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1.
When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary
condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and
extreme eigenvalue already reaching O(N
5 for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained
by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping
and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N
4. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial
differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison
verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the
pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
2.
Boris N. Khoromskij 《Numerical Linear Algebra with Applications》1996,3(2):91-111
In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N logp N) and O(N log2 N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations. 相似文献
3.
I. A. Blatov N. V. Dobrobog 《Computational Mathematics and Mathematical Physics》2010,50(9):1476-1493
The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval [−1, 1] is examined.
The highest derivative in this equation appears with a small parameter ɛ2 (ɛ ∈ (0, 1]). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints.
An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process,
the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To
find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown
to converge “conditionally ɛ-uniformly” to some limit partition for which the error estimate O(N
−2ln3
N) is proved. The main results are obtained under the assumption that ɛ ≪ N
−1, where N is number of grid nodes; thus, conditional ɛ-uniform convergence is dealt with. The proofs use the Galerkin projector and
its property to be quasi-optimal. 相似文献
4.
Sven Beuchler Clemens Pechstein Daniel Wachsmuth 《Computational Optimization and Applications》2012,51(2):883-908
We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal
domains by the boundary concentrated finite element method. We prove that the discretization error ||u*-uh*||L2(G)\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)} decreases like N
−1, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N
−3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The
paper is complemented with numerical results. 相似文献
5.
Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics
Bernard Bialecki Graeme Fairweather David B. Knudson D. Abram Lipman Que N. Nguyen Weiwei Sun Gadalia M. Weinberg 《Numerical Algorithms》2009,52(1):1-23
Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of
linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve
Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N
2logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding
discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices
of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs.
Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110). 相似文献
6.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2010,50(4):633-645
A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered.
The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation
parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem.
Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux
grids are constructed that converge ɛ-uniformly at a rate of O(N
1−2ln2
N
1 + N
2−2), where N
1 + 1 and N
2 + 1 are the number of mesh points on the x
1-axis and the minimal number of mesh points on a unit interval of the x
2-axis respectively. The normalized difference derivatives ɛ
k
(∂
k
/∂x
1
k
)u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer,
and the derivatives along the boundary layer (∂
k
/∂
x
2
k
)u(x) (k = 1, 2) converge ɛ-uniformly at the same rate. 相似文献
7.
Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N2). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
8.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2010,50(12):2003-2022
For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where
ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary
of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed
for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme
within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform
grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O(N
−2ln2
N + N
0−1), where N + 1 and N
0 + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique
to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly
at the rate of O(N
−4ln4
N + N
0−2). For fixed values of the parameter, the convergence rate is O(N
−4 + N
0−2). 相似文献
9.
Bernard Bialecki 《Numerical Algorithms》1994,8(2):167-184
Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N
2log2
N) andO(N
2log2log2
N) arithmetic operations, respectively. 相似文献
10.
In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N
−
a disjoint union, where N
+ and N
−
are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting. 相似文献
11.
Zhimin Zhang 《Numerical Methods for Partial Differential Equations》2002,18(3):374-395
Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N?1 ln (N + 1))p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001 相似文献
12.
Numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials. When M points are used in the discretization of the boundary and N time steps are computed, an amount of work of the order O(N2M2) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NM), and we observe speedups of five orders of magnitude for large-scale problems. Thus, the method makes it possible to solve the heat equation by potential theory in practical situations. 相似文献
13.
Günter Berger 《Mathematische Nachrichten》2000,220(1):11-32
Spectral properties of strongly elliptic operators of second order on bounded snowflake domains without W12–extension property are investigated. We prove that the operators have a pure point spectrum and the asymptotic eigenvalue distributions for the counting function N(λ) are of Weyl type. It is shown that the remainder estimate of N(λ) for Dirichlet and Neumann boundary conditions depends on the inner and outer Minkowski dimension of the boundary ∂Ω, respectively. 相似文献
14.
The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ3û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H1(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H1(Ωˆ) is proved to be 𝒪(h+N−1) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments. 相似文献
15.
《复变函数与椭圆型方程》2012,57(11):969-979
It is shown that a domain in C N with piecewise smooth boundary (and also of some more general shape) is a domain of holomorphy, provided the Levi form at every regular point is positively semidefinite and the tangent cone is convex at every point outside a boundary subset of zero Hausdorff (2N-2)-dimensional measure. 相似文献
16.
Boris N. Khoromskij 《Constructive Approximation》2011,34(2):257-280
In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional
applications. We describe the multifolding or quantics-based tensor approximation method of O(dlog N)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1,…,N}⊗d
, or briefly N-d tensors of size N
d
, and to the respective discretized differential-integral operators in ℝ
d
. As the basic approximation result, we prove that a complex exponential sampled on an equispaced grid has quantics rank 1.
Moreover, a Chebyshev polynomial, sampled over a Chebyshev Gauss–Lobatto grid, has separation rank 2 in the quantics tensor
format, while for the polynomial of degree m over a Chebyshev grid the respective quantics rank is at most 2m+1. For N-d tensors generated by certain analytic functions, we give a constructive proof of the O(dlog Nlog ε
−1)-complexity bound for their approximation by low-rank 2-(dlog N) quantics tensors up to the accuracy ε>0. In the case ε=O(N
−α
), α>0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/αlog 2
ε
−1). From numerical examples presented here, we observe that the quantics tensor method has proved its value in application
to various function related tensors/matrices arising in computational quantum chemistry and in the traditional finite element
method/boundary element method (FEM/BEM). The tool apparently works. 相似文献
17.
We study the approximate controllability of a stationary Stokes system with linearized convection in a bounded domain of N. The control acts on a part of the boundary and the velocity field is observed on an interior curve (N=2) or surface (N=3). We establish the L
2-approximate controllability under certain compatibility conditions and suitable geometrical assumptions on the curve or surface. We build controls of minimal L
2-norm by duality. To compute the control, we propose a numerical method, based on duality techniques, consisting in the minimization of a nonquadratic functional coupled to a Stokes system. It is tested in several situations leading to interesting numerical results. 相似文献
18.
We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when
various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems
in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional
problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can
be employed in an MDA with a resulting computational cost of O(N
2 logN) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices
arising in corresponding two–point boundary value problems in one space dimension. In many important cases, these eigensystems
are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite
difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation,
finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily
on the Dirichlet problem for Poisson’s equation in the unit square, sketch extensions to other boundary conditions and to
more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems
in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising
from discretizations of non-separable boundary value problems. 相似文献
19.
In this paper we consider a general nonlinear boundary value problem for second-order differential inclusions. We prove two
existence theorems, one for the ``convex' problem and the other for the ``nonconvex' problem. Then we show that the solution
set of the latter is dense in the C
1
(T,R
N
) -norm to the solution set of the former (relaxation theorem). Subsequently for a Dirichlet boundary value problem we prove the
existence of extremal solutions and we show that they are dense in the solutions of the convexified problem for the C
1
(T,R
N
) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray—Schauder
principle.
Accepted 18 September 1997 相似文献
20.
We study the Dirichlet problem for the parabolic equation ut = Δum, m > 0, in a bounded, non-cylindrical and non-smooth domain Ω
N + 1, N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent
is critical as in the classical theory of the one-dimensional heat equation ut = uxx. 相似文献