A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear,Dispersive Equations

The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

Error estimates of H^1-Galerkin mixed finite element method for Schrdinger equation

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.

A POSTERIORI ERROR ESTIMATES OF FINITEELEMENT METHOD FOR PARABOLIC PROBLEMS

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.

Error Estimates for Mixed Finite Element Methods for Sobolev Equation

The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space Vh × Wh H(div; × L2(). Optimal order estimates are obtained for the approximation of u, ut, the associated velocity p and divp respectively in L(0,T;L2()), L(0,T;L2()), L(0,T;L2()2), and L2(0, T; L2()). Quasi-optimal order estimates are obtained for the approximations of u, ut in L(0, T; L()) and p in L(0,T; L()2).

ANALYSIS OF AUGMENTED THREE-FIELD MACRO-HYBRID MIXED FINITE ELEMENT SCHEMES

On the basis of composition duality principles, augmented three-field macrohybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladysenskaja-Babulka-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decompositions. Then, through compositional dualization macro-hybrid mixed problems are obtained, with internal boundary dual traces as Lagrange multipliers. Also, ＂mass＂ preconditioned aug- mentation of three-field formulations are derived, stabilizing macro-hybrid mixed finite element schemes and rendering possible speed up of rates of convergence. Dual mixed incompressible Darcy flow problems illustrate the theory throughout the paper.

EXPLICIT ERROR ESTIMATE FOR THE NONCONFORMING WILSON'S ELEMENT

In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson＇s element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.

EXPLICIT ERROR ESTIMATES FOR MIXED AND NONCONFORMING FINITE ELEMENTS

In this paper, we study the explicit expressions of the constants in the error estimates of the lowest order mixed and nonconforming finite element methods. We start with an explicit relation between the error constant of the lowest order Raviart-Thomas interpolation error and the geometric characters of the triangle. This gives an explicit error constant of the lowest order mixed finite element method. Furthermore, similar results can be ex- tended to the nonconforming P1 scheme based on its close connection with the lowest order Raviart-Thomas method. Meanwhile, such explicit a priori error estimates can be used as computable error bounds, which are also consistent with the maximal angle condition for the optimal error estimates of mixed and nonconforming finite element methods.

Enriched goal-oriented error estimation for fracture problems solved by continuum-based shell extended finite element method

An enriched goal-oriented error estimation method with extended degrees of freedom is developed to estimate the error in the continuum-based shell extended finite element method. It leads to high quality local error bounds in three-dimensional fracture mechanics simulation which involves enrichments to solve the singularity in crack tip. This enriched goal-oriented error estimation gives a chance to evaluate this continuum- based shell extended finite element method simulation. With comparisons of reliability to the stress intensity factor calculation in stretching and bending, the accuracy of the continuum-based shell extended finite element method simulation is evaluated, and the reason of error is discussed.

COARSE-MESH-ACCURACY IMPROVEMENT OF BILINEAR Q_4-PLANE ELEMENT BY THE COMBINED HYBRID FINITE ELEMENT METHOD

The combined hybrid finite element method is of an intrinsic mechanism of enhancing coarse-mesh-accuracy of lower order displacement schemes. It was confirmed that the combined hybrid scheme without energy error leads to enhancement of accuracy at coarse meshes, and that the combination parameter plays an important role in the enhancement. As an improvement of conforming bilinear Q(4)-plane element, the combined hybrid method adopted the most convenient quadrilateral displacements-stress mode, i.e.,the mode of compatible isoparametric bilinear displacements and pure constant stresses. By adjusting the combined parameter, the optimized version of the combined hybrid element was obtained and numerical tests indicated that this parameter-adjusted version behaves much better than Q(4)-element and is of high accuracy at coarse meshes. Due to elimination of stress parameters at the elemental level, this combined hybrid version is of the same computational cost as that of Q(4)-element.

Goal-oriented error estimation applied to direct solution of steady-state analysis with frequency-domain finite element method

Based on the concept of the constitutive relation error along with the residuals of both the origin and the dual problems, a goal-oriented error estimation method with extended degrees of freedom is developed. It leads to the high quality locM error bounds in the problem of the direct-solution steady-state dynamic analysis with a frequency-domain finite element, which involves the enrichments with plural variable basis functions. The solution of the steady-state dynamic procedure calculates the harmonic response directly in terms of the physical degrees of freedom in the model, which uses the mass, damping, and stiffness matrices of the system. A three-dimensional finite element example is carried out to illustrate the computational procedures.

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

Recovery Type A Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.

A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM

The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.

A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper,we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions.We then set up its weak formulation and the finite element approximation scheme.Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L^(2)norms.Furthermore some numerical tests are presented to verify the theoretical results.

A Posteriori Error Estimates of Triangular Mixed Finite Element Methods for Semilinear Optimal Control Problems

In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.