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Title: Compatible meshfree discretization of surface PDEs

Abstract

Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in Rd. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. As a result, we provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.

Authors:
 [1];  [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1528998
Report Number(s):
SAND-2019-3154J
Journal ID: ISSN 2196-4378; 673630
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Computational Particle Mechanics
Additional Journal Information:
Journal Volume: 7; Journal Issue: 2; Journal ID: ISSN 2196-4378
Publisher:
Springer Nature
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 97 MATHEMATICS AND COMPUTING; Generalized moving least squares; Compatible discretization; Surface PDE; Meshfree

Citation Formats

Trask, Nathaniel Albert, and Kuberry, Paul Allen. Compatible meshfree discretization of surface PDEs. United States: N. p., 2019. Web. doi:10.1007/s40571-019-00251-2.
Trask, Nathaniel Albert, & Kuberry, Paul Allen. Compatible meshfree discretization of surface PDEs. United States. doi:10.1007/s40571-019-00251-2.
Trask, Nathaniel Albert, and Kuberry, Paul Allen. Sat . "Compatible meshfree discretization of surface PDEs". United States. doi:10.1007/s40571-019-00251-2. https://www.osti.gov/servlets/purl/1528998.
@article{osti_1528998,
title = {Compatible meshfree discretization of surface PDEs},
author = {Trask, Nathaniel Albert and Kuberry, Paul Allen},
abstractNote = {Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in Rd. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. As a result, we provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.},
doi = {10.1007/s40571-019-00251-2},
journal = {Computational Particle Mechanics},
number = 2,
volume = 7,
place = {United States},
year = {2019},
month = {6}
}

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