Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.     

油水驱动半正定问题迎风混合元格式及其数值分析

研究了不可压缩油水两相渗透流驱动问题.在扩散矩阵仅是半正定的假设条件下,提出了迎风混合元方法.混合元方法近似压力方程,饱和度方程的对流项用Godunov迎风格式来处理,扩散项则用推广的混合元来逼进,并推导出格式的误差估计.此种格式的优越性表现在两个方面:首先是饱和度方程的扩散矩阵仅是半正定的;二是摒弃了特征格式所限制的周期性条件,更适用于实际问题.     

Spectral-finite element method for solving three-dimensional unsteady Navier-Stokes equations

This paper is devoted to the combined Fourier spectral and finite element approximations of three-dimensional, semi-periodic, unsteady Navier-Stokes equations. Fourier spectral method and finite element method are employed in the periodic and non-periodic directions respectively. A class of fully discrete schemes are constructed with artificial compression. Strict error estimations are proved. The analysis shows also that the classical two-dimensional velocity-pressure elements can be readily extended to solving such three-dimensional semi-periodic problems, provided they satisfy the two-dimensional “inf-suf” condition.     

The Two-dimensional Electromagnetic Scattering from Periodic Chiral Structures and Its Finite Element Approximation

In this paper, we consider the electromagnetic scattering from periodic chiral structures. The structure is periodic in one direction and invariant in another direction. The electromagnetic fields in the chiral medium are governed by the Maxwell equations together with the Drude-Born-Fedorov equations. We simplify the problem to a two-dimensional scattering problem and we show that for all but possibly a discrete set of wave numbers, there is a unique quasi-periodic weak solution to the diffraction problem. The diffraction problem can be solved by finite element method. We also establish uniform error estimates for the finite element method and the error estimates when the truncation of the nonlocal transparent boundary operators takes place.     

定常不可压Navier—Stokes方程迎风格式

本文利用齐次定解条件对定常不可压Navier—Stokes方程的非线性项进行处理,给出了相应的一种迎风Galerkin有限元算法;针对这种迎风Galerkin有限元算法,在迎风参数满足一定条件下,利用其三项式具有的一些很好性质,更简单地证明了该问题解的存在唯一性。     

Analysis of a domain decomposition method for the nearly elastic wave equations based on mixed finite element methods

This paper concerns the numerical solution of the nearly elasticwave equations with the first-order absorbing boundary condition;these equations describe the motion of a nearly elastic solidin the frequency domain. Two mixed finite elements, the Johnson-Mercierelement and the Arnold-Douglas-Gupta element, are adapted andanalyzed for the problem. The resulting mixed finite elementequations are complex-valued and are neither Hermitian nor definite.As a result, most standard iterative methods fail to convergefor the systems. To solve the mixed finite element equations,a parallelizable domain decomposition iterative method is proposed.The convergence of the method is demonstrated and a rate ofconvergence of the form 1 - Ch is derived. These results arevalid for the case when the original domain is decomposed intosubdomains which consist of an individual element associatedwith the above two mixed finite elements.     

A two‐level finite element method for the stationary Navier‐Stokes equations based on a stabilized local projection

In this article, we propose a two‐level finite element method to analyze the approximate solutions of the stationary Navier‐Stokes equations based on a stabilized local projection. The local projection allows to circumvent the Babuska‐Brezzi condition by using equal‐order finite element pairs. The local projection can be used to stabilize high equal‐order finite element pairs. The proposed method combines the local projection stabilization method and the two‐level method under the assumption of the uniqueness condition. The two‐level method consists of solving a nonlinear equation on the coarse mesh and solving a linear equation on fine mesh. The nonlinear equation is solved by the one‐step Newtonian iteration method. In the rest of this article, we show the error analysis of the lowest equal‐order finite element pair and provide convergence rate of approximate solutions. Furthermore, the numerical illustrations coincide with the theoretical analysis expectations. From the view of computational time, the results show that the two‐level method is effective to solve the stationary Navier‐Stokes equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A finite element method for elliptic problems with rapidly oscillating coefficients

