共查询到20条相似文献,搜索用时 15 毫秒
1.
In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L 2-error estimates are derived, when the initial data is in L 2. A superconvergence phenomenon is also observed, which is then used to prove L ∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data. 相似文献
2.
A. A. Amosov 《Journal of Mathematical Sciences》2011,176(3):361-408
We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value
problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields
of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate
the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates
for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε
3/4) for asymptotic approximations. Bibliography: 9 titles. 相似文献
3.
《Numerical Functional Analysis & Optimization》2013,34(7-8):891-909
ABSTRACT A posteriori error estimates for semidiscrete finite element methods for a nonlinear parabolic initial-boundary value problem are considered. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element. The convergence results improve previous results where unnecessary assumptions are imposed on the approximate solution and the elliptic projection of the exact solution. 相似文献
4.
P. Danumjaya Amiya K. Pani 《Numerical Methods for Partial Differential Equations》2012,28(4):1227-1251
Mixed finite element methods are applied to a fourth order reaction diffusion equation with different types of boundary conditions. Some a priori bounds are established with the help of Lyapunov functional. The semidiscrete schemes are derived using C0‐piecewise linear finite elements in spatial direction and error estimates are obtained. The semidiscrete problem is then discretized in the temporal direction using backward Euler method and the wellposedness of the completely discrete scheme is discussed. Finally, a priori error estimates are established. While deriving a priori error estimates, Gronwall's lemma is applied and the constants involved in the error bounds do not depend exponentially on $\frac{1}{\gamma}$, where γ is a parameter appeared in the fourth order derivative. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012 相似文献
5.
We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L 2(Ω) and H 1(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose. 相似文献
6.
In this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L∞(L2). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L∞(H1) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O(e?γ0t) with γ0 > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings. 相似文献
7.
A family of elliptic optimal control problems with pointwise constraints on control and state is considered. We are interested
in approximation of the optimal solution by a finite element discretization of the involved partial differential equations.
The discretization error for a problem with mixed state constraints is estimated in the semidiscrete case and in the fully
discrete scheme with the convergence of order h|ln h| and h
1/2, respectively. However, considering the unregularized continuous problem and the discrete regularized version, and choosing
suitable relation between the regularization parameter and the mesh size, i.e., ε∼h
2, a convergence order arbitrary close to 1, i.e., h
1−β
is obtained. Therefore, we benefit from tuning the involved parameters. 相似文献
8.
Ch. G. Makridakis 《Numerische Mathematik》1992,61(1):235-260
Summary We construct and analyze finite element methods for approximating the equations of linear elastodynamics, using mixed elements for the discretization of the spatial variables. We consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth-order in time fully discrete approximations.L
2 optimal-order error estimates are proved for the approximations of displacement and stress.Work supported in part by the Hellenic State Scholarship Foundation 相似文献
9.
In this exposition we study the finite element methods for second-order semilinear parabolic interface problems in two dimensional convex polygonal domains with smooth interface. Both semidiscrete and fully discrete schemes are analyzed. Optimal order error estimates in the L2(0, T; H1(Ω))-norm are established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
Tianliang Hou 《Numerical Methods for Partial Differential Equations》2013,29(5):1675-1693
In this article, we investigate the L∞(L2) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
11.
Sudhakar Chaudhary Vimal Srivastava V. V. K. Srinivas Kumar Balaji Srinivasan 《Numerical Methods for Partial Differential Equations》2017,33(3):786-813
In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L2 and H1 norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017 相似文献
12.
《Numerical Methods for Partial Differential Equations》2018,34(1):336-356
In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H1‐norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis. 相似文献
13.
Miloslav Feistauer Václav Kučera Karel Najzar Jaroslava Prokopová 《Numerische Mathematik》2011,117(2):251-288
The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary
convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly
perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in
general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary
penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates
in “L
2(L
2)”- and “DG”-norm formed by the “L
2(H
1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration
has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect
to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour
in time as a polynomial of degree ≤ q. 相似文献
14.
Thanh Tran Thanh‐Binh Duong 《Numerical Methods for Partial Differential Equations》2005,21(3):521-535
A posteriori error estimates for semidiscrete finite element methods for a nonlinear Sobolev equation are considered. The error estimates are obtained by solving local nonlinear or linear pseudo‐parabolic equations for corrections to the solution on each element. The ratios of these estimates and the true errors are proved to converge to 1, implying that the estimates can be used as indicators in adaptive schemes for the problem. Numerical results underline our theoretical results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
15.
An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition. 相似文献
16.
The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear
nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately
in space and time using, in general, different nonconforming space grids on different time levels and different polynomial
degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty
approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L
2(L
2)”-and “
”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization,
we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution,
the errors are of order O(h
p
+ τ
q
). The estimates hold true even in the hyperbolic case when ɛ = 0. 相似文献
17.
Tlili Kadri Noomen Khiari Faycal Abidi Khaled Omrani 《Numerical Methods for Partial Differential Equations》2008,24(6):1501-1516
L∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
18.
Interior estimates are proved in the L ∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations. 相似文献
19.
Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow
of closed curves in
\mathbbRd, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of
the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time
semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition
to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the
plane and in higher codimension that satisfies various boundary conditions. 相似文献
20.
Ajit Patel Amiya K. Pani Neela Nataraj 《Numerical Methods for Partial Differential Equations》2008,24(6):1460-1484
In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L∞(L2) and L∞(H1)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 相似文献