A spectral mimetic leastsquares method
We present a spectral mimetic leastsquares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a firstorder 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 leastsquares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the fourfield leastsquares functional by spectral spaces compatible with the differential operators leads to a leastsquares 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:

^{[1]};
^{[2]}
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Delft Univ. of Technology, Delft (The Netherlands)
 Publication Date:
 Report Number(s):
 SAND20141827J
Journal ID: ISSN 08981221; PII: S0898122114004623
 Grant/Contract Number:
 AC0494AL85000
 Type:
 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 08981221
 Publisher:
 Elsevier
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; leastsquares; mimetic methods; algebraic topology; spectral elements; geometric localization
 OSTI Identifier:
 1140934
Bochev, Pavel, and Gerritsma, Marc. A spectral mimetic leastsquares method. United States: N. p.,
Web. doi:10.1016/j.camwa.2014.09.014.
Bochev, Pavel, & Gerritsma, Marc. A spectral mimetic leastsquares method. United States. doi:10.1016/j.camwa.2014.09.014.
Bochev, Pavel, and Gerritsma, Marc. 2014.
"A spectral mimetic leastsquares 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 leastsquares method},
author = {Bochev, Pavel and Gerritsma, Marc},
abstractNote = {We present a spectral mimetic leastsquares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a firstorder 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 leastsquares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the fourfield leastsquares functional by spectral spaces compatible with the differential operators leads to a leastsquares 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 = {2014},
month = {9}
}