## 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 -∇ ^{2}φ+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:

- 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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