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1.
In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for
non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal
order H
1-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption
相似文献
2.
Chunjia Bi 《Numerical Methods for Partial Differential Equations》2007,23(1):220-233
In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H1 and W1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the Lp and W1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
3.
Richard Ewing Raytcho Lazarov Yanping Lin 《Numerical Methods for Partial Differential Equations》2000,16(3):285-311
In this article, we study finite volume element approximations for two‐dimensional parabolic integro‐differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher‐order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H1‐ and L2‐norm and are superconvergent in a discrete H1‐norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L∞‐ and W1,∞‐norms. These results are also new for finite volume element methods for elliptic and parabolic equations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 285–311, 2000 相似文献
4.
In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption. 相似文献
5.
P. Chatzipantelidis R. D. Lazarov V. Thome 《Numerical Methods for Partial Differential Equations》2004,20(5):650-674
We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L2 and H 1 . The convergence rate in the L∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 相似文献
6.
《Journal of Computational and Applied Mathematics》2002,146(2):373-386
The finite volume element method is a discretization technique for partial differential equations, but in general case the coefficient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modified finite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coefficient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term. 相似文献
7.
Zhaoliang Meng Ruifang Wang 《Numerical Methods for Partial Differential Equations》2019,35(3):1152-1163
In this paper, a new nonparametric nonconforming pyramid finite element is introduced. This element takes the five face mean values as the degrees of the freedom and the finite element space is a subspace of P2. Different from the other nonparametric elements, the basis functions of this new element can be expressed explicitly without solving linear systems locally, which can be achieved by introducing a new reference pyramid. To evaluate the integration, a class of new quadrature formulae with only two/three equally weighted points on pyramid are constructed. We present the error estimation in the presence of quadrature formulae. Numerical results are shown to confirm the optimality of the convergence order for the second order elliptic problems. 相似文献
8.
Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse
and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems
and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint
and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem
on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear
elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic
problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms.
A set of numerical examples are presented to confirm the estimates.
The work is supported by the National Natural Science Foundation of China (Grant No: 10601045). 相似文献
9.
Manicka Dhanasekar Jianjun Han Qinghua Qin 《Finite Elements in Analysis and Design》2006,42(14-15):1314-1323
Much work on special elements that simplify geometrical modelling of structures containing holes, cracks and/ or inclusions has been reported extensively in the literature. This paper presents a hybrid-Trefftz element containing elliptic hole formulated using Hellinger–Reissner principle by employing trial functions based on the mapping technique and the Cauchy integral method. The element presented in this paper could be regarded as an improved formulation over Piltner [Special finite elements with holes and internal cracks, Int. J. Numer. Methods Eng. 21 (1985) 1471–1485] element because the chosen trail functions in this paper have provided relatively more stable solutions. The use of the element with other ordinary displacement-based finite elements has also yielded very accurate solutions even when very coarse meshes relative to the size of the elliptic hole have been used. 相似文献
10.
Xiaoming He Tao Lin Yanping Lin 《Numerical Methods for Partial Differential Equations》2008,24(5):1265-1300
This article discusses a bilinear immersed finite element (IFE) space for solving second‐order elliptic boundary value problems with discontinuous coefficients (interface problem). This is a nonconforming finite element space and its partition can be independent of the interface. The error estimates for the interpolation of a Sobolev function indicate that this IFE space has the usual approximation capability expected from bilinear polynomials. Numerical examples of the related finite element method are provided. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 相似文献
11.
In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method (FVE) based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems. 相似文献
12.
This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem. 相似文献
13.
Alan Demlow. 《Mathematics of Computation》2004,73(248):1623-1653
In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.
14.
Jichun Li 《Numerical Methods for Partial Differential Equations》2006,22(4):884-896
By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
15.
In this paper, first we discuss a technique to compare finite volume method and some well-known finite element methods, namely the dual mixed methods and nonconforming primal methods, for elliptic equations. These both equivalences are exploited to give us a posteriori error estimator for finite volume methods. This estimator is explicitly given, easy to compute and asymptotically exact without any regularity of the solution in unstructured grids. 相似文献
16.
The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis. 相似文献
17.
Chunjia Bi Yanping Lin Min Yang 《Numerical Methods for Partial Differential Equations》2013,29(4):1097-1120
In this article, we consider the finite volume element method for the monotone nonlinear second‐order elliptic boundary value problems. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz‐continuous, and with the minimal regularity assumption on the exact solution, that is, u∈H1(Ω), we show that the finite volume element method has a unique solution, and the finite volume element approximation is uniformly convergent with respect to the H1 ‐norm. If u∈H1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate \begin{align*}\mathcal{O}(h^{\epsilon})\end{align*} in the H1 ‐norm. Moreover, we propose a natural and computationally easy residual‐based H1 ‐norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
18.
Sarvesh Kumar Neela Nataraj Amiya K. Pani 《Numerical Methods for Partial Differential Equations》2009,25(6):1402-1424
In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L2 and broken H1‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
19.
JiangChengshun 《高校应用数学学报(英文版)》1999,14(4):440-450
Abstract. This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms. A Crank-Nicolson approximation for this kind of equations is presented. By using the elliptic Ritz Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H1-norm error estimate are demonstrated. 相似文献
20.
Z. Li T. Lin Y. Lin R. C. Rogers 《Numerical Methods for Partial Differential Equations》2004,20(3):338-367
This article discusses an immersed finite element (IFE) space introduced for solving a second‐order elliptic boundary value problem with discontinuous coefficients (interface problem). The IFE space is nonconforming and its partition can be independent of the interface. The error estimates for the interpolation of a function in the usual Sobolev space indicate that this IFE space has an approximation capability similar to that of the standard conforming linear finite element space based on body‐fit partitions. Numerical examples of the related finite element method based on this IFE space are provided. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 338–367, 2004 相似文献