共查询到20条相似文献,搜索用时 15 毫秒
1.
B. Bialecki G. Fairweather A. Karageorghis Q.N. Nguyen 《BIT Numerical Mathematics》2008,48(3):449-472
We formulate new optimal quadratic spline collocation methods for the solution of various elliptic boundary value problems
in the unit square. These methods are constructed so that the collocation equations can be solved using a matrix decomposition
algorithm. The results of numerical experiments exhibit the expected optimal global accuracy as well as superconvergence phenomena.
AMS subject classification (2000) 65N35, 65N22 相似文献
2.
Since the accuracy of finite element solutions of partial differential equations is generally mesh dependent, especially when
solutions have singularities and discontinuities, a proper mesh generation is often important and sometimes crucial for an
accurate numerical approximation of such problems. In this paper, the mesh transformation method is applied to the boundary
value problems of elliptic partial differential equations, and it is proved that the method leads to the optimal finite element
solutions.
AMS subject classification (2000) 73C50, 65K10, 65N12, 65N30 相似文献
3.
Holger Wendland 《BIT Numerical Mathematics》2007,47(2):455-468
In this paper, we study the stability of symmetric collocation methods for boundary value problems using certain positive
definite kernels. We derive lower bounds on the smallest eigenvalue of the associated collocation matrix in terms of the separation
distance. Comparing these bounds to the well-known error estimates shows that another trade-off appears, which is significantly
worse than the one known from classical interpolation. Finally, we show how this new trade-off can be overcome as well as
how the collocation matrix can be stabilized by smoothing.
AMS subject classification (2000) 65N12, 65N15, 65N35 相似文献
4.
Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite
element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems
as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears
even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true.
In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization
problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence
of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures.
AMS subject classification (2000) 65K10,65N30 相似文献
5.
Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically
by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually
employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be
“close enough” to the dynamics of the continuous system (which is typically easier to analyze) provided that small – hence
many – time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report
results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process
can be improved to better reflect the actual properties sought.
In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend
themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms
of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling
approach may possibly lead to algorithms with improved efficiency.
AMS subject classification (2000) 65L05, 65M32, 65N21, 65N22, 65D18 相似文献
6.
Some three-scale finite element discretization schemes are proposed and analyzed in this paper for a class of elliptic eigenvalue
problems on tensor product domains. With these schemes, the solution of an eigenvalue problem on a fine grid may be reduced
to the solutions of eigenvalue problems on a relatively coarse grid and some partially mesoscopic grids, together with the
solutions of linear algebraic systems on a globally mesoscopic grid and several partially fine grids. It is shown theoretically
and numerically that this type of discretization schemes not only significantly reduce the number of degrees of freedom but
also produce very accurate approximations.
AMS subject classification (2000) 65N15, 65N25, 65N30, 65N50 相似文献
7.
A multigrid scheme for the solution of constrained optimal control problems discretized by finite differences is presented. This scheme is based on a new relaxation procedure that satisfies the given constraints pointwise on the computational grid. In applications, the cases of distributed and boundary control problems with box constraints are considered. The efficient and robust computational performance of the present multigrid scheme allows to investigate bang-bang control problems.AMS Subject Classification: 49J20, 65N06, 65N12, 65N55Supported in part by the SFB 03 “Optimization and Control” 相似文献
8.
High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting
the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of
the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation
problems are taken into account to emphasize the main features of the proposed methods.
AMS subject classification (2000) 65L10, 65L12, 65L50 相似文献
9.
10.
The celebrated classical sampling theorem is used to compute approximate values of the eigenvalues of Dirac systems with eigenvalue
parameter in the boundary conditions. We deal with problems with an eigenparameter in one or two boundary conditions. The
error analysis is established considering both truncation and amplitude errors associated with the sampling theorem. We indicate
the role of the amplitude error as well as other parameters in the method via illustrative examples.
AMS subject classification (2000) 34L16, 65L15, 94A20 相似文献
11.
The penetration function measures the effect of the boundary data on the energy of the solution of a second order linear elliptic
PDE taken over an interior subdomain. Here the coefficients of the PDE are functions of position and often represent the material
properties of non homogeneous media with microstructure. The penetration function is used to assess the accuracy of global-local
approaches for recovering local solution features from coarse grained solutions such as those delivered by homogenization
theory.
AMS subject classification (2000) 65N15, 78M40 相似文献
12.
Arne Barinka Stephan Dahlke Wolfgang Dahmen 《Advances in Computational Mathematics》2006,24(1-4):5-34
Recently adaptive wavelet methods have been developed which can be shown to exhibit an asymptotically optimal accuracy/work
balance for a wide class of variational problems including classical elliptic boundary value problems, boundary integral equations
as well as certain classes of noncoercive problems such as saddle point problems. A core ingredient of these schemes is the
approximate application of the involved operators in standard wavelet representation. Optimal computational complexity could
be shown under the assumption that the entries in properly compressed standard representations are known or computable in
average at unit cost. In this paper we propose concrete computational strategies and show under which circumstances this assumption
is justified in the context of elliptic boundary value problems.
Dedicated to Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 41A25, 41A46, 65F99, 65N12, 65N55.
