## This content will become publicly available on April 16, 2020

## Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods

## Abstract

The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite-difference-like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements. The GD basis can be used to define the solution space for a discontinuous Galerkin finite element discretization of partial differential equations. In this work we propose two approaches to handling complex geometries within this setting: (1) using nonconforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial elements. In both cases the (semidiscrete) discontinuous Galerkin method is provably energy stable even when variational crimes are committed. Additionally, for both element types a weight-adjusted mass matrix is used, which ensures that only the reference mass matrix must be inverted. We also present sufficient conditions on the treatment of metric terms for the curvilinear, nonconforming GD elements to ensure that the scheme is both constant preserving and conservative. Numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes.

- Authors:

- Naval Postgraduate School, Monterey, CA (United States)
- Southern Methodist Univ., Dallas, TX (United States)
- Rensselaer Polytechnic Inst., Troy, NY (United States)

- Publication Date:

- Research Org.:
- Rensselaer Polytechnic Inst., Troy, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1507993

- Grant/Contract Number:
- SC0017626

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Kozdon, Jeremy, Wilcox, Lucas, Hagstrom, Thomas, and Banks, Jeffrey. Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.04.031.
```

```
Kozdon, Jeremy, Wilcox, Lucas, Hagstrom, Thomas, & Banks, Jeffrey. Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods. United States. doi:10.1016/j.jcp.2019.04.031.
```

```
Kozdon, Jeremy, Wilcox, Lucas, Hagstrom, Thomas, and Banks, Jeffrey. Tue .
"Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods". United States. doi:10.1016/j.jcp.2019.04.031.
```

```
@article{osti_1507993,
```

title = {Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods},

author = {Kozdon, Jeremy and Wilcox, Lucas and Hagstrom, Thomas and Banks, Jeffrey},

abstractNote = {The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite-difference-like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements. The GD basis can be used to define the solution space for a discontinuous Galerkin finite element discretization of partial differential equations. In this work we propose two approaches to handling complex geometries within this setting: (1) using nonconforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial elements. In both cases the (semidiscrete) discontinuous Galerkin method is provably energy stable even when variational crimes are committed. Additionally, for both element types a weight-adjusted mass matrix is used, which ensures that only the reference mass matrix must be inverted. We also present sufficient conditions on the treatment of metric terms for the curvilinear, nonconforming GD elements to ensure that the scheme is both constant preserving and conservative. Numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes.},

doi = {10.1016/j.jcp.2019.04.031},

journal = {Journal of Computational Physics},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {4}

}