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Title: Final Report of the Project "From the finite element method to the virtual element method"

Abstract

The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.

Authors:
 [1];  [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1415356
Report Number(s):
LA-UR-17-30453
DOE Contract Number:  
AC52-06NA25396
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics; virtual element method

Citation Formats

Manzini, Gianmarco, and Gyrya, Vitaliy. Final Report of the Project "From the finite element method to the virtual element method". United States: N. p., 2017. Web. doi:10.2172/1415356.
Manzini, Gianmarco, & Gyrya, Vitaliy. Final Report of the Project "From the finite element method to the virtual element method". United States. doi:10.2172/1415356.
Manzini, Gianmarco, and Gyrya, Vitaliy. Wed . "Final Report of the Project "From the finite element method to the virtual element method"". United States. doi:10.2172/1415356. https://www.osti.gov/servlets/purl/1415356.
@article{osti_1415356,
title = {Final Report of the Project "From the finite element method to the virtual element method"},
author = {Manzini, Gianmarco and Gyrya, Vitaliy},
abstractNote = {The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.},
doi = {10.2172/1415356},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Wed Dec 20 00:00:00 EST 2017},
month = {Wed Dec 20 00:00:00 EST 2017}
}

Technical Report:

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