2,290 research outputs found
Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
In this paper, we revisit approximation properties of piecewise polynomial
spaces, which contain more than but not . We
develop more accurate upper and lower error bounds that are sharper than those
used in literature. These new error bounds, especially the lower bounds are
particular useful to finite element methods. As an important application, we
establish sharp lower bounds of the discretization error for Laplace and
-th order elliptic eigenvalue problems in various finite element spaces
under shape regular triangulations, and investigate the asymptotic convergence
behavior for large numerical eigenvalue approximations.Comment: 17 Pages, 0 Fogures. arXiv admin note: text overlap with
arXiv:1106.439
Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schr\"{o}dinger Equations with an Inverse Square Potential
In this article, we study numerical approximation of eigenvalue problems of
the Schr\"{o}dinger operator .
There are three stages in our investigation: We start from a ball of any
dimension, in which case the exact solution in the radial direction can be
expressed by Bessel functions of fractional degrees. This knowledge helps us to
design two novel spectral methods by modifying the polynomial basis to fit the
singularities of the eigenfunctions. At the second stage, we move to circular
sectors in the two dimensional setting. Again the radial direction can be
expressed by Bessel functions of fractional degrees. Only in the tangential
direction some modifications are needed from stage one. At the final stage, we
extend the idea to arbitrary polygonal domains. We propose a mortar spectral
element approach: a polygonal domain is decomposed into several sub-domains
with each singular corner including the origin covered by a circular sector, in
which origin and corner singularities are handled similarly as in the former
stages, and the remaining domains are either a standard quadrilateral/triangle
or a quadrilateral/triangle with a circular edge, in which the traditional
polynomial based spectral method is applied. All sub-domains are linked by
mortar elements (note that we may have hanging nodes). In all three stages,
exponential convergence rates are achieved. Numerical experiments indicate that
our new methods are superior to standard polynomial based spectral (or spectral
element) methods and -adaptive methods. Our study offers a new and
effective way to handle eigenvalue problems of the Schr\"{o}dinger operator
including the Laplacian operator on polygonal domains with reentrant corners
An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries
We propose and analyze an efficient spectral-Galerkin approximation for the
Maxwell transmission eigenvalue problem in spherical geometry. Using a vector
spherical harmonic expansion, we reduce the problem to a sequence of equivalent
one-dimensional TE and TM modes that can be solved individually in parallel.
For the TE mode, we derive associated generalized eigenvalue problems and
corresponding pole conditions. Then we introduce weighted Sobolev spaces based
on the pole condition and prove error estimates for the generalized eigenvalue
problem. The TM mode is a coupled system with four unknown functions, which is
challenging for numerical calculation. To handle it, we design an effective
algorithm using Legendre-type vector basis functions. Finally, we provide some
numerical experiments to validate our theoretical results and demonstrate the
efficiency of the algorithms.Comment: 22 pages, 8 figure
A Multilevel Correction Scheme for Nonsymmetric Eigenvalue Problems by Finite Element Methods
A multilevel correction scheme is proposed to solve defective and nodefective
of nonsymmetric partial differential operators by the finite element method.
The method includes multi correction steps in a sequence of finite element
spaces. In each correction step, we only need to solve two source problems on a
finer finite element space and two eigenvalue problems on the coarsest finite
element space. The accuracy of the eigenpair approximation is improved after
each correction step. This correction scheme improves overall efficiency of the
finite element method in solving nonsymmetric eigenvalue problems.Comment: 17 pages, 16 figure
Finite volume schemes of any order on rectangular meshes
In this paper, we analyze vertex-centered finite volume method (FVM) of any
order for elliptic equations on rectangular meshes. The novelty is a unified
proof of the inf-sup condition, based on which, we show that the FVM
approximation converges to the exact solution with the optimal rate in the
energy norm. Furthermore, we discuss superconvergence property of the FVM
solution. With the help of this superconvergence result, we find that the FVM
solution also converges to the exact solution with the optimal rate in the
-norm. Finally, we validate our theory with several numerical experiments.Comment: 15 page
Optimal loss-carry-forward taxation for L\'{e}vy risk processes stopped at general draw-down time
Motivated by Kyprianou and Zhou (2009), Wang and Hu (2012), Avram et al.
