208 research outputs found
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
An a posteriori error estimate for the Stokes-Brinkman problem in a polygonal domain
summary:We derive a residual based a posteriori error estimate for the Stokes-Brinkman problem on a two-dimensional polygonal domain. We use Taylor-Hood triangular elements. The link to the possible information on the regularity of the problem is discussed
A posteriori error estimates in the maximum norm for parabolic problems
We derive a posteriori error estimates in the
norm for approximations of solutions to
linear para bolic equations. Using the elliptic reconstruction technique
introduced by Makridakis and Nochetto and heat kernel estimates for linear
parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for
semidiscrete finite element approximations. We then establish a posteriori
bounds for a fully discrete backward Euler finite element approximation. The
elliptic reconstruction technique greatly simplifies our development by allow\
ing the straightforward combination of heat kernel estimates with existing
elliptic maximum norm error estimators
The conforming virtual element method for polyharmonic and elastodynamics problems: a review
In this paper, we review recent results on the conforming virtual element
approximation of polyharmonic and elastodynamics problems. The structure and
the content of this review is motivated by three paradigmatic examples of
applications: classical and anisotropic Cahn-Hilliard equation and phase field
models for brittle fracture, that are briefly discussed in the first part of
the paper. We present and discuss the mathematical details of the conforming
virtual element approximation of linear polyharmonic problems, the classical
Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1912.0712
Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis
We propose a strategy for approximating Pareto optimal sets based on the
global analysis framework proposed by Smale (Dynamical systems, New York, 1973,
pp. 531-544). The method highlights and exploits the underlying manifold
structure of the Pareto sets, approximating Pareto optima by means of
simplicial complexes. The method distinguishes the hierarchy between singular
set, Pareto critical set and stable Pareto critical set, and can handle the
problem of superposition of local Pareto fronts, occurring in the general
nonconvex case. Furthermore, a quadratic convergence result in a suitable
set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
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