On the coupled NBEM and FEM for a class of nonlinear exterior Dirichlet problem in R~(2*)

In this paper, based on the natural boundary reduction advanced by Feng and Yu, we couple the finite element approach with the natural boundary element method to study the weak solvability and Galerkin approximation of a class of nonlinear exterior boundary value problems. The analysis is mainly based on the variational formulation with constraints. We prove the error estimate of the finite element solution and obtain     

On the coupled NBEM and FEM for a class of nonlinear exterior Dirichlet problem in <Emphasis Type="Italic">R</Emphasis><Superscript>2</Superscript>

In this paper, based on the natural boundary reduction advanced by Feng and Yu, we couple the finite element approach with the natural boundary element method to study the weak solvability and Galerkin approximation of a class of nonlinear exterior boundary value problems. The analysis is mainly based on the variational formulation with constraints. We prove the error estimate of the finite element solution and obtain the asymptotic rate of convergence. Finally, we also give a numerical example.     

解一类半线性外边值问题的自然边界元与有限元耦合法

In this paper, we combine a finite element approach with the natural boundary element method to stduy the weak solvability and Galerkin approximations of a class of semilinear exterior boundary value problems. Our analysis is mainly based on the variational formulation with constraints. We discuss the error estimate of the finite element solution and obtain the asymptotic rate of convergence O（h^n) Finally, we also give two numerical examples.     

n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.     

Mathematical Analysis and Numerical Simulation of a Nonlinear Thermoelastic System

Abstract

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.     

The elliptic Cauchy problem revisited: Control of boundary data in natural norms

In this note, we prove error estimates in natural norms on the approximation of the boundary data in the elliptic Cauchy problem, for the finite element method first analysed in E. Burman, Error estimates for stabilised finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I 352 (7–8) (2014) 655–659.     

Natural Boundary Integral Method and its New Development

In this paper, the natural boundary integral method, and some related methods, including coupling method of the natural boundary elements and finite elements, which is also called DtN method or the method with exact artificial boundary conditions, domain decomposition methods based on the natural boundary reduction, and the adaptive boundary element method with hyper-singular a posteriori error estimates, are discussed.     

An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

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.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

Finite volume element method for monotone nonlinear elliptic problems

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∈H^1+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     

基于Monte-Carlo随机有限元方法的随机边界条件下自然对流换热不确定性研究

为分析边界条件不确定性对方腔内自然对流换热的影响,发展了一种求解随机边界条件下自然对流换热不确定性传播的Monte-Carlo随机有限元方法.通过对输入参数场随机边界条件进行Karhunen-Loeve展开及基于Latin(拉丁)抽样法生成边界条件随机样本,数值计算了不同边界条件随机样本下方腔内自然对流换热流场与温度场,并用采样统计方法计算了随机输出场的平均值与标准偏差.根据计算框架编写了求解随机边界条件下方腔内自然对流换热不确定性的MATLAB随机有限元程序,分析了随机边界条件相关长度与方差对自然对流不确定性的影响.结果表明:平均温度场及流场与确定性温度场及流场分布基本相同;随机边界条件下Nu数概率分布基本呈现正态分布,平均Nu数随着相关长度和方差增加而增大;方差对自然对流换热的影响强于相关长度的影响.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

     

Two-scale sparse finite element approximations

To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general setting.Over any tensor product domain ?R~d with d = 2,3,we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper approximations.As applications,we obtain some new efficient finite element discretizations for the two classes of problem:The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.     

Superconvergence of the - version of the finite element method in one dimension

In this paper, we investigate the superconvergence properties of the h-p version of the finite element method (FEM) for two-point boundary value problems. A postprocessing technique for the h-p finite element approximation is analyzed. The analysis shows that the postprocess improves the order of convergence. Furthermore, we obtain asymptotically exact a posteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.     

On Exact Results in the Finite Element Method

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u. We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

样条有限元

本文用三次B样条变分方法解规则区域上板梁组合弹性结构的平衡问题.推导出了适用于各种边界条件的统一计算格式,便于在计算机上实现.与通常有限元相比,具有计算量少、精确度高等显著特点.文中对自然边界条件作为约束条件的影响给予了考虑,并以板的弯曲问题为例说明影响极微.给出了几个数值的例子.