NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

A normal compliance contact problem in viscoelasticity: An a posteriori error analysis and computational experiments

In this work, the numerical approximation of a viscoelastic contact problem is studied. The classical Kelvin-Voigt constitutive law is employed, and contact is assumed with a deformable obstacle and modelled using the normal compliance condition. The variational formulation leads to a nonlinear parabolic variational equation. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. A priori error estimates recently proved for this problem are recalled. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and other parabolic equations. Upper and lower error bounds are proved. Finally, some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

BOUNDARY ELEMENT ANALYSIS FOR A CLASS OF MULTI-DIMENSIONAL LINEAR PARABOLIC EQUATIONS

In this paper,by me as of beundary element method,we try to deal with the initial -boundary value problem for a class of linear parunolic equations,which is a linear heat conduction equation. We tresent a boundary integral equation for the solution to the problem and its variational formalation The well-posedness of the variational formulation is proved. And the error estimates for the approsutate solutions are provided. The results of this paper are more general than those of[1]     

Plate Contact问题的混合有限元逼近

论文考虑了Plate Contact问题的混合有限元逼近,其变分问题为第二类四阶椭圆变分不等问题.首先,根据正则化方法,得到原问题的正则化问题.再根据网格依赖范数技巧,考虑了正则化问题的Ciarlet-Raviart混合有限元逼近,并证明了真解与逼近解之间的误差估计.最后通过数值算例验证了理论分析的结果.     

凹角区域双曲外问题的精确人工边界条件

研究一类凹角区域双曲型外问题的数值方法.先用Newmark方法对时间进行离散化,在每个时间步求解一个椭圆外问题.然后引入人工边界,并获得精确的人工边界条件.给出半离散化问题的变分问题,证明了变分问题的适定性,并给出了误差估计.最后给出数值例子,以示该方法的可行性与有效性.     

A power penalty approach to a mixed quasilinear elliptic complementarity problem

In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust.

     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

NUMERICAL ANALYSIS OF A CONTACT PROBLEM IN RATE-TYPE VISCOPLASTICITY

In this paper, we consider numerical approximations of a contact problem in rate-type viscoplasticity. The contact conditions are described in term of a subdifferential and include as special cases some classical frictionless boundary conditions. The contact problem consists of an evolution equation coupled with a time-dependent variational inequality. Error estimates for both spatially semi-discrete and fully discrete solutions are derived and some convergence results are shown. Under appropriate regularity assumptions on the exact solution, error estimates are obtained.     

Error estimates for three-dimensional Stokes problem with non-standard boundary conditions

This work is devoted to the optimal and a posteriori error estimates of the Stokes problem with some non-standard boundary conditions in three dimensions. The variational formulation is decoupled into a system for the velocity and a Poisson equation for the pressure. The velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a posteriori estimates.     

VARIATIONAL DISCRETIZATION OF A CONTROL-CONSTRAINED PARABOLIC BANG-BANG OPTIMAL CONTROL PROBLEM

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional.For the numerical treatment,we use variational discretization of its Tikhonov regularization:For the state and the adjoint equation,we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space.We prove a-priori estimates for the error between the discretized regularized problem and the limit problem.Since these estimates are not robust if the regularization parameter tends to zero,we establish robust estimates,which--depending on the problem's regularity——enhance the previous ones.In the special case of bang-bang solutions,these estimates are further improved.A numerical example confirms our analytical findings.     

Galerkin Methods for Nonlinear Ordinary Differential Equation with Impulses

Methods based on fixed mesh variational formulations for ordinary differential equations in presence of a possibly infinite number of impulses on the righthand side are presented. A simple transformation allows us to show that the problem can be treated as an ordinary differential equation. Existence and uniqueness results for the solution and approximation schemes with their error estimates are obtained.     

Numerical solutions of some parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we introduce a variational form by defining a new function. Both continuous and discrete Galerkin procedures are illustrated in this paper. The error estimates are also derived.     

On Vector Helmholtz Equation with a Coupling Boundary Condition

The Helmholtz equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown.In this paper, we study the vector Helmholtz problem in domains of both C~(1,1)and Lipschitz.We es- tablish a rigorous variational analysis such as equivalence,existence and uniqueness. And we propose finite element approximations based on the uncoupled solutions.Fi- nally we present a convergence analysis and error estimates.     

无穷凹角区域各向异性问题的自然边界元与有限元耦合法

本文研究无穷凹角区域上一类各向异性问题的自然边界元与有限元耦合法.利用自然边界归化原理,获得圆弧或椭圆弧人工边界上的自然积分方程,给出了耦合的变分形式及其数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.     

多孔直杆扭转问题的边界元分析

多孔直杆弹性扭转问题导致一个Poisson方程的非局部边值问题,相应的积分方程既包含未知函数又包含若干个未知数,并带有约束条件,我们通过把问题化为若干个无约束的变分问题证明了变分解的唯一性,并对近似解进行了误差分析,得到了最佳的误差估计。     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

Error estimates for finite element method solution of the Stokes problem in the primitive variables

Summary In this paper we derive error estimates for a class of finite element approximation of the Stokes equation. These elements, popular among engineers, are conforming lagrangian both in velocity and pressure and therefore based on a mixed variational principle. The error estimates are established from a new Brezzi-type inequality for this kind of mixed formulation. The results are true in 2 or 3 dimensions.     

A Galerkin boundary element method based on interpolatory Hermite trigonometric wavelets

A Galerkin boundary element method based on interpolatory Hermite trigonometric wavelets is presented for solving 2-D potential problems defined inside or outside of a circular boundary in this paper. In this approach, an equivalent variational form of the corresponding boundary integral equation for the potential problem is used; the trigonometric wavelets are employed as trial and test functions of the variational formulation. The analytical formulae of the matrix entries indicate that most of the matrix entries are naturally zero without any truncation technique and the system matrix is a block diagonal matrix. Each block consists of four circular submatrices. Hence the memory spaces and computational complexity of the system matrix are linear scale. This approach could be easily coupled into domain decomposition method based on variational formulation. Finally, the error estimates of the approximation solutions are given and some test examples are presented.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.