Finite element methods for Timoshenko beam,circular arch and Reissner-Mindlin plate problems

In this paper some finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems are discussed. To avoid locking phenomenon, the reduced integration technique is used and a bubble function space is added to increase the solution accuracy. The method for Timoshenko beam is aligned with the Petrov-Galerkin formulation derived in Loula et al. (1987) and can be naturally extended to solve the circular arch and the Reissner-Mindlin plate problems. Optimal order error estimates are proved, uniform with respect to the small parameters. Numerical examples for the circular arch problem shows that the proposed method compares favorably with the conventional reduced integration method.     

ABSOLUTE STABLE HOMOTOPY FINITE ELEMENT METHODS FOR CIRCULAR ARCH PROBLEM AND ASYMPTOTIC EXACTNESS POSTERIORI ERROR ESTIMATE

In this paper, HFEM is proposed to investigate the circular arch problem. Optimal error estimates are derived, some superconvergence results are established, and an asymptotic exactness posteriori error estimator is presented. In contrast with the classical displacement variational method, the optimal convergence rate for displacement is uniform to the small parameter. In contrast with classical mixed finite element methods, our results are free of the strict restriction on h(the mesh size) which is preserved by all the previous papers. Furtheremore we introduce an asymptotic exactness posteriori error estimator based on a global superconvergence result which is discovered in this kind of problem for the first time.     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

Approximation of general arch problems by straight beam elements

Summary In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.     

LINEAR FINITE ELEMENT APPROXIMATIONS FOR THE TIMOSHENKO BEAM AND THE SHALLOW ARCH PROBLEMS

1. Introductionln this paper we examine the linear finite element discretization of the Timoshenko beamand the shalIow arch problems. The thickness of beam and arch appears parametrically inthe model and locking phenomenon may occur due to the smaIl param…     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

Superconvergence of mixed finite element methods for optimal control problems

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

     

Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

In this paper, we investigate the superconvergence of fully discrete splitting positive definite mixed finite element (MFE) methods for parabolic optimal control problems. For the space discretization, the state and co-state are approximated by the lowest order Raviart–Thomas MFE spaces and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. We derive the superconvergence between the projections of exact solutions and numerical solutions or the exact solutions and postprocessing numerical solutions for the control, state and co-state. A numerical example is provided to validate the theoretical results.     

DOMAIN DECOMPOSITION METHODS WITH NONMATCHING GRIDS FOR THE UNILATERAL PROBLEM

1. IntroductionDomain decomposition methods (DDMs) with nonmatching grids, which have been deveLoped in recent years3 are a quite new class of nonconforming DDMs. As this kind of DDMscan be aPplied to solving many practical problems which can't be handled…     

Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations

In this paper, we investigate the L ^??(L ₂)-error estimates and superconvergence of the semidiscrete mixed finite elementmethods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state 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 approximation of both state and control. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

Superconvergence for rectangular mixed finite elements

Summary In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL ² between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.     

The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property: where denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.     

Superconvergent biquadratic finite volume element method for two-dimensional Poisson’s equations

In this paper, a kind of biquadratic finite volume element method is presented for two-dimensional Poisson’s equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes. The method can be effectively implemented by alternating direction technique. It is proved that the method has optimal energy norm error estimates. The superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique. Finally, a numerical example verifies the theoretical results and illustrates the effectiveness of the method.     

FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. The optimal and superconvergence error estimates for this new method are derived both in space and in time. Also, a class of new error estimates of convergence and superconvergence for the time-continuous finite element method is demonstrated in which there are no time derivatives of the exact solution involved, such that these estimates can be bounded by the norms of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.     

Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for three dimensional first order cuboid Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence is derived by a standard postprocessing. For first order nonconforming finite element methods of three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the validity of the theoretical results.     

Superconvergence of immersed finite element methods for interface problems

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions.     

Locking and robustness in the finite element method for circular arch problem

Summary. In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order , and is uniformly robust with order for which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. Received June 5, 1992 / Revised version received May 17, 1994     

最小二乘法及光滑问题不规则剖分上有限元解的超收敛分析

对有限元近似解提出一种通用的超收敛框架,该框架是对有限元解在另一有限维空间中作最小二乘逼近,中证明新构造的逼近解具有局部和整体上的超收敛,与所有已知的超收敛结果不同的是,该框架给出的超收敛结果对区域的有限元剖分没有附加任何一致性或对称性要求,这种得用小二乘作超收剑的技巧可以很简单地推广到混合有限元法,斯托克斯方程及重调方程的有限元法。     

SUPERCONVERGENCE OF GENERALIZED DIFFERENCE METHOD FOR ELLIPTIC BOUNDARY VALUE PROBLEM

Some superconvergence results of generalized difference solution for elliptic boundary value problem are given. It is shown that optimal points of the stresses for generalized difference method are the same as that for finite element method.