多孔介质驱动问题的混合元最小二乘特征有限元方法及其收敛分析

１．引言多孔介质二相驱动问题的数学模型是偶合的非线性偏微分方程组的初边值问题．该问题可转化为压力方程和浓度方程［１－４］．浓度方程一般是对流占优的对流扩散方程,它的对流速度依赖于比浓度方程的扩散系数大得多的Ｆａｒｃｙ速度．因此Ｄａｒｃｙ速度的求解精度直接影响着浓度的求解精度．为了提高速度的求解精度,７０年代Ｐ．Ａ．Ｒａｖｉａｔ和Ｊ．Ｍ．Ｔｈｏｍａｓ提出混合有限元方法［５］．Ｊ．ＤｏｕｇｌａｓＪｒ,Ｔ．Ｆ．Ｒｕｓｓｅｌｌ,Ｒ．Ｅ．Ｅｗｉｎｇ,Ｍ．Ｆ．Ｗｈｅｅｌｅｒ［１］－［４］,［９］,［１２］袁…     

A formulation of the eddy current problem in the presence of electric ports

The time-harmonic eddy current problem with either voltage or current intensity excitation is considered. We propose and analyze a new finite element approximation of the problem, based on a weak formulation where the main unknowns are the electric field in the conductor, a scalar magnetic potential in the insulator and, for the voltage excitation problem, the current intensity. The finite element approximation uses edge elements for the electric field and nodal elements for the scalar magnetic potential, and an optimal error estimate is proved. Some numerical results illustrating the performance of the method are also presented.     

Numerical analysis of relaxed micromagnetics by penalised finite elements

Summary. Some micromagnetic phenomena in rigid (ferro-)magnetic materials can be modelled by a non-convex minimisation problem. Typically, minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy. Solutions exist in a generalised sense and the observed microstructure can be described in terms of Young measures. A relaxation by convexifying the energy density resolves the essential macroscopic information. The numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations. The lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed. An a priori and a posteriori error analysis is presented for a penalised version of the side-restriction that the modulus of the magnetic field is bounded pointwise. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient. Received June 24, 1999 / Revised version received August 24, 2000 / Published online May 4, 2001     

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.     

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

Summary It is shown that a boundary-value problem based on a holonomic elastic-plastic constitutive law may be formulated equivalently as a variational inequality of the second kind. A regularised form of the problem is analysed, and finite element approximations are considered. It is shown that solutions based on finite element approximation of the regularised problem converge.     

Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems

Summary The equivalence in a Hilbert space of variational and weak formulations of linear elliptic boundary value problems is well known. This same equivalence is proved here for mildly nonlinear problems where the right hand side of the differential equation involves the solution function. A finite element approximation to the solution of the weak problem ina finite dimensional subspace of the original Hilbert space is defined. An inequality bounding the error in this approximation over all functions of the space is derived, and in particular this holds for an interpolant to the weak solution. Thus this inequality, together with previously known, interpolation error bounds, produces a bound on the finite element solution to this nonlinear problem. An example of a mildly nonlinear Poisson problem is given.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

Finite Element Approximation of Eigenvalue Problem for a Coupled Vibration Between Acoustic Field and Plate

1.IntroductionInthispaper,westudyanumericalmethodtocalculateeigen-frequenciesofacoupledvibrationbetweenacousticfieldandplate.Atypicalapplicationofthisresearchistoreduceanoiseinsideacarcausedbyanengineorothersourcesofthesound.OurstudywasmotivatedbytheworkofHagiwaraetal.15].Thebackgroundoftheresearchandsomeapplicationscanbeseenin[5].Werestrictourresearchtotheproblemswhereexactsolutionscanbegiveninaspecialcase.Themainfeatureofourresearchisthemathematicallyrigorousapproachtotheproblem.Weformulat…     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A Multiscale Approach for the Modelling of Failure

In the present contribution a multi–scale – or rather two–scale – framework for the modelling of propagating discontinuities is introduced. The method is based on the Variational Multiscale Method. The displacement field is additively decomposed into a coarse– and a fine–scale part. This kinematic assumption implies a separation of the weak form in two equations, corresponding to the coarse–scale and the fine–scale problem. Both scales are discretized by means of finite elements. On the fine–scale, due to a much finer discretization, a heteregeneous mesostructure and propagating mesocracks can be considered. The propagation of the mesocracks is simulated independently of the underlying finite element mesh by discontinuous elements. The performance of the multi–scale approach is shown by a numerical example. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of nonconforming finite elements to solving advection—diffusion problems

The paper investigates some nonconforming finite elements and nonconforming finite element schemes for solving an advection—diffusion equation. This investigation is aimed at finding new schemes for solving parabolic equations. The study uses a finite element method, variational-difference schemes, and test calculations. Two types of schemes are examined: one is obtained with the help of the Bubnov—Galerkin method from a weak problem determination (nonmonotone scheme), and the other one is a monotone up-stream scheme obtained from an approximate weak problem determination with a special approximation of the skew-symmetric terms.     

