10,066 research outputs found
A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems
We consider a two-point boundary value problem involving a Riemann-Liouville
fractional derivative of order \al\in (1,2) in the leading term on the unit
interval . Generally the standard Galerkin finite element method can
only give a low-order convergence even if the source term is very smooth due to
the presence of the singularity term x^{\al-1} in the solution
representation. In order to enhance the convergence, we develop a simple
singularity reconstruction strategy by splitting the solution into a singular
part and a regular part, where the former captures explicitly the singularity.
We derive a new variational formulation for the regular part, and establish
that the Galerkin approximation of the regular part can achieve a better
convergence order in the , H^{\al/2}(0,1) and -norms
than the standard Galerkin approach, with a convergence rate for the recovered
singularity strength identical with the error estimate. The
reconstruction approach is very flexible in handling explicit singularity, and
it is further extended to the case of a Neumann type boundary condition on the
left end point, which involves a strong singularity x^{\al-2}. Extensive
numerical results confirm the theoretical study and efficiency of the proposed
approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea
An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion
model with a Caputo fractional derivative of order in time,
which is often used to describe anomalous diffusion processes in heterogeneous
media. The nonlocality of the fractional derivative requires storing all the
solutions from time zero. The proposed scheme is based on continuous piecewise
linear finite elements, L1 time stepping, and proper orthogonal decomposition
(POD). By constructing an effective reduced-order scheme using problem-adapted
basis functions, it can significantly reduce the computational complexity and
storage requirement. We shall provide a complete error analysis of the scheme
under realistic regularity assumptions by means of a novel energy argument.
Extensive numerical experiments are presented to verify the convergence
analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure
Correction of high-order BDF convolution quadrature for fractional evolution equations
We develop proper correction formulas at the starting steps to restore
the desired -order convergence rate of the -step BDF convolution
quadrature for discretizing evolution equations involving a fractional-order
derivative in time. The desired -order convergence rate can be
achieved even if the source term is not compatible with the initial data, which
is allowed to be nonsmooth. We provide complete error estimates for the
subdiffusion case , and sketch the proof for the
diffusion-wave case . Extensive numerical examples are provided
to illustrate the effectiveness of the proposed scheme.Comment: 22 pages, 3 figure
Superactivation of monogamy relations for nonadditive quantum correlation measures
We investigate the general monogamy and polygamy relations satisfied by
quantum correlation measures. We show that there exist two real numbers
and such that for any quantum correlation measure ,
is monogamous if and polygamous if for a
given multipartite state . For , we show that the
monogamy relation can be superactivated by finite copies
of for nonadditive correlation measures. As a detailed example, we use
the negativity as the quantum correlation measure to illustrate such
superactivation of monogamy properties. A tighter monogamy relation is
presented at last
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