# A spectral mimetic least-squares method

## Abstract

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are also satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Delft Univ. of Technology, Delft (The Netherlands)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1140934

- Report Number(s):
- SAND-2014-1827J

Journal ID: ISSN 0898-1221; PII: S0898122114004623

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computers and Mathematics with Applications (Oxford)

- Additional Journal Information:
- Journal Name: Computers and Mathematics with Applications (Oxford); Journal Volume: 68; Journal Issue: 11; Journal ID: ISSN 0898-1221

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; least-squares; mimetic methods; algebraic topology; spectral elements; geometric localization

### Citation Formats

```
Bochev, Pavel, and Gerritsma, Marc.
```*A spectral mimetic least-squares method*. United States: N. p., 2014.
Web. doi:10.1016/j.camwa.2014.09.014.

```
Bochev, Pavel, & Gerritsma, Marc.
```*A spectral mimetic least-squares method*. United States. doi:10.1016/j.camwa.2014.09.014.

```
Bochev, Pavel, and Gerritsma, Marc. Mon .
"A spectral mimetic least-squares method". United States.
doi:10.1016/j.camwa.2014.09.014. https://www.osti.gov/servlets/purl/1140934.
```

```
@article{osti_1140934,
```

title = {A spectral mimetic least-squares method},

author = {Bochev, Pavel and Gerritsma, Marc},

abstractNote = {We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are also satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.},

doi = {10.1016/j.camwa.2014.09.014},

journal = {Computers and Mathematics with Applications (Oxford)},

number = 11,

volume = 68,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}

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