Finite Element Approximation of Hemivariational Inequalities

The paper deals with a finite element approximation of elliptic and parabolic variational inequalities. Elliptic hemivariational inequalities are equivalently expressed as a system consisting of one equation and one inclusion for a couple of unknowns, namely a primal variable u and an element belonging to a multivalued mapping at u. Both components of the solution are approximated independently each other by a finite element method. Parabolic inequalities are transformed into a system of elliptic ones by using an appropriate time discretization. A numerical experiment is realized by using non-smooth optimization methods.     

Extrapolation of Finite Element Approximation in a Rectangular Domain

Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution $u$ but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0＜\alpha＜1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.     

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.     

三维有限元本征值的外推

通过二维和三维积分恒等式,探讨泊松方程本征值问题三角线元和四面体线元Richardson外推的可行性.理论分析表明,如果剖分为均匀一致和拟一致,外推均可将解的精度提高二阶.     

Finite Element Approximation of Vector-Valued Hemivariational Problems

In this paper we develop a finite element approximationfor vector-valuedhemivariational inequalities.This class of hemivariational problems wasintroducedin [12],[13]. We study two differentproblems: unconstrained oneand constrained one witha nonempty, closed, convex constraint set K.We shall show firstly that the discrete problemsare solvable by usingconsequences of Kakutanifixed point theorem and secondly that the solutionsof the discrete problemsare close on subsequences to the continuous ones.     

非协调有限元与本征值下界

与协调有限元相比,非协调有限元常常给出本征值下界.这种现象已经由很多数值例子观察到.但是理论上的证明最近才做的更多.以Morley非协调元作为更简单的范例,来说明这种理论为何成功.     

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

An Approximation of Three-Dimensional Semiconductor Devices by Mixed Finite Element Method and Characteristics-Mixed Finite Element Method

The mathematical model for semiconductor devices in three space dimensions are numerically discretized. The system consists of three quasi-linear partial differential equations about three physical variables: the electrostatic potential, the electron concentration and the hole concentration. We use standard mixed finite element method to approximate the elliptic electrostatic potential equation. For the two convection-dominated concentration equations, a characteristics-mixed finite element method is presented. The scheme is locally conservative. The optimal $L^2$-norm error estimates are derived by the aid of a post-processing step. Finally, numerical experiments are presented to validate the theoretical analysis.     

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.     

h — P Finite Element Approximation for Pull-Potential Electronic Structure Calculations

The （continuous） finite element approximations of different orders for the computation of the solution to electronic structures were proposed in some papers and the performance of these approaches is becoming appreciable and is now well understood. In this publication, the author proposes to extend this discretization for full-potential electronic structure calculations by combining the refinement of the finite element mesh, where the solution is most singular with the increase of the degree of the polynomial approximations in the regions where the solution is mostly regular. This combination of increase of approximation properties, done in an a priori or a posteriori manner, is well-known to generally produce an optimal exponential type convergence rate with respect to the number of degrees of freedom even when the solution is singular. The analysis performed here sustains this property in the case of Hartree-Fock and Kohn-Sham problems.     

管道Bingham流的变网格有限元法

本文讨论了管道Bingham流的变网格有限元逼近,并在合理的正则性假设下,得到了确定模意义下的误差估计。     

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.     

Finite Element Approximation of Conformal Mappings to Unbounded Jordan Domains

In two-dimensional Euclidean space, a simply connected unbounded domain whose boundary is a Jordan curve is called an unbounded Jordan domain. In this article, we discuss a piecewise linear finite element approximation of conformal mappings from the unit disk of the plane to unbounded Jordan domains. Some convergence results and error analysis are presented. Numerical examples are also given.     

Finite Element Approximation of Fluid-Structure Interaction with Coupled Wave Propagation

In this contribution, a concept of coupling the fluid-structure interaction (FSI) with an ultrasonic wave propagation is proposed. It is referred to as the extended Fluid-Structure Interaction (eXFSI) problem. The eXFSI is a one-directional coupling of typical FSI problem with an ultrasonic wave propagation in fluid-solid and their interaction (WpFSI). Notably, the WpFSI is a strongly coupled problem of acoustic and elastic wave equations and automatically adopts the boundary conditions from the FSI problem at each time step. To the best of our knowledge, such a model is novel in the literature. The principal aim we pursue is to explore and develop concept of the efficient numerical solution of the eXFSI problem. The solution of the eXFSI and WpFSI models constitute an important part of on-live and off-live structural health monitoring (SHM) systems design for composite material and lightweight structures. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and Application of the Initially Rigid Cohesive Finite Element Method

The presented procedure for cohesive crack propagation is based on an adaptive finite element (FE) implementation, which enables the introduction of cohesive surfaces in dependence on the current crack state. In contrast to already existing formulations, the focus of the present model lies on failure processes that can be described at quasi-static conditions within an implicit framework. Furthermore, an extension for mesh independent crack propagation in terms of an additional mesh adaptive formulation is presented. By the evaluation of the failure criterion considering the preferred crack direction, a new crack tip coordinate is computed and the discretization is accordingly adjusted. The remaining mesh is modified for the new boundary representation. The application of the proposed method is shown by the numerical investigation of a concrete fracture specimen from an experimental research project. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)