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Compatible meshfree discretization of surface PDEs

  • Nathaniel TraskEmail author
  • Paul Kuberry
Article
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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 \(\mathbb {R}^d\). 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. 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.

Keywords

Generalized moving least squares Compatible discretization Surface PDE Meshfree 

Notes

Acknowledgements

The authors acknowledge support under the Sandia National Laboratories LDRD program, and thank Dr. Pavel Bochev for reviewing an early draft of the work. The views expressed in the article do not necessarily represent the views of the US Department of Energy or the US Government.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply  2019

Authors and Affiliations

  1. 1.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA

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