Superconvergence of a discontinuous finite element method for a nonlinear ordinary differential equation

In this paper, n-degree discontinuous finite element method with interpolated coefficients for an initial value problem of nonlinear ordinary differential equation is introduced and analyzed. By using the finite element projection for an auxiliary linear problem as comparison function, an optimal superconvergence , at (n + 1)-order characteristic points in each element respectively is proved. Finally the theoretic results are tested by a numerical example.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L^∞(L²) for velocity and concentration and the super convergence in L^∞(H¹) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.     

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β ∈ (0, 1) and 2α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax-Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.     

The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter ? ? 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h_i = x_i+1 − x_i, with 0 = x₀ < x₁ < ? < x_N = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small ?, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L², for large as well as small values of N, and over a wide range of values of ?. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness.     

浅水中污染物扩散的自适应有限元分析法

介绍浅水中污染物扩散分析中的有限元法.分析包括两个部分:1)流场速度、水面高度的计算;2)根据扩散模型计算污染物浓度场.联合使用了自适应网格技术以期提高解的精度,同时减少计算时间和计算机内存的消耗.通过几个有已知解的实例验证了有限元公式和计算机程序.最后,使用这种联合方法分析泰国Chao Phraya河附近海湾中的污染物扩散.     

Numerical modelling of aeroelastic behaviour of an airfoil in viscous incompressible flow

In this paper, the problem of the numerical approximation of a two-dimensional incompressible viscous fluid flow interacting with a flexible structure is considered. Due to high Reynolds numbers in the range 10⁴ − 10⁶ the turbulent character of the flow is considered and modelled with the aid of Reynolds equations coupled with the k − ω turbulence model. The structure motion is described by a system of ordinary differential equations for three degrees of freedom: vertical displacement, rotation and rotation of the aileron. The problem is discretized in space by the Galerkin Least-Squares stabilized finite element method and the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian method.     

FROM ENERGY IMPROVEMENT TO ACCURACYENHANCEMENT： IMPROVEMENT OF PLATE BENDING ELEMENTS BY THE COMBINED HYBRID METHOD

By following the geometric point of view in mechanics, a novel expression of the combined hybrid method for plate bending problems is introduced to clarify its intrinsic mechanism of enhancing coarse-mesh accuracy of conforming or nonconforming plate elements. By adjusting the combination parameter α∈(0, 1) and adopting appropriate bending moments modes, reduction of energy error for the discretized displacement model leads to enhanced numerical accuracy. As an application, improvement of Adini‘s rectangle is discussed. Numerical experiments show that the combined hybrid counterpart of Adini‘s element is capable of attaining high accuracy at coarse meshes.     

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on and meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only (N ^3/2) degrees of freedom compared with the (N ²) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method. This work was supported by the National Natural Science Foundation of China (10701083 and 10425105), the Chinese National Basic Research Program (2005CB321704) and the Boole Centre for Research in Informatics at National University of Ireland Cork.     

A Least‐Squares Formulation for the Shallow Water Equations with Small Viscosity

A least‐squares finite element method for the shallow water equations with viscosity parameter μ > 0 is proposed and studied. The shallow water equations are reformulated as a first order system by adding a new variable for the velocity flux. The reformulated first order system is combined with a characteristic‐based time discretization and a least squares approach. For the correct boundary treatment in the limit case μ → 0, a trace theorem is presented. For the numerical computation of the velocity, the finite element spaces introduced recently by Mardal, Tai and Winther (SIAM Journal on Numerical Analysis 40, pp. 1605–1631) are used. The degrees of freedom in these spaces can be identified with the normal and tangential components, respectively. Numerical results for some test examples are shown. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element method for two‐phase immiscible flow

An explicit finite element method for numerically solving the two‐phase, immiscible, incompressible flow in a porous medium in two space dimensions is analyzed. The method is based on the use of a mixed finite element method for the approximation of the velocity and pressure a discontinuous upwinding finite element method for the approximation of the saturation. The mixed method gives an approximate velocity field in the precise form needed by the discontinuous method, which is trivially conservative and fully parallelizable in computation. It is proven that it converges to the exact solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 407–416, 1999     

Mathematical framework for predicting solar thermal build-up of spectrally selective coatings at the Earth’s surface

A transient finite element thermal model is formulated valid for surface coatings on any substrate material and based on the continuum conduction equations with solar loading as a heat source. The model allows cooling to be applied at outer surfaces of the body, by natural convection and accounts for ambient radiative heat loss. Hemispherical spectral reflectivities are obtained for various polymer-based coatings on a steel substrate using spectrophotometers in the 0.1 μm to 25 μm wavelengths. A time-dependent solar irradiation energy source (blackbody equivalent) is applied to an object with spectrally diffuse outer surfaces, and the incoming heat flux is split by a band approximation into reflected and absorbed energy and finally integrated over the complete spectrum to provide thermal source terms for the finite element model.     