In this paper, we consider solving second-order elliptic problems with rapidly oscillating coefficients. Under the assumption that the oscillating coefficients are periodic, on the basis of classical homogenization theory, we present a finite element method whose key is to combine a numerical approximation of the 1-order approximate solution of those equations and a numerical approximation of the classical boundary corrector of those equations from different meshes exploiting the need for different levels of resolution. Numerical experiments are included to illustrate the competitive behavior of the proposed finite element method.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

不可压缩流问题低次元稳定有限体积数值方法研究

分析了R^d,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈Hd,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈H¹情况下,本文方法得到的解与稳定有限元方法解之间具有O(h1情况下,本文方法得到的解与稳定有限元方法解之间具有O(h²)阶超收敛阶结果,且稳定有限体积方法取得了与稳定有限元方法相同的收敛速度,与稳定有限元方法比较,稳定有限体积方法计算简单高效,同时保持物理守恒,因此在实际应用中具有很好的潜力。     

Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations

A multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth-order nonautonomous differential equations related either to the extended Fisher-Kolmogorov or to the Swift-Hohenberg equation is proved. Variational approach is used. Some numerical solutions are calculated via the finite element method.     

Numerical solution of nonlinear diffraction problems

An adaptive finite element method is developed for solving Maxwell's equations in a nonlinear periodic structure. The medium or computational domain is truncated by a perfect matched layer (PML) technique. Error estimates are established. Numerical examples are provided, which illustrate the efficiency of the method.     

Stabilized finite element approximations for the Reissner–Mindlin plate

Based on the Helmholtz decomposition of the transverse shear strain, Brezzi and Fortin in [5] introduced a three-stage algorithm for approximating the Reissner–Mindlin plate model with clamped boundary conditions and established uniform error estimates in the plate thickness. The first- and third-stage involve approximating two simple Poisson equations and the second-stage approximating a perturbed Stokes equation. Instead of using the mixed finite element method which is subject to the inf–sup condition, we consider a stabilized finite element approximation to such perturbed Stokes equations. Optimal error estimates independent of thickness of the plate are obtained for such equations. Then error analysis is established for the whole system.     

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.     

Asymptotic-numerical derivation of the Robin type coupling conditions for the macroscopic pressure at a reservoir–capillaries interface

In this article the Stokes equations are considered in a domain simulating a capillary bed system. The capillaries are supposed to be thin, parallel and periodic. An asymptotic approximation is constructed. The macroscopic pressure satisfies a Robin interface condition whose coefficients are calculated numerically through a finite element approximation of a boundary layer problem, which is inspired to a domain decomposition technique.     

The primal mixed finite element method and the LBB condition

Spurious or kinematic modes have posed a major obstacle to the implementation of the mixed finite element method. This research shows that spurious modes resulting from the approximation spaces not satisfying the LBB condition do not prevent a well posed problem. When the LBB condition is not satisfied, the resulting matrix equations are singular. A direct solution method is presented for the efficient solution of the possibly singular equations. Orthogonal flux basis functions are introduced to simplify the problem. Then the solution procedure is based on nested domain decomposition. This solution procedure is shown to be competitive with direct solution methods for the displacement finite element method. Examples are included to demonstrate various aspects of the LBB condition and the solution procedure.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

A Positivity-preserving Finite Element Method for Quantum Drift-diffusion Model

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.     

Unstructured finite volume method for matrix free explicit solution of stress-strain fields in two dimensional problems with curved boundaries in equilibrium condition

In this paper, a plane stress structural solver which uses a matrix free unstructured finite volume method based on Galerkin approach is introduced for solution of weak form of two dimensional Cauchy equations on linear triangular element meshes. The developed shape function free Galerkin finite volume structural solver explicitly computes stresses and displacements in cartesian coordinate directions for the two dimensional solid mechanic problems in equilibrium condition. The accuracy of the introduced algorithm is assessed by comparison of computed results of two plane-stress cases with curved boundaries under uniformly distributed loads with available analytical solutions. The results of the introduced method are presented in terms of stress and strain contours and its effective parameters on convergence behaviour to equilibrium condition are assessed.     

The numerical solution of a periodic parabolic problem subject to a nonlinear boundary condition

The numerical solution by finite differences of a periodic parabolic problem subject to a nonlinear boundary condition is considered. It is shown that Newton's method can be used to solve the nonlinear equations provided a suitable initial approximation is known, and a method for constructing this first approximation is given.