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Title: Annotations on the virtual element method for second-order elliptic problems

Abstract

This document contains working annotations on the Virtual Element Method (VEM) for the approximate solution of diffusion problems with variable coefficients. To read this document you are assumed to have familiarity with concepts from the numerical discretization of Partial Differential Equations (PDEs) and, in particular, the Finite Element Method (FEM). This document is not an introduction to the FEM, for which many textbooks (also free on the internet) are available. Eventually, this document is intended to evolve into a tutorial introduction to the VEM (but this is really a long-term goal).

Authors:
 [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:
1338710
Report Number(s):
LA-UR-16-29660
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, partial differential equations

Citation Formats

Manzini, Gianmarco. Annotations on the virtual element method for second-order elliptic problems. United States: N. p., 2017. Web. doi:10.2172/1338710.
Manzini, Gianmarco. Annotations on the virtual element method for second-order elliptic problems. United States. doi:10.2172/1338710.
Manzini, Gianmarco. Tue . "Annotations on the virtual element method for second-order elliptic problems". United States. doi:10.2172/1338710. https://www.osti.gov/servlets/purl/1338710.
@article{osti_1338710,
title = {Annotations on the virtual element method for second-order elliptic problems},
author = {Manzini, Gianmarco},
abstractNote = {This document contains working annotations on the Virtual Element Method (VEM) for the approximate solution of diffusion problems with variable coefficients. To read this document you are assumed to have familiarity with concepts from the numerical discretization of Partial Differential Equations (PDEs) and, in particular, the Finite Element Method (FEM). This document is not an introduction to the FEM, for which many textbooks (also free on the internet) are available. Eventually, this document is intended to evolve into a tutorial introduction to the VEM (but this is really a long-term goal).},
doi = {10.2172/1338710},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Jan 03 00:00:00 EST 2017},
month = {Tue Jan 03 00:00:00 EST 2017}
}

Technical Report:

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  • 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.
  • A new class of hybrid finite element methods for the numerical analysis of second-order elliptic boundary-value problems is presented. The methods are characterized by the use of particular solutions of the differential equation being solved, in contrast to conventional hybrid methods, in which polynomial approximations are used. As a model problem the Dirichlet problem for Laplace's equation is studied. A priori error estimates are derived, and the results of numerical experiments are presented. 3 figures, 1 table.
  • Mixed-hybrid finite element approximations are described for second-order elliptic boundary-value problems in which independent approximations are used for the solution and its gradient on the interior of an element and the trace of the gradients on the boundary of the element. This lead to nonconforming finite elements. The independent boundary approximations are introduced by means of Lagrange multipliers. Error estimates are derived a priori. Several other finite element models are also obtained as special cases. (auth)
  • The discretization uses nodal basis functions and the preconditioner arises. We present preconditioners for a symmetric, positive definite linear system arising from the finite element discretization of a second order elliptic problem in three dimensions. The discretization uses nodal basis functions and the preconditioner arises from the use of hierarchical basis functions. We show that the condition number of the linear hierarchical basis coefficient matrix {cflx A} scaled by a coarse grid operator is O(N{sup 1/3} log N{sup 1/3}) when uniform tetrahedral refinement is used, where N is the number of unknowns. If additional diagonal scaling by levels is appliedmore » in the fine grid, a condition number of O(N{sup 1/3}) is obtained. The same result is obtained if {cflx A} is scaled by its block diagonal. Moreover, we show that any other block diagonal scaling of {cflx A} will yield a condition number that grows at least as O(N{sup 1/3}). These results compare favorably with the condition number of O(N{sup 2/3}) of the nodal coefficient matrix. We provide numerical results that confirm this theory. These results are extensions of those obtained by Yserentant for two dimensional problems. We also extend the analysis of the linear preconditioner to the case of non-uniform refinement. 74 refs., 78 figs., 14 tabs.« less
  • The bulk of the effort was devoted to the problem of the finite element method and related questions. The program included work on developing mathematical principles for the next generation of finite-element computer codes. Principal results of the work are listed and briefly explained. References to the literature where complete discussion appears are given. (RWR)