41,744 research outputs found
Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation
We show that if a numerical method is posed as a sequence of operators acting
on data and depending on a parameter, typically a measure of the size of
discretization, then consistency, convergence and stability can be related by a
Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if
and only if it is stable. We define consistency as convergence on a dense
subspace and stability as discrete well-posedness. In some applications
convergence is harder to prove than consistency or stability since convergence
requires knowledge of the solution. An equivalence theorem can be useful in
such settings. We give concrete instances of equivalence theorems for
polynomial interpolation, numerical differentiation, numerical integration
using quadrature rules and Monte Carlo integration.Comment: 18 page
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
Regular polynomial interpolation and approximation of global solutions of linear partial differential equations
We consider regular polynomial interpolation algorithms on recursively
defined sets of interpolation points which approximate global solutions of
arbitrary well-posed systems of linear partial differential equations.
Convergence of the 'limit' of the recursively constructed family of
polynomials to the solution and error estimates are obtained from a priori
estimates for some standard classes of linear partial differential equations,
i.e. elliptic and hyperbolic equations. Another variation of the algorithm
allows to construct polynomial interpolations which preserve systems of linear
partial differential equations at the interpolation points. We show how this
can be applied in order to compute higher order terms of WKB-approximations of
fundamental solutions of a large class of linear parabolic equations. The error
estimates are sensitive to the regularity of the solution. Our method is
compatible with recent developments for solution of higher dimensional partial
differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo,
and has obvious applications to mathematical finance and physics.Comment: 28 page
On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators
This paper addresses the problem of computing the eigenvalues lying in the
gaps of the essential spectrum of a periodic Schrodinger operator perturbed by
a fast decreasing potential. We use a recently developed technique, the so
called quadratic projection method, in order to achieve convergence free from
spectral pollution. We describe the theoretical foundations of the method in
detail, and illustrate its effectiveness by several examples.Comment: 17 pages, 2 tables and 2 figure
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