A finite volume element formulation and error analysis for the non-stationary conduction–convection problem

The non-stationary conduction–convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element (FVE) method. A fully discrete FVE formulation and the error estimates between the fully discrete FVE solutions and the accuracy solution are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the FVE formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non-stationary conduction–convection problem.     

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.     

An optimal order convergence for a weak formulation of the compressible Stokes system with inflow boundary condition

Summary. We formulate the compressible Stokes system given in (1.1) into a (new) weak formulation (2.1). A finite element method for this is presented. Existence and uniqueness of the finite element method is shown. An optimal error estimate for the numerical approximation is obtained. Numerical examples are given, showing its efficiency and rates of convergence of the approximate solutions that results from the discrete problem (3.1). Received October 20, 1996 / Revised version received January 21, 1999 / Published online: April 20, 2000     

Coupled time-domain analysis with boundary and finite elements

A methodology for the combination of boundary and finite element discretizations for the numerical analysis of time-dependent problems is presented. The interface conditions arising from the partitioning of the problem are incorporated in a weak form by means of Lagrange multiplier fields and, therefore, allow for nonconform interface discretizations. The resulting system matrices have the same saddle point structure as in the FETI method. Possible applications of the proposed method are the dynamic analysis of soil-structure interaction and similar wave propagation phenomena in unbounded media. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Galerkin/least-square finite element formulation for nearly incompressible elasticity/stokes flow

A Galerkin/least-square finite element formulation (GLS) is used to study mixed displacement-pressure formulation of nearly incompressible elasticity. In order to fully incorporate the effect of the residual-based stabilized term to the weak form, the second derivatives of shape functions were also derived and accounted, which can accurately discretize the residual term and improve the GLS method as well as the Petrov–Galerkin method. The numerical studies show that improved stabilized method can effectively remove volumetric locking problem for incompressible elasticity and stabilize the pressure field for stokes flow. When apply GLS to study material nonlinearity, the derivative of tangent modulus at the integration point will be required. Both advantage and disadvantage of using GLS method for nearly incompressible elasticity/stokes flow were demonstrated.     

Error estimates for the approximate identification of a constant coefficient from boundary flux data

A priori error estimates are derived for the simplest finite difference and finite element approximations to an inverse problem in which it is desired to identify an unknown constant coefficient in a differential equation whose general form is known.     

A modified numerical manifold method for simulation of finite deformation problem

With many people contributing to its modifications and advancements, the numerical manifold method (NMM) is now recognized as an efficient tool to solve the continuum–discontinuum coupling problem in geotechnical engineering. However, false solutions have been found when modeling finite deformation problems using the original NMM. Based on the finite deformation theory, a modified version of NMM is derived from the weak form of conservation of momentum and the corresponding traction boundary condition. By taking the dual cover system as the displacement approximation, the governing equations of the modified NMM are formulated. A comparison of the governing equations of the original NMM and modified NMM illustrates the reason that the original NMM is not suitable for simulation of finite deformation problems. Three numerical examples are investigated to verify the capability of proposed method to predict static and dynamic finite deformation response. Numerical results show that the modified NMM eliminates the errors caused by large rotation and large strain, and obtains a good agreement with analytical solutions and the finite element method.     

A finite element convergence analysis for 3D Stokes equations in case of variational crimes

We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.     

A weak Galerkin finite element method for the Oseen equations

In this paper, a weak Galerkin finite element method for the Oseen equations of incompressible fluid flow is proposed and investigated. This method is based on weak gradient and divergence operators which are designed for the finite element discontinuous functions. Moreover, by choosing the usual polynomials of degree i ≥ 1 for the velocity and polynomials of degree i ? 1 for the pressure and enhancing the polynomials of degree i ? 1 on the interface of a finite element partition for the velocity, this new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained by defining a weak convection term. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the Oseen problem.