Non-linear analysis and calculation of the performance of a shelving protection system by FEM

The aim of this paper consists on the study, analysis and calculation of the efficiency of a shelving protection system by means of the finite element method (FEM). These shelving protection systems are intended to prevent the eventual damage due to the impacts of transport elements in motion, such as: forklifts, dumpers, hand pallet trucks, and so on. The impact loads may threaten the structural integrity of the shelving system. The present structural problem is highly non-linear, due to the simultaneous presence of the following nonlinearities: material non-linearity (plasticity in this case), geometrical non-linearity (large displacements) and contact-type boundary conditions (between the rigid body and the protection system). A total of 48 different FEM models are built varying the thickness of the steel plate (4, 5 and 6 mm), the impact height (0.1, 0.2, 0.3 and 0.4 m) and the impact direction (head-on collision and side impact). Once the models are solved, the stress distribution, the overall displacements and the absorbed impact energy were calculated. In order to determine the best shelving protection’s candidate, some constraints must be taken into account: the maximum allowable stress (235 MPa), the maximum displacement (0.05 m) and the absorbed impact energy (400 J according to the European Standard Rule PREN-15512). Finally, the most important results are shown and conclusions of this study are exposed.     

Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint

In this paper, we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite elements and the control is discretized by piecewise constant functions. We prove the superconvergence error estimate of h² in L²-norm between the approximated solution and the interpolation of the exact control variable. Moreover, by postprocessing technique, we find that the projection of the discrete adjoint state is superclose (in order h²) to the exact control variable.     

Multiscale analysis and numerical algorithm for the Schrödinger equations in heterogeneous media

In solid state physics, the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation (EMA) and its extension the multiband k · p method (see [9]). They have been particularly successful in the case of heterostructures (see, e.g. [4], [9] and [11]). This paper discusses the multiscale analysis of the Schrödinger equation with rapidly oscillating coefficients. The new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions, the definition of boundary layer solutions, and higher-order correctors. Consequently, a multiscale finite element method and some numerical results are presented. As one of the main results of this paper, we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of Γ point, from the viewpoint of mathematics.     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

A study on 3-D stress distributions in the bi-adhesively bonded T-joints

In this study, the normal (σ_x, σ_y, σ_z) and shear stress (τ_xy, τ_yz, τ_zx) distributions occurring in a bi-adhesively bonded T-joint with was investigated via a non-linear three dimensional finite element analysis. For this purpose, first of all, using 2024-T3 aluminum alloy as the adherend and the support, a two-part paste (DP 460) and a film type (SBT 9244) as adhesive, two different types of T-joint samples (single-adhesively bonded T-joint and bi-adhesively bonded T-joint) were produced for experimental studies. After experimental studies on the three different T-joint types were conducted, stress analyses in the T-joints were performed with a three-dimensional finite element analysis by considering the geometrical non-linearity and the material non-linearities of the adhesive (DP460 and SBT9244) and adherend (AA2024-T3). Finally, for a given adherend, the lower the stiffness of the adhesive used in the overlap, the lower the stress concentration, leading to potentially higher joint strength. The use of relatively low stiffness adhesives at the ends of the overlap in a bi-adhesive can decrease the stress concentration and, therefore, potentially lead to higher joint strength.     

流线扩散有限元方法在分层网格上的收敛性分析

本文在分层网格上分析了采用线性元的流线扩散有限元方法求解一维对流扩散型奇异摄动问题的一致收敛性.在ε≤N~(-1)的前提下,可以证明在SD范数下的一致误差估计为O(N~(-1)(log 1/ε)~2)在数值算例部分对理论结果进行了验证.     

Mixed finite element methods for general quadrilateral grids

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Π_h satisfies ∇ · Π_h = P_hdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble-Hilbert lemma for vector variables. We show that a local H(div)-projection reproducing certain polynomials suffices to yield an optimal L²-error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart-Thomas element of lowest order. Numerical experiments are presented to verify our theory.     

Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Solutions of a semilinear elliptic boundary value problem, (with bounded below) can be put into a one-to-one correspondence with zeros of a function . Often d is small. The function is called the bifurcation function. It can also be shown that the eigenvalues of the matrix characterize the stability properties of the solutions of the elliptic problem as rest points of . A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given. This revised version was published online in June 2006 with corrections to the Cover Date.