This work has been supported in part by the Deutsche Forschungsgemeinschaft SFB 401, the first and third author are supported
in part by the European Community's Human Potential Programme under contract HPRN-CT-202-00286 (BREAKING COMPLEXITY). The
second author acknowledges the financial support provided through the European Union's Human Potential Programme, under contract
HPRN-CT-2002-00285 (HASSIP) and through DFG grant DA 360/4–1. 相似文献
13.
Tobin A. Driscoll Folkmar Bornemann Lloyd N. Trefethen 《BIT Numerical Mathematics》2008,48(4):701-723
In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations
of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to
be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described
in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun
system in object-oriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.
AMS subject classification (2000) 65L10, 65M70, 65N35 相似文献
14.
We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient
problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced
via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of
the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers,
where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly
correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions.
AMS subject classification (2000) 65F10, 65N22, 65N30, 65N55 相似文献
15.
K. Wright 《BIT Numerical Mathematics》2007,47(1):197-212
Various adaptive methods for the solution of ordinary differential boundary value problems using piecewise polynomial collocation
are considered. Five different criteria are compared using both interval subdivision and mesh redistribution. The methods
are all based on choosing sub-intervals so that the criterion values have (approximately) equal values in each sub-interval.
In addition to the main comparison it is shown by example that at least when accuracy is low then equidistribution may not
give a unique solution.
The main results that using interval size times maximum residual as criterion gives very much better results than using maximum
residual itself. It is also shown that a criterion based on a global error estimate while giving very good results in some
cases, is unsatisfactory in other cases. The other criteria considered are that given by De Boor and the last Chebyshev series
coefficient.
AMS subject classification (2000) 65L10, 65L50, 65L60 相似文献
16.
Yiorgos-Sokratis Smyrlis 《BIT Numerical Mathematics》2006,46(1):163-194
We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In
particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to
an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm
chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete
Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence
of the method in the case of the Laplace’s equation with Dirichlet boundary data in the disk. We develop an alternative way
of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present
numerical experiments suggesting that introduction of Sobolev weights improves the approximation.
AMS subject classification (2000) 35E05, 35J25, 65N12, 65N15, 65N35, 65T50 相似文献
17.
Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation,
applied to isoparametric d-linear quadrilateral finite element approximation of functionals of multi scale solutions to second order elliptic partial
differential equations in bounded domains of ℝd. In contrast to the usual aim to derive an a posteriori error estimate, this work derives, as the mesh size tends to zero,
a uniformly convergent error expansion for the error density, with computable leading order term. It is shown that the optimal
adaptive isotropic mesh uses a number of elements proportional to the d/2 power of the
quasi-norm of the error density; the same error for approximation with a uniform mesh requires a number of elements proportional
to the d/2 power of the larger L1 norm of the same error density. A point is that this measure recognizes different convergence rates for multi scale problems,
although the convergence order may be the same. The main result is a proof that the adaptive algorithm based on successive
subdivisions of elements reduces the maximal error indicator with a factor or stops with the error asymptotically bounded
by the tolerance using the optimal number of elements, up to a problem independent factor. An important step is to prove uniform
convergence of the expansion for the error density, which is based on localized averages of second order difference quotients
of the primal and dual finite element solutions. The averages are used since the difference quotients themselves do not converge
pointwise for adapted meshes. The proof uses weak convergence techniques with a symmetrizer for the second order difference
quotients and a splitting of the error into a dominating contribution, from elements with no hanging nodes or edges on the
initial mesh, and a remaining asymptotically negligible part. Numerical experiments for an elasticity problem with a crack
and different variants of the averages show that the algorithm is useful in practice also for relatively large tolerances,
much larger than the small tolerances needed to theoretically guarantee that the algorithm works well.
AMS subject classification (2000) 65N12, 65N30, 65N50 相似文献
18.
Bivariate least squares approximation with linear constraints 总被引:1,自引:1,他引:0
In this article linear least squares problems with linear equality constraints are considered, where the data points lie on
the vertices of a rectangular grid. A fast and efficient computational method for the case when the linear equality constraints
can be formulated in a tensor product form is presented. Using the solution of several univariate approximation problems the
solution of the bivariate approximation problem can be derived easily.
AMS subject classification (2000) 65D05, 65D07, 65D10, 65F05, 65F20 相似文献
19.
The eigenvalue problems are considered for the fractional ordinary differential equations with different classes of boundary conditions including the Dirichlet, Neumann, Robin boundary conditions and the periodic boundary condition. The eigenvalues and eigenfunctions are characterized in terms of the Mittag–Leffler functions. The eigenvalues of several specified boundary value problems are calculated by using MATLAB subroutine for the Mittag–Leffler functions. When the order is taken as the value 2, our results degenerate to the classical ones of the second-ordered differential equations. When the order α satisfies 1 < α < 2 the eigenvalues can be finitely many. 相似文献
20.
The paper addresses bivariate surface fitting problems, where data points lie on the vertices of a rectangular grid. Efficient
and stable algorithms can be found in the literature to solve such problems. If data values are missing at some grid points,
there exists a computational method for finding a least squares spline by fixing appropriate values for the missing data.
We extended this technique to arbitrary least squares problems as well as to linear least squares problems with linear equality
constraints. Numerical examples are given to show the effectiveness of the technique presented.
AMS subject classification (2000) 65D05, 65D07, 65D10, 65F05, 65F20 相似文献