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Title: Conforming and nonconforming virtual element methods for elliptic problems

Abstract

Here we present, in a unified framework, new conforming and nonconforming virtual element methods for general second-order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and nonsymmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal H 1- and L 2-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.

Authors:
 [1];  [2];  [1]
  1. Univ. of Leicester, Leicester (United Kingdom)
  2. 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 Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1331260
Report Number(s):
LA-UR-15-23951
Journal ID: ISSN 0272-4979
Grant/Contract Number:
AC52-06NA25396
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
IMA Journal of Numerical Analysis
Additional Journal Information:
Journal Name: IMA Journal of Numerical Analysis; Journal ID: ISSN 0272-4979
Publisher:
Oxford University Press/Institute of Mathematics and its Applications
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; mathematics; elliptic problems; virtual element methods; polygonal and polyhedral meshes; convection-diffusion-reaction equations

Citation Formats

Cangiani, Andrea, Manzini, Gianmarco, and Sutton, Oliver J. Conforming and nonconforming virtual element methods for elliptic problems. United States: N. p., 2016. Web. doi:10.1093/imanum/drw036.
Cangiani, Andrea, Manzini, Gianmarco, & Sutton, Oliver J. Conforming and nonconforming virtual element methods for elliptic problems. United States. doi:10.1093/imanum/drw036.
Cangiani, Andrea, Manzini, Gianmarco, and Sutton, Oliver J. 2016. "Conforming and nonconforming virtual element methods for elliptic problems". United States. doi:10.1093/imanum/drw036. https://www.osti.gov/servlets/purl/1331260.
@article{osti_1331260,
title = {Conforming and nonconforming virtual element methods for elliptic problems},
author = {Cangiani, Andrea and Manzini, Gianmarco and Sutton, Oliver J.},
abstractNote = {Here we present, in a unified framework, new conforming and nonconforming virtual element methods for general second-order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and nonsymmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal H1- and L2-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.},
doi = {10.1093/imanum/drw036},
journal = {IMA Journal of Numerical Analysis},
number = ,
volume = ,
place = {United States},
year = 2016,
month = 8
}

Journal Article:
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  • This thesis deals with the condition numbers and singular value distribution of the preconditioned operators B[sub h][sup [minus]1]A[sub h] and A[sub h]B[sub h][sup [minus]1], where A[sub h] and B[sub h] are nonconforming finite element discretizations of second-order elliptic operators A and B. It generalizes the works of Manteuffel and Parter, Goldstein, Manteuffel and Parter, as well as Parter and Wong. Three nonconforming finite element methods are considered. The first two methods, the penalty method and the method of nearly-zero boundary conditions, deal with Dirichlet boundary conditions on curved domains. It is shown that if the leading part of A ismore » a smooth function times the leading part of B and if the boundary condition of B is the same as that of A, then the L[sup 2]-condition number of A[sub h] B[sub h][sup [minus]1] is bounded independent of the mesh size h, and its L[sup 2]-singular values cluster and fill-in some bounded, estimable interval. If the boundary condition of the adjoint of B is the same as that of the adjoint of A, then the above conclusion holds for B[sub h][sup [minus]1]A[sub h] instead of A[sub h]B[sub h][sup [minus]1]. If B is self-adjoint and positive definite, then the B[sub h]-condition number of B[sub h][sup [minus]1]A[sub h] is bounded independent of h and the B[sub h]-singular values of B[sub h][sup [minus]1]A[sub h] cluster in some bounded, estimable interval. None of the results require full H[sup 2]-regularity nor uniform grids. The third method uses the P[sub 1] nonconforming finite elements, which consist of piecewise linear functions continuous only at mid-points of interelement boundaries. The generalized results of Parter and Wong are valid for this method with quasi-uniform grids and simpler boundary conditions. Through these three particular examples, one has learned what difficulties to anticipate when dealing with nonconforming finite element methods in general.« less
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