非线性双曲型方程的变网格有限元法

对一类非线性双曲型方程给出了两种变网格有限元逼近格式 .在一定条件下 ,得到了最优 H 1模误差估计     

Finite Element Simulations with Adaptively Moving Mesh for the Reaction Diffusion System

A moving mesh method is proposed for solving reaction-diffusion equations. The finite element method is used to solving the partial different equation system, and an efficient scheme is applied to implement mesh moving. In the practical calculations, the moving mesh step and the problem equation solver are performed alternatively. Serveral numerical examples are presented, including the Gray-Scott, the Activator-Inhibitor and a case with a growing domain. It is illustrated numerically that the moving mesh method costs much lower, compared with the numerical schemes on a fixed mesh. Even in the case of complex pattern dynamics described by the reaction-diffusion systems, the adapted meshes can capture the details successfully.     

一类非线性抛物型方程组的交替方向变网格有限元方法

本文对一类非线性抛物型方程组提出并分析了一类全离散交替方向变网格有限元格式，且在相当一般的情况下得到了最佳的L^2模误差估计。     

边界元法在带可动边界的瞬时热传导问题中的应用

该文利用边界无法对二维瞬时传热和相变问题进行求解。采用空间二次曲线单元对边界进行离散化,并采用时间相关基本解正确处理了相应边界积分。计算表明该文数值格式稳定,结果与已有文献相比较吻合一致,且该文模式时段可放大,从而提高了计算效能和稳定性。     

Adaptive Cross Approximation in an elastodynamic Boundary Element formulation

Elastodynamic phenomena can be effectively analyzed by using the Boundary Element Method (BEM), especially in unbounded media. However, for the simulation of such problems, beside others, two difficulties restrict the BEM to rather small or medium–sized problems. Firstly, one has to deal with dense matrices and secondly the treatment of the kernel functions is very costly. Several approaches have been developed to overcome these drawbacks. Approaches, such as Fast Multipole and Panel Clustering etc. gain their efficiency basically from an analytic kernel approximation. The main difficulty of these methods is that the so called degenerate kernel has to be known explicitly. Hence, the present work focuses on a purely algebraic approach, the adaptive cross approximation (ACA). By means of a geometrical clustering and a reliable admissibility condition, first, a so called hierarchical matrix structure is set up. Then each admissible block can be represented by a low–rank approximation. The advantage of the ACA is based on the fact that only a few of the original matrix entries have to be generated. As will be shown numerically, the presented approach is suitable for an efficient simulation of elastodynamic problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive Finite Element Methods with convergence rates

Summary. Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ² by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H¹ norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n1, it can be approximated to accuracy O(n^–s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.Mathematics Subject Clasification (2000): 65N30, 65Y20, 65N12, 65N50, 68W40, 68W25.This work has been supported by the Office of Naval Research Contract Nr. N00014-03-10051, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation Grants DMS 0221642, DMS 9872890 the Deutsche Forschungsgemeinschaft grant SFB 401, the European Communitys Human Potential Programme under Contract HPRN-CT-2002-00286, Breaking Complexity.     

一种h型自适应有限单元

ｈ型自适应有限单元在网格局部细划时．会产生非常规节点，从而破坏了一般意义上的单元连续性假定．本文利用参照节点对非常规单元进行坐标和位移插值．为保证单元之间坐标和位移的连续性，本文提出了一组修正的形函数，常用的形函数是它的一个特例．本方法应用于有限元程序时，除形函数外无须做任何改动．算例表明水文的方法具有方法简单、精度高、自由度少、计算量小等优点．     

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。     

EXTRAPOLATION OF BILINEAR FINITE ELEMENT SOLUTION WITH LOCAL REFINEMENT MESH

For the less smooth solution caused by the reentrant domain it is shown that one step of extrapolation increases the order of bilinear finite element solution from 2 to 3 when the rectangular mesh satisfies certain local refinement condition.     

A Finite Element that Satisfies Natural Boundary Conditions Exactly

Blending function interpolation is used to construct finiteelements that match both essential and natural boundary conditionsexactly along the entire boundary. The convergence of the methodis analyzed as well as the effect on the energy functional offorcing the admissible functions to satisfy the natural conditions.     

Large Scale Simulation with Scaled Boundary Finite Element Method

Nowadays scientific and engineering applications often require wave propagation in infinite or unbounded domains. In order to model such applications we separate our model into near-field and far-field. The near-field is represented by the well-known finite element method (FEM), whereas the far-field is mapped by a scaled boundary finite element (SBFE) approach. This latter approach allows wave propagation in infinite domains and suppresses the reflection of waves at the boundary, thus being a suitable method to model wave propagation to infinity. It is non-local in time and space. From a computational point of view, those characteristics are a drawback because they lead to storage consuming calculations with high computational time-effort. The non-locality in space causes fully populated unit-impulse acceleration influence matrices for each time step, leading to immense storage consumption for problems with a large number of degrees of freedom. Additionally, a different influence matrix has to be assembled for each time step which yields unacceptable storage requirements for long simulation times. For long slender domains, where many nodes are rather far from each other and where the influence of the degrees of freedom of those distant nodes is neglectable, substructuring represents an efficient method to reduce storage requirements and computational effort. The presented simulation with substructuring still yields satisfactory results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Variational Inequality Formulation of an Electrochemical Machining Moving Boundary Problem and its Approximation by the Finite Element Method

An electrochemical machining moving boundary problem is formulated,after a change of variable, as an elliptic variational inequality.The unknown anode surface may now be found by solving just oneelliptic free boundary problem. The variational inequality isapproximated by the finite element method and numerical resultsare presented.     

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.     

Convergence Rates for an Adaptive Dual Weighted Residual Finite Element Algorithm

Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation,

Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value Problems in Non-smooth Domains

This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and non-intersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second-order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of ‘actual’ errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence.     

The Boundary Finite Element Method for the Efficient Computation of Orders and Modes of Three-Dimensional Stress Singularities

In this contribution the B oundary F inite E lement M ethod (BFEM) is employed for the computation of the orders of stress singularities for several three-dimensional stress concentration problems in linear elastic fracture mechanics. The BFEM combines the advantages of both the FEM and the BEM: while only a discretization on a structural boundary is required, the actual surface mesh consists of standard displacement based finite elements. In contrast to the BEM, no fundamental solution is required. The BFEM is an ef.cient analysis tool which leads to highly accurate results with significantly lesser computational effort when compared to e.g. standard FEM procedures. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

Finite Element Eigenvalue Computation on Domains with Reentrant Corners Using Richardson Extrapolation

In the presence of reentrant corners or changing boundary conditions, standard finite element schemes have only a reduced order of accuracy even at interior nodal points. This pollution effect can be completely described in terms of asymptotic expansions of the error with respect to certain fractional powers of the mesh size. Hence, eliminating the leading pollution terms by Richardson extrapolation may locally increase the accuracy of the scheme. It is shown here that this approach also gives improved approximations for eigenvalues and eigenfunctions which are globally defined quantities.