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Title: Analysis and computation of a least-squares method for consistent mesh tying

Abstract

We report in the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test [T.A. Laursen, M.W. Heinstein, Consistent mesh-tying methods for topologically distinct discretized surfaces in non-linear solid mechanics, Internat. J. Numer. Methods Eng. 57 (2003) 1197–1242]. This paper presents a theoretical and computational study of a least-squares method for mesh tying [P. Bochev, D.M. Day, A least-squares method for consistent mesh tying, Internat. J. Numer. Anal. Modeling 4 (2007) 342–352], applied to the partial differential equation -∇ 2φ+αφ=f. We prove optimal convergence rates for domains represented as overlapping subdomains and show that the least-squares method passes a patch test of the order of the finite element space by construction. To apply the method to subdomain configurations with gaps and overlaps we use interface perturbations to eliminate the gaps. Finally, theoretical error estimates are illustrated by numerical experiments.

Authors:
 [1];  [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Computational Mathematics and Algorithms
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1426952
Report Number(s):
SAND-2007-1510J
Journal ID: ISSN 0377-0427; 526844
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 218; Journal Issue: 1; Journal ID: ISSN 0377-0427
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Finite elements; Mesh tying; Least-squares; First-order elliptic systems

Citation Formats

Day, David, and Bochev, Pavel. Analysis and computation of a least-squares method for consistent mesh tying. United States: N. p., 2007. Web. doi:10.1016/j.cam.2007.04.049.
Day, David, & Bochev, Pavel. Analysis and computation of a least-squares method for consistent mesh tying. United States. doi:10.1016/j.cam.2007.04.049.
Day, David, and Bochev, Pavel. Tue . "Analysis and computation of a least-squares method for consistent mesh tying". United States. doi:10.1016/j.cam.2007.04.049. https://www.osti.gov/servlets/purl/1426952.
@article{osti_1426952,
title = {Analysis and computation of a least-squares method for consistent mesh tying},
author = {Day, David and Bochev, Pavel},
abstractNote = {We report in the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test [T.A. Laursen, M.W. Heinstein, Consistent mesh-tying methods for topologically distinct discretized surfaces in non-linear solid mechanics, Internat. J. Numer. Methods Eng. 57 (2003) 1197–1242]. This paper presents a theoretical and computational study of a least-squares method for mesh tying [P. Bochev, D.M. Day, A least-squares method for consistent mesh tying, Internat. J. Numer. Anal. Modeling 4 (2007) 342–352], applied to the partial differential equation -∇2φ+αφ=f. We prove optimal convergence rates for domains represented as overlapping subdomains and show that the least-squares method passes a patch test of the order of the finite element space by construction. To apply the method to subdomain configurations with gaps and overlaps we use interface perturbations to eliminate the gaps. Finally, theoretical error estimates are illustrated by numerical experiments.},
doi = {10.1016/j.cam.2007.04.049},
journal = {Journal of Computational and Applied Mathematics},
number = 1,
volume = 218,
place = {United States},
year = {2007},
month = {7}
}

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