共查询到20条相似文献,搜索用时 31 毫秒
1.
Thomas Apel Thomas G. Flaig Serge Nicaise 《Numerical Functional Analysis & Optimization》2013,34(2):153-176
The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h 2 + τ k ) in the energy norm and O((h 2 + τ k )(h 2 + τ)) in the L 2(Q)-norm. 相似文献
2.
A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem
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Xianbing Luo 《Numerical Methods for Partial Differential Equations》2016,32(5):1331-1356
In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h2 + k2) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L2‐norm. To derive this result, we also get the error estimate (convergent order O(h2 + k2)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016 相似文献
3.
It is proved that the Bergman type operatorT, is a bounded projection from the pluriharmonic Bergman spaceL
p
(B)∩h(B) onto Bergman spaceL
p
(B) ∩
H(B) for 0p 1 ands (p1-1)(n+1). As an application it is shown that the Gleason’s problem can be solved in Bergman space LP(B)∩H(B) for 0p 1.
Project supported by the National Natural Science Foundation of China (Grant No. 19871081) and the Doctoral Program Foundation
of the State Education Commission of China. 相似文献
4.
W. K. Zahra 《Numerical Algorithms》2009,52(4):561-573
In this paper, we developed numerical methods of order O(h
2) and O(h
4) based on exponential spline function for the numerical solution of class of two point boundary value problems over a Semi-infinite
range. The present approach gives better approximations over all the existing finite difference methods. Properties of the
infinite linear system are established. Convergence analysis and a bound on the approximate solution are discussed. Test problem
with various kinds of boundary conditions is included to illustrate the practical usefulness and superiority of our methods. 相似文献
5.
The wellposedness problem for an anisotropic incompressible viscous fluid in R3,rotating around a vector B(t,x):=(b1(t,x),b2(t,x),b3(t,x)),is studied.The global wellposedness in the homogeneous case (B... 相似文献
6.
Michael Bildhauer Martin Fuchs Victor Osmolovskii 《Mathematical Methods in the Applied Sciences》2002,25(2):149-178
We consider the problem of minimizing among functions u:?d?Ω→?d, u∣?Ω=0, and measurable subsets E of Ω. Here fh+, f? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of fh+ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
7.
Xuan-ru Lu Guang-Hua Gao Zhi-Zhong Sun 《Numerical Methods for Partial Differential Equations》2023,39(1):447-480
In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ2 + h2) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results. 相似文献
8.
EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS 总被引:1,自引:0,他引:1
Du Zengji Lin Xiaojie Ge Weigao 《高校应用数学学报(英文版)》2005,20(3):323-330
This paper is concerned with the following second-order vector boundary value problem :x^R=f(t,Sx,x,x'),0〈t〈1,x(0)=A,g(x(1),x'(1))=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method. 相似文献
9.
Uwe Khler 《中国科学A辑(英文版)》2005,48(2):145-154
Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem. 相似文献
10.
The semi‐linear equation −uxx − ϵuyy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h4) finite difference method, which has local truncation error O(h2) at the mesh points neighboring the boundary and O(h4) at most interior mesh points. It is proved that the finite difference method is O(h4) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000 相似文献
11.
Yasser Khalili Nematollah Kadkhoda Dumitru Baleanu 《Mathematical Methods in the Applied Sciences》2020,43(12):7143-7151
In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h1,h2, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H1,H2, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra. 相似文献
12.
Gershon Kedem Seymour V. Parter Michael Steuerwalt 《Studies in Applied Mathematics》1980,63(2):119-146
Consider the boundary value problem εy″ =(y2 ? t2)y′, ?1 ?t?0, y(?1) = A, y(0) = B. We discuss the multiplicity of solutions and their limiting behavior as ε→+0+ for certain choices of A and B. In particular, when A = 1, B = 0, a bifurcation analysis gives a detailed and fairly complete analysis. The interest here arises from the complexity of the set of "turning points." 相似文献
13.
