共查询到20条相似文献，搜索用时 0 毫秒
1.
A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory.
2.
In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper. 相似文献
3.
Michael Holst Ryan Szypowski Yunrong Zhu 《Numerical Methods for Partial Differential Equations》2013,29(5):1729-1748
In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
4.
In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330–349, 2007 相似文献
5.
Qingshan Chen 《Numerical Methods for Partial Differential Equations》2017,33(4):1159-1182
This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017 相似文献
6.
Seick Kim 《Transactions of the American Mathematical Society》2008,360(11):6031-6043
Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on . In particular, in the case when they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.
7.
Lutz Angermann 《Numerical Methods for Partial Differential Equations》2002,18(2):241-259
This article investigates Petrov‐Galerkin discretizations of operator equations with linearly stable operators, where the residual does not belong to the annihilator W of the discrete test space Wh. Conforming and nonconforming methods are considered separately, and for the treatment of the nonconforming situation the concept of elliptic lifting is introduced. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 241–259, 2002; DOI 10.1002/num.1005 相似文献
8.
Mahboub Baccouch 《Numerical Methods for Partial Differential Equations》2021,37(1):505-532
In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal. 相似文献
9.
Mark Steinhauer Jos Miguel Urbano Juha Videman 《Mathematical Methods in the Applied Sciences》2006,29(11):1339-1348
This note bridges the gap between the existence and regularity classes for the third‐grade Rivlin–Ericksen fluid equations. We obtain a new global a priori estimate, which conveys the precise regularity conditions that lead to the existence of a global in time regular solution. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
10.
Noelia Bazarra José R. Fernández Antonio Magaña Ramón Quintanilla 《Mathematical Methods in the Applied Sciences》2020,43(5):2759-2771
In this paper, we numerically analyse a phase-lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an existence and uniqueness result and an energy decay property. Then, using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations are introduced. A discrete stability property is proved, and a priori error estimates are obtained, from which the linear convergence of the approximation is derived. Finally, some one-dimensional numerical simulations are described to demonstrate the accuracy of the approximation and the behaviour of the solution. 相似文献
11.
一维非齐次BBM方程初边值问题的整体吸引子 总被引：1，自引：0，他引：1
研究了一维非齐次BBM方程ut-uxxt-(x^2n 1)x=f(u) γuxx g(x),ux(0,t)=0,u(1,5)=0,u(x,0)=u0(x).的初值边界问题，利用Sobolev插值不等式，对解做关于时间t的一致性先验估计，证明了该问题的整体吸引子的存在性． 相似文献
12.
Jens Wirth 《Mathematical Methods in the Applied Sciences》2004,27(1):101-124
We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+t vt=0, x∈?n, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain Lp–Lq estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of μand that μ=2 is critical. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
13.
Sadek Gala 《数学物理学报(B辑英文版)》2010,30(5):1413-1418
In this note, we prove that Xr （0 〈 r 〈 1） norm of the vorticity controls the blow-up phenomena of strong solutions to the Navier-Stokes equations in R3. 相似文献
14.
15.
Hai Bi Zhengxia Li Yidu Yang 《Numerical Methods for Partial Differential Equations》2016,32(2):399-417
Based on the work of Xu and Zhou [Math Comput 69 (2000) 881–909], we propose and analyze in this article local and parallel finite element algorithms for the Steklov eigenvalue problem. We also prove a local error estimate which is suitable for the case that the locally refined region contains singular points lying on the boundary of domain, which is an improvement of the existing results. Numerical experiments are reported finally to validate our theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 399–417, 2016 相似文献
16.
This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates. 相似文献
17.
In this paper, we prove the global existence and asymptotic behavior, as time tends to infinity, of solutions in Hi (i=1, 2) to the initial boundary value problem of the compressible Navier–Stokes equations of one‐dimensional motion of a viscous heat‐conducting gas in a bounded region with a non‐autonomous external force and a heat source. Some new ideas and more delicate estimates are used to prove these results. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
18.
19.
N. V. Krylov 《偏微分方程通讯》2016,41(4):644-664
Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of (t, x) bounded away from 0. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations. 相似文献
20.
研究了一类广义长短波方程组在无界区域上的解的长时间行为.通过引进加权空间,应用内插不等式和在加权空间的先验估计,证明了长短波方程组整体吸引子的存在性. 相似文献