(2017), Li et al. (2017) and Wang and Zhou (2018), we consider in this paper
the problem of maximizing the expected accumulated discounted tax payments of
an insurance company, whose reserve process (before taxes are deducted) evolves
as a spectrally negative L\'{e}vy process with the usual exclusion of negative
subordinator or deterministic drift. Tax payments are collected according to
the very general loss-carry-forward tax system introduced in Kyprianou and Zhou
(2009). To achieve a balance between taxation optimization and solvency, we
consider an interesting modified objective function by considering the expected
accumulated discounted tax payments of the company until the general draw-down
time, instead of until the classical ruin time. The optimal tax return function
together with the optimal tax strategy is derived, and some numerical examples
are also provided
An (curl)-conforming finite element in 2D and its applications to the quad-curl problem
In this paper, we first construct the (curl)-conforming finite elements
both on a rectangle and a triangle. They possess some fascinating properties
which have been proven by a rigorous theoretical analysis. Then we apply the
elements to construct a finite element space for discretizing quad-curl
problems. Convergence orders in the (curl) norm and in
the (curl) norm are established. Numerical experiments are provided to
confirm our theoretical results
Superconvergence of Immersed Finite Element Methods for Interface Problems
In this article, we study superconvergence properties of immersed finite
element methods for the one dimensional elliptic interface problem. Due to low
global regularity of the solution, classical superconvergence phenomenon for
finite element methods disappears unless the discontinuity of the coefficient
is resolved by partition. We show that immersed finite element solutions
inherit all desired superconvergence properties from standard finite element
methods without requiring the mesh to be aligned with the interface. In
particular, on interface elements, superconvergence occurs at roots of
generalized orthogonal polynomials that satisfy both orthogonality and
interface jump conditions
Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations
We present an efficient algorithm for the evaluation of the Caputo fractional
derivative of order , which can be
expressed as a convolution of with the kernel . The
algorithm is based on an efficient sum-of-exponentials approximation for the
kernel on the interval with a uniform absolute
error , where the number of exponentials needed
is of the order
.
As compared with the direct method, the resulting algorithm reduces the
storage requirement from to and the overall
computational cost from to with the
total number of time steps. Furthermore, when the fast evaluation scheme of the
Caputo derivative is applied to solve the fractional diffusion equations, the
resulting algorithm requires only storage and
work with the total number of points in space;
whereas the direct methods require ) storage and work.
The complexity of both algorithms is nearly optimal since is
of the order for or for for
fixed accuracy . We also present a detailed stability and error
analysis of the new scheme for solving linear fractional diffusion equations.
The performance of the new algorithm is illustrated via several numerical
examples. Finally, the algorithm can be parallelized in a straightforward
manner.Comment: 21 pages, 5 figure
On -Convergence of PSWFs and A New Well-Conditioned Prolate-Collocation Scheme
The first purpose of this paper is to provide a rigorous proof for the
nonconvergence of -refinement in -approximation by the PSWFs, a
surprising convergence property that was first observed by Boyd et al [J. Sci.
Comput., 2013]. The second purpose is to offer a new basis that leads to
spectral-collocation systems with condition numbers independent of the
intrinsic bandwidth parameter and the number of collocation points. In
addition, this work gives insights into the development of effective spectral
algorithms using this non-polynomial basis. We in particular highlight that the
collocation scheme together with a very practical rule for pairing up
significantly outperforms the Legendre polynomial-based method (and likewise
other Jacobi polynomial-based method) in approximating highly oscillatory
bandlimited functions.Comment: 23 pages, 17 figure
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