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Title: 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:
 [1];  [1];  [2];  [3]
  1. Naval Postgraduate School, Monterey, CA (United States)
  2. Southern Methodist Univ., Dallas, TX (United States)
  3. 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)
OSTI Identifier:
1507993
Grant/Contract Number:  
SC0017626
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 392; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
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. https://doi.org/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. https://doi.org/10.1016/j.jcp.2019.04.031. https://www.osti.gov/servlets/purl/1507993.
@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 = 392,
place = {United States},
year = {2019},
month = {4}
}

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Works referencing / citing this record:

hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
journal, October 2019