R. K. Mohanty M. K. Jain Dinesh Kumar 《Numerical Methods for Partial Differential Equations》2000,16(4):408-415
We report a new two‐level explicit finite difference method of O(kh2 + h4) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000 相似文献
14.
M. Bildhauer M. Fuchs V. G. Osmolovskii 《Mathematical Methods in the Applied Sciences》2002,25(4):289-308
We consider the problem of minimizing 0<p<1, h∈?, σ>0, among functions u:?d?Ω→?d, u∣?Ω=0, and measurable characteristic functions χ:Ω→?. Here ?+h, ??, denote quadratic potentials defined on the space of all symmetric d×d matrices, h is the minimum energy of ?+h and ε(u) denotes the symmetric gradient of the displacement field. An equilibrium state û, χ?, of I [·,·,h, σ] is termed one‐phase if χ?≡0 or χ?≡1, two‐phase otherwise. We investigate the way in which the distribution of phases is affected by the choice of the parameters h and σ. Copyright 2002 John Wiley & Sons, Ltd. 相似文献
15.
Yuan Xu 《Constructive Approximation》2001,17(3):383-412
Generalized classical orthogonal polynomials on the unit ball B
d
and the standard simplex T
d
are orthogonal with respect to weight functions that are reflection-invariant on B
d
and, after a composition, on T
d
, respectively. They are also eigenfunctions of a second-order differential—difference operator that is closely related to
Dunkl's h -Laplacian for the reflection groups. Under a proper limit, the generalized classical orthogonal polynomials on B
d
converge to the generalized Hermite polynomials on R
d
, and those on T
d
converge to the generalized Laguerre polynomials on R
d
+
. The latter two are related to the Calogero—Sutherland models associated to the Weyl groups of type A and type B .
February 14, 2000. Date revised: July 26, 2000. Date accepted: August 4, 2000. 相似文献
16.
Guang‐Hua Gao Zhi‐Zhong Sun 《Numerical Methods for Partial Differential Equations》2013,29(5):1459-1486
This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ2 + h4) for interior mesh point approximation and O(τ2 + h3) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ2 + h4) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ2 + h3.5) while the numerical accuracy is O(τ2 + h4), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ2 + h2.5), while the actual numerical accuracy is O(τ2 + h3). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ2 + h4) and O(τ2 + h3), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ2 + h4) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
17.
We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials A ,φ in the Lorenz gauge, we establish boundary conditions on the potentials themselves rather than on the field quantities. This permits the development of a well‐conditioned second‐kind Fredholm integral equation that has no spurious resonances, avoids low‐frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, φscat, is determined entirely by the incident scalar potential φinc. Likewise, the unknown vector potential defining the scattered field, A scat is determined entirely by the incident vector potential Ainc. This decoupled formulation is valid not only in the static limit but for arbitrary ω ≥ 0$. © 2016 Wiley Periodicals, Inc. 相似文献
18.
Lei Zhao Zhi‐zhong Sun Jian‐ming Liu 《Numerical Methods for Partial Differential Equations》2006,22(3):744-760
In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ2 + h2), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
19.
Shaowei Chen Shujie Li 《Calculus of Variations and Partial Differential Equations》2006,27(1):105-123
In this paper we study the following problem:with periodic nonlinearity g, where and λ2 is the second eigenvalue of −Δ, on H
1
0(B). We proved that the problem has infinitely many solutions under some additional conditions on g and h. The method we used is a new variational reduction method.
Mathematics Subject Classi cation (2000) 35J20, 35J70 相似文献
20.
Sergey K. Bagdasarov 《Journal of Approximation Theory》1997,90(3):340-378
The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblemwhereω(t) is a concave modulus of continuity,r, m: 1?m?r,are integers, andB?B0(r, m, ω). We show that theextremal functionsZBhaver+1 points of alternance andthe full modulus of continuity ofZ(r)B: ω(Z(r)B; t)=ω(t) for allt∈[0, 1]. This generalizesthe Karlin's result on the extremality of classical Zolotarevpolynomials in the problem () forω(t)=tand allB?Br. 相似文献