## This content will become publicly available on October 29, 2020

# Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods

## Abstract

In this paper, 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 methods, weak Galerkin methods, and hybridized versions 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.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Univ. of Texas, Austin, TX (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC); National Science Foundation (NSF)

- OSTI Identifier:
- 1595039

- Report Number(s):
- SAND-2018-11044J

Journal ID: ISSN 1064-8275; 669660

- Grant/Contract Number:
- AC04-94AL85000; SC0018147; NSF-DMS1620352; NA-0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 41; Journal Issue: 5; Journal ID: ISSN 1064-8275

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Wildey, Tim, Muralikrishnan, Sriramkrishnan, and Bui-Thanh, Tan. Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods. United States: N. p., 2019.
Web. doi:10.1137/18M1193505.
```

```
Wildey, Tim, Muralikrishnan, Sriramkrishnan, & Bui-Thanh, Tan. Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods. United States. doi:10.1137/18M1193505.
```

```
Wildey, Tim, Muralikrishnan, Sriramkrishnan, and Bui-Thanh, Tan. Tue .
"Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods". United States. doi:10.1137/18M1193505.
```

```
@article{osti_1595039,
```

title = {Unified Geometric Multigrid Algorithm for Hybridized High-Order Finite Element Methods},

author = {Wildey, Tim and Muralikrishnan, Sriramkrishnan and Bui-Thanh, Tan},

abstractNote = {In this paper, 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 methods, weak Galerkin methods, and hybridized versions 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.},

doi = {10.1137/18M1193505},

journal = {SIAM Journal on Scientific Computing},

number = 5,

volume = 41,

place = {United States},

year = {2019},

month = {10}

}