5,275 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
Determining Projections and Functionals for Weak Solutions of the Navier-Stokes Equations
In this paper we prove that an operator which projects weak solutions of the
two- or three-dimensional Navier-Stokes equations onto a finite-dimensional
space is determining if it annihilates the difference of two "nearby" weak
solutions asymptotically, and if it satisfies a single appoximation inequality.
We then apply this result to show that the long-time behavior of weak solutions
to the Navier-Stokes equations, in both two- and three-dimensions, is
determined by the long-time behavior of a finite set of bounded linear
functionals. These functionals are constructed by local surface averages of
solutions over certain simplex volume elements, and are therefore well-defined
for weak solutions. Moreover, these functionals define a projection operator
which satisfies the necessary approximation inequality for our theory. We use
the general theory to establish lower bounds on the simplex diameters in both
two- and three-dimensions. Furthermore, in the three dimensional case we make a
connection between their diameters and the Kolmogoroff dissipation small scale
in turbulent flows.Comment: Version of frequently requested articl
Approximation of Eigenfunctions in Kernel-based Spaces
Kernel-based methods in Numerical Analysis have the advantage of yielding
optimal recovery processes in the "native" Hilbert space \calh in which they
are reproducing. Continuous kernels on compact domains have an expansion into
eigenfunctions that are both -orthonormal and orthogonal in \calh
(Mercer expansion). This paper examines the corresponding eigenspaces and
proves that they have optimality properties among all other subspaces of
\calh. These results have strong connections to -widths in Approximation
Theory, and they establish that errors of optimal approximations are closely
related to the decay of the eigenvalues.
Though the eigenspaces and eigenvalues are not readily available, they can be
well approximated using the standard -dimensional subspaces spanned by
translates of the kernel with respect to nodes or centers. We give error
bounds for the numerical approximation of the eigensystem via such subspaces. A
series of examples shows that our numerical technique via a greedy point
selection strategy allows to calculate the eigensystems with good accuracy
Some Error Analysis on Virtual Element Methods
Some error analysis on virtual element methods including inverse
inequalities, norm equivalence, and interpolation error estimates are presented
for polygonal meshes which admits a virtual quasi-uniform triangulation
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