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.     

On the approximate solution of integro-differential equations arising in oscillating magnetic fields

In this work, we propose the Shannon wavelets approximation for the numerical solution of a class of integro-differential equations which describe the charged particle motion for certain configurations of oscillating magnetic fields. We show that using the Galerkin method and the connection coefficients of the Shannon wavelets, the problem is transformed to an infinite algebraic system, which can be solved by fixing a finite scale of approximation. The error analysis of the method is also investigated. Finally, some numerical experiments are reported to illustrate the accuracy and applicability of the method.     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

Vibration analysis of quasi-symmetric structures

An efficient computational procedure is presented for the free vibration analysis of structures with unsymmetric geometry. The procedure is based on approximating the unsymmetric vibrational response of the structure by a linear combination of a few symmetric and antisymmetric modes (global approximation vectors), each obtained using approximately half the degrees of freedom of the original model. The three key elements of the procedure are: (a) use of mixed finite element models having independent shape functions for the internal forces (stress resultants) and generalized displacements, with the internal forces allowed to be discontinuous at interelement boundaries, (b) operator splitting, or additive decomposition of the different arrays in the governing finite element equations to delineate the contributions to the symmetric and antisymmetric response vectors, and (c) use of a reduction method through successive application of the finite element method and the classical Bubnov-Galerkin technique. The finite element method is first used to generate a few symmetric and antisymmetric global approximation response vectors. Then, the classical Bubnov-Galerkin technique is used to substantially reduce the size of the eigenvalue problem.

An initial set of global approximation vectors is selected to be a few symmetric and antisymmetric eigenvectors, and their various-order derivatives with respect to a tracing parameter identifying all the correction terms to the symmetric (and antisymmetric) eigenvectors. A modified (improved) set of approximation vectors is obtained by using the inverse iteration procedure. The effectiveness of the proposed procedure is demonstrated by means of a numerical example.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

Numerical solutions of backward stochastic differential equations: A finite transposition method

In this Note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.     

Numerical analysis of the stochastic Stokes equations of Wick type

We propose a finite element method for the numerical solution of the stochastic Stokes equations of the Wick type. We give existence and uniqueness results for the continuous problem and its approximation. Optimal error estimates are derived and algorithmic aspects of the method are discussed. Our method will reduce the problem of solving stochastic Stokes equations to solving a set of deterministic ones. Moreover, one can reconstruct particular realizations of the solution directly from Wiener chaos expansions once the coefficients are available. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

An unfitted interface penalty method for the numerical approximation of contrast problems

We aim to approximate contrast problems by means of a numerical scheme which does not require that the computational mesh conforms with the discontinuity between coefficients. We focus on the approximation of diffusion-reaction equations in the framework of finite elements. In order to improve the unsatisfactory behavior of Lagrangian elements for this particular problem, we resort to an enriched approximation space, which involves elements cut by the interface. Firstly, we analyze the H¹-stability of the finite element space with respect to the position of the interface. This analysis, applied to the conditioning of the discrete system of equations, shows that the scheme may be ill posed for some configurations of the interface. Secondly, we propose a stabilization strategy, based on a scaling technique, which restores the standard properties of a Lagrangian finite element space and results to be very easily implemented. We also address the behavior of the scheme with respect to large contrast problems ending up with a choice of Nitsche?s penalty terms such that the extended finite element scheme with penalty is robust for the worst case among small sub-elements and large contrast problems. The theoretical results are finally illustrated by means of numerical experiments.     

Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ${\sqrt{\delta t}}In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ?{dt}{\sqrt{\delta t}}. Finally, some numerical experiments that confirm the theoretical analysis are presented.     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

Well-posedness and finite element approximation of time dependent generalized bioconvective flow

We consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro-organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro-organisms. We show the existence and uniqueness of the weak solution of the system in two dimensions and construct numerical approximations based on the finite element method, for which we obtain error estimates. In addition, we conduct several numerical experiments to demonstrate the accuracy of the numerical method and perform simulations of the bioconvection pattern formations based on realistic model parameters to demonstrate the validity of the proposed numerical algorithm.     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

l)ThisworkwassupportedbyNWOthroughgrantIBo7-3Go12.BOUNDAarv^LUEPRoBLEMFORELLIPTICEQUMIONwiTHMIXEDBOUNDAavCONDITION1.IntroductionInthispedwesketchavarietyofspecialmethodswhichareusedforconstructinge-unifornilyconvergelltschemes-WeshaJldemonstrateamethodwhichachieveshaprovedaccuracyforsolvingsingularlyperturbedb0undaryvalueproblemforeiliPicequatiouswithparabolicboundarylayers-InSecti0n4weshallintroduceanaturalclass,B,oftritefferenceschemes,inwhich(bytheabovementi0nedaP…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.     

Méthode des éléments finis multi-échelles pour le problème de Stokes

This work consists of a numerical study of a multi-scale finite element method for a Stokes-type problem with highly oscillating coefficients. The objective of this method is to capture the multi-scale structure of the solution via local basis functions calculated in advance, which contain the essential information on small scales.