution 0.96 | 0.89 | 0.13 | asserted 1.00 |
| s1 | The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. | contribution 0.91 | 0.42 | 0.11 | asserted 1.00 |
| s2 | For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. | background 0.84 | 0.23 | 0.03 | asserted 1.00 |
| s3 | The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. | method 0.68 | 0.88 | 0.24 | asserted 1.00 |
| s4 | Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. | contribution 0.35 | 0.85 | 0.19 | asserted 1.00 |
| s5 | Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. | method 0.62 | 0.68 | 0.06 | asserted 0.97 |
| s6 | We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. | method 0.63 | 0.97 | 0.07 | asserted 1.00 |
| s7 | We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm. | method 0.63 | 0.96 | 0.06 | asserted 0.94 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.
Human review
How the literature uses it
3 arXiv papers cite this in 8 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 2 papers
-
Tables [ref] and [ref] show the number of iterations taken by ML- and EML-preconditioned GMRES, and in Table [ref] we compare them with those taken by GMRES preconditioned by one v-cycle of geometric multigrid proposed in [ref] .
A Multilevel Approach for Trace System in HDG Discretizations · Example II: Discontinuous highly heterogeneous permeability · 0.98
-
Our approach differs from the more algebraic method presented in [ref] , which constructs a multigrid hierarchy directly on the trace-space defined on the skeleton of the mesh; this is achieved by using $hp$-refinement and statically condensing out unknowns on fine edges which are not represented on the coarser mesh.
Multigrid preconditioners for the hybridized Discontinuous Galerkin discretisation of the shallow water equations · Introduction · 1.00
Cites it as support 1 paper
-
This is also supported by our previous work on geometric multigrid methods [ref] for elliptic PDEs, where we compared few different smoothers and found block-Jacobi to be a competitive choice.
A Multilevel Approach for Trace System in HDG Discretizations · Combining multilevel approaches with domain decomposition methods · 0.99
Cites it as background 2 papers
-
Multigrid methods for nonconforming hybrid discretisations have been studied in the literature, mainly for hdg [ref] and hho [ref] , but also for wg [ref] , dpg [ref] , or hybridised Raviart–Thomas / Douglas–Marini methods [ref] .
Geometric Multigrid solvers for Hybrid High-Order methods on polytopal meshes · Introduction · 0.99
-
A unified geometric multigrid algorithm based on Dirichlet-to-Neumann maps was developed in [ref] for hybridized methods including HDG.
A Multilevel Approach for Trace System in HDG Discretizations · A brief review on solvers/preconditioners for HDG trace system · 0.95
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.