The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.
Abstract
We develop and analyze a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, that use arbitrarily regular discrete spaces V{sub h} {contained_in} C{sup {alpha}} {element_of} N. The degrees of freedom are (a) solution and derivative values of various degree at suitable nodes and (b) solution moments inside polygons. The convergence of the method is proven theoretically and an optimal error estimate is derived. The connection with the Mimetic Finite Difference method is also discussed. Numerical experiments confirm the convergence rate that is expected from the theory.
 Authors:

 Los Alamos National Laboratory
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 DOE/LANL
 OSTI Identifier:
 1046508
 Report Number(s):
 LAUR1222977
TRN: US201215%%642
 DOE Contract Number:
 AC5206NA25396
 Resource Type:
 Technical Report
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICAL METHODS AND COMPUTING; CONVERGENCE; DEGREES OF FREEDOM; DIFFUSION; FINITE DIFFERENCE METHOD; FINITE ELEMENT METHOD
Citation Formats
Manzini, Gianmarco. The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.. United States: N. p., 2012.
Web. doi:10.2172/1046508.
Manzini, Gianmarco. The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.. United States. doi:10.2172/1046508.
Manzini, Gianmarco. Fri .
"The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.". United States. doi:10.2172/1046508. https://www.osti.gov/servlets/purl/1046508.
@article{osti_1046508,
title = {The Mimetic Finite Element Method and the Virtual Element Method for elliptic problems with arbitrary regularity.},
author = {Manzini, Gianmarco},
abstractNote = {We develop and analyze a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, that use arbitrarily regular discrete spaces V{sub h} {contained_in} C{sup {alpha}} {element_of} N. The degrees of freedom are (a) solution and derivative values of various degree at suitable nodes and (b) solution moments inside polygons. The convergence of the method is proven theoretically and an optimal error estimate is derived. The connection with the Mimetic Finite Difference method is also discussed. Numerical experiments confirm the convergence rate that is expected from the theory.},
doi = {10.2172/1046508},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2012},
month = {7}
}
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