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Title: A high-order staggered meshless method for elliptic problems

Abstract

Here, we present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $$O(h^{m})$$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.

Authors:
 [1];  [2];  [2]
  1. Brown Univ., Providence, RI (United States)
  2. 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 Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1369444
Report Number(s):
SAND-2016-0864J
Journal ID: ISSN 1064-8275; 643552
Grant/Contract Number:
AC04-94AL85000
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 39; Journal Issue: 2; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; generalized moving least squares; primal-dual grid methods; compatible discretizations; mixed methods; div-grad system

Citation Formats

Trask, Nathaniel, Perego, Mauro, and Bochev, Pavel Blagoveston. A high-order staggered meshless method for elliptic problems. United States: N. p., 2017. Web. doi:10.1137/16M1055992.
Trask, Nathaniel, Perego, Mauro, & Bochev, Pavel Blagoveston. A high-order staggered meshless method for elliptic problems. United States. doi:10.1137/16M1055992.
Trask, Nathaniel, Perego, Mauro, and Bochev, Pavel Blagoveston. Tue . "A high-order staggered meshless method for elliptic problems". United States. doi:10.1137/16M1055992. https://www.osti.gov/servlets/purl/1369444.
@article{osti_1369444,
title = {A high-order staggered meshless method for elliptic problems},
author = {Trask, Nathaniel and Perego, Mauro and Bochev, Pavel Blagoveston},
abstractNote = {Here, we present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.},
doi = {10.1137/16M1055992},
journal = {SIAM Journal on Scientific Computing},
number = 2,
volume = 39,
place = {United States},
year = {Tue Mar 21 00:00:00 EDT 2017},
month = {Tue Mar 21 00:00:00 EDT 2017}
}

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