ON THE COUPLING OF BOUNDARY INTEGRAL AND FINITE ELEMENT METHODS FOR SIGNORINI PROBLEMS

1.IntroductionPartialdifferentialequationssubjecttounilateralboundaryconditionsareusuallycalledSignoriniproblemsintheliterature.TheseproblemshavebeenstudiedbymanyauthodssincetheappearenceofthehistoricalpaperbyA.Signoriniin1933[25].Signoriniproblemsaroseinmanyareasofapplicationse.g.,theelasticitywithunilateralconditions[lo],thefluidmechnicsproblemsinmediawithsemipermeableboundaries[8,12],theelectropaintprocess[1]etc.Fortheexistence,uniquenessandregularityresultsforSignorinitypeproblemswerefer…     

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of convection-diffusion type of third order Ordinary Differential Equations (ODEs) in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. In order to get a numerical solution for the derivative of the solution, the domain is divided into two regions namely inner region and outer region. The shooting method is applied to the inner region while standard finite difference scheme (FD) is applied for the outer region. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing.     

Numerical treatment of singularly perturbed two point boundary value problems

We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

Numerical solution of singular perturbation problems via deviating arguments

We present a new approach to numerically solving linear, singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in the implicit form is derived. Then, the outer region problem is solved as a two point boundary value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, a new inner region problem is obtained and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Some numerical examples have been solved to demonstrate the applicability of the method.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Numerical simulation and convergence analysis for a system of nonlinear singularly perturbed differential equations arising in population dynamics

In this article, we consider a system of nonlinear singularly perturbed differential equations with two different parameters. To solve this system, we develop a weighted monotone hybrid scheme on a nonuniform mesh. The proposed scheme is a combination of the midpoint scheme and the upwind scheme involving the weight parameters. The weight parameters enable the method to switch automatically from the midpoint scheme to the upwind scheme as the nodal points start moving from the inner region to the outer region. The nonuniform mesh in particular the adaptive grid is constructed using the idea of equidistributing a positive monitor function involving the solution gradient. The method is shown to be second order convergent with respect to the small parameters. Numerical experiments are presented to show the robustness of the proposed scheme and indicate that the estimate is optimal.     

Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media

We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell's equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order formulation for the electric field with an ordinary differential equation for the electric polarization added as an auxiliary constraint. The finite element method uses linear finite elements in space for the electric field as well as the electric polarization, and a theta scheme for the time discretization. Numerical experiments suggest the method is unconditionally stable for both Debye and Lorentz models. We compare the stability and phase error properties of the method presented here with those of finite difference methods that have been analyzed in the literature. We also conduct numerical simulations that verify the stability and dispersion properties of the scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stokes型积分-微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法

在半离散格式下.研究了Stokes型积分一微分方程的Crouzeix-Raviart型非协调三角形各向异性有限元方法,在不需要传统Ritz-Volterra投影下,通过辅助空间等新的技巧得到了与传统有限元方法相同的误差估计.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

四阶抛物偏微分方程的H1-Galerkin混合元方法及数值模拟

到目前为止, H¹-Galerkin 混合有限元方法研究的问题仅局限于二阶发展方程. 然而对于高阶发展方程, 特别是重要的四阶发展方程问题的研究却没有出现. 本文首次提出四阶发展方程的H¹-Galerkin 混合有限元方法, 为了给出理论分析的需要, 我们考虑四阶抛物型发展方程. 通过引进三个适当的中间辅助变量, 形成四个一阶方程组成的方程组系统, 提出四阶抛物型方程的H¹-Galerkin 混合有限元方法. 得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计, 并采用迭代方法证明了全离散格式的稳定性. 最后, 通过数值例子验证了提出算法的可行性. 在一维情况下我们能够同时得到未知纯量函数、一阶导数、负二阶导数和负三阶导数的最优逼近解, 这一点是以往混合元方法所不能得到的.     

A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems

This paper deals with the study on system of reaction diffusion differential equations for Robin or mixed type boundary value problems (MBVPs). A cubic spline approximation has been used to obtain the difference scheme for the system of MBVPs, on a piecewise uniform Shishkin mesh defined in the whole domain. It has been shown that our proposed scheme, i.e., central difference approximation for outer region with cubic spline approximation for inner region of boundary layers, leads to almost second order parameter uniform convergence whereas the standard method i.e., the forward-backward approximation for mixed boundary conditions with central difference approximation inside the domain leads to almost first order convergence on Shishkin mesh. Numerical results are provided to show the efficiency and accuracy of these methods.     

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

In this study, both the dual reciprocity boundary element method and the differential quadrature method are used to discretize spatially, initial and boundary value problems defined by single and system of nonlinear reaction–diffusion equations. The aim is to compare boundary only and a domain discretization method in terms of accuracy of solutions and computational cost. As the time integration scheme, the finite element method is used achieving solution in terms of time block with considerably large time steps. The comparison between the dual reciprocity boundary element method and the differential quadrature method solutions are made on some test problems. The results show that both methods achieve almost the same accuracy when they are combined with finite element method time discretization. However, as a method providing very good accuracy with considerably small number of grid points differential quadrature method is preferrable.     

A coupling of FEM-BEM for a kind of Signorini contact problem

In this paper, we consider a kind of coupled nonlinear problem with Signorini contact conditions. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive an asymptotic error estimate of the approximation to the coupled FEM-BEM variational inequality. Then we design an iterative method for solving the coupled system, in which only three standard subproblems without involving any boundary integral equation are solved. It will be shown that the convergence speed of this iteration method is independent of the mesh size.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004     

On a Class of Coupled Variational Formulations for a Nonlinear Parabolic Equation

51.Introducti0nSince198O)stheoriesandapplicationsofboundaryelementmethods(BEM)orboundaryintegralmethods(BIM)havemadegreatsuccessesfortheparaboliclnit1alboundaryvalueproblems(seeL1-12j),andtheapproachhasbeenappliedtonumericalsolutionsofinitialboundaryva1ueproblemssuccessfully(seeL1-5j'L8j).Thepropertiesofboundaryelementoperatorshavebeenstudiedbyboundaryintegralmethodsbymanyauthors(see.[4j,L6J'[7j'L12J).Theseresultsprovideabasisforconvergencesanderrorestimatesfornumericalapproximationofbou…     

Hessian型方程Neumann边值问题的梯度估计

通过构造辅助函数,利用基本对称函数的性质以及函数在极大值点的性质,得到Hessian型方程S_k(D~2u-A(x,u,Du))=B(x,u)的梯度内估计,构造不同的辅助函数,分近边、边界和内部3种情形讨论该方程Neumann边值问题,进而得到全局梯度估计.