热敏电阻问题的有限差分解法及其收敛性分析

The thermistor problem is an initial-boundary value problem of coupled nonlinear differential equations.The nonlinear PDEs consist of a heat equation with the Joule heating as a source and a current conservation equation with temperature-dependent electrical conductivity.This problem has important applications in industry.In this paper,A new finite difference scheme is proposed on nonuniform rectangular partition for the thermistor problem.In the theoretical analyses,the second-order error estimates are obtained for electrical potential in discrete L2 and H1 norms,and for the temperature in L2 norm.In order to get these second-order error estimates,the Joule heating source is used in a changed equivalent form.     

Numerical Analysis of Nonstationary Thermistor Problem

1.AMathematicalModelandaDiscreteSchemeThemodelofanonstationaxythermistorproblemisderivedfromtheconservationlawsofcurrentandenergy(see[l][2]l3]):Findapair{W,u}suchthatV-(a(u)VW)=oinQT=flX(O,T),(1.l)W=Woonoflx(O,T),(1.2)ut-bu=a(U)IVWl'inQT,(1'3)u=oonoflx(O,T),(1.5)u(x,o)=uo(x)infl(1.5)whereflCR"(N21)isaboundeddomain,occupiedbythethermistor;W=yt(x,t),u=u(x,t)aredistributionsoftheelectricalpotentialandthetemperatureinfl,respectively;J(u)isthetemperaturedependentelectricalconductivity;a…     

二阶常微初值问题的时间间断最小二乘有限元法

采用时间间断最小二乘线性有限元方法求解二阶常微分方程初值问题.利用回收技巧及离散Gronwall引理证明了方法的稳定性.通过引入有限元空间上的范数,给出了方法在该范数意义下丰满的误差估计.数值实验验证了理论分析结果.     

An FEM approximation for a fourth-order variational inequality of second kind

A fourth-order variational inequality of the second kind arising in a plate frictional bending problem is considered. By using regularization method, the original problem can be formulated as a differentiable variational equation, and the corresponding discrete FEM variational equation is presented afterwards. Abstract error estimates and error estimates of the approximation are derived in terms of energy norm and L^2-norm.     

Approximation of Derivative for a Singularly Perturbed Second-Order ODE of Robin Type with Discontinuous Convection Coefficient and Source Term

In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.AMS subject classifications: 65L10, CR G1.7     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

河床流体模型方程的稳定化有限差分法及非线性稳定性分析

1 引言本文考虑如下问题: μ(x+2π,t)=μ(x,t), x∈R,t∈[0,τ], (1.2) μ(x,0) =μ_0(x) β,ε,σ∈R,ε,σ>0. (1.3) 该模型描述河床流体流动,其中μ(x,t)为实值函数,它代表河床流体中微粒沉淀(concen—tration)在空间方向上的周期小扰动。G.H.Ganser和D.A.Drew用摄动法对该问题进行了分析,认为该问题是非线性不稳定的。 数值研究表明,对该问题,采用通常的差分方法和Galerkin有限元是不稳定的。文     

NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

We consider numerical methods to solve the Allen-Cahn equation using the second-order Crank-Nicolson scheme in time and the second-order central difference approach in space.The existence of the finite difference solution is proved with the help of Browder fixed point theorem.The difference scheme is showed to be unconditionally convergent in L∞ norm by constructing an auxiliary Lipschitz continuous function.Based on this result,it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size.The numerical experiments also verify the reliability of the method.     

Determination of an unknown function in a parabolic inverse problem by Sinc‐collocation method

In this article, an inverse problem of determining an unknown time‐dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution‐type. By using Sinc‐collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1584–1598, 2010     

Existence for the thermoelastic thermistor problem

We study a new model for the thermistor problem that consists of a system of dynamic thermoelasticity equations of displacement and a stationary charge conservation equation of electrical current. The heat source generated by Joule heating is quadratic in the gradient of the electrical potential. This system is nonlinear and degenerate since the electrical conductivity is assumed to be temperature dependent and vanishes at large temperatures. We establish the existence of a very weak solution, the so-called “capacity solution” for the model. The proof is based on regularization and time retarding.     

Modified High-order Upwind Method for Convection Diffusion Equation

Abstract In this paper, we study the high-order upwind finite difference method for steady convection-diffusionproblems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference schemeon nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is inconversation form, satisfies maximum value principle and has second-order error estimates in discrete H~1 norm.To illustrate our conclusion, several numerical examples are given.     

New locally conservative finite element methods on a rectangular mesh

A new family of locally conservative, finite element methods for a rectangular mesh is introduced to solve second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations. Next, this approach is exploited to approximate Stokes equations. Numerical results are presented for various examples including the lid driven-cavity problem.     

Large time step maximum norm regularity of L-stable difference methods for parabolic equations

We establish almost sharp maximum norm regularity properties with large time steps for L-stable finite difference methods for linear second-order parabolic equations with spatially variable coefficients. The regularity properties for first and second spatial differences of the numerical solution mimic those of the continuous problem, with logarithmic factors in second differences. The regularity results for the inhomogeneous problem imply that the uniform rate of convergence of the numerical solution and its differences is controlled only by the maximum norm of the local truncation error.     

椭圆型问题一类广义差分法的L~2模误差估计

1.引 言 广义差分法作为处理偏微分方程的离散技术,能够保持质量,动量,能量等物理量的守恒.广义差分法（有些文献称为box method[3];finite volume element method[4],[5],[6]）利用在对偶剖分体积单元积分原始方程,并将近似解限制于某一有限元空间而得到离散方程.因此,它在局部区域保持了原始方程的物理守恒性和其他重要特性.从而被广泛地应用于数值求解数学物理方程,特别是计算流体力学和热传导问题[11]. 对广义差分法的研究已有许多文献,专著[10]有详细的介绍.早期的工作主要考虑标准的重心对偶剖分.近年来Cai et,al[4],[5],[6],在某些假定下对较一般的对偶剖分给出了能量模误差估计,Huang and Xi[9]去掉了文献[6]中的这些限制.Chou,Li[8]和Li,     

On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second-order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.     

Difference schemes for the dirichlet problem for an elliptical equation with discontinuous coefficients in an arbitrary region on a regular grid

Exact difference scheme operators are used to construct a difference scheme for a second-order elliptical equation with discontinuous coefficients. The solution of the scheme converges to the solution of the original problem at a rate O(h^1/2) in the grid norm W₂ ¹().Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 3–7, 1985     

L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes

This paper is concerned with the finite volume element methods on quadrilateral mesh for second-order elliptic equation with variable coefficients. An error estimate in L ² norm is shown on the quadrilateral meshes consisting of h ²-parallelograms. Superconvergence of numerical solution is also derived in an average gradient norm on h ²-uniform quadrilateral meshes. Numerical examples confirm our theoretical conclusions.     

对流占优扩散方程的特征有限体积元方法

考虑对流占优扩散方程初边值问题的特征有限体积元方法,并给出特征有限体积元解的误差分析.理论分析表明特征有限体积元解具有最优阶L~2和H~1模误差估计.数值算例说明此方法是有效的.     

A posteriori error estimates for optimal distributed control governed by the evolution equations

We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the L²-norm in the space-time cylinder combined with a special time weighted energy norm.     

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

<正>This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes,one is analogous to Douglas finite difference scheme with second-order splitting error,the other two schemes have third-order splitting error,and the last one is an extended LOD scheme.The L~2 norm and H~1 semi-norm error estimates are obtained for the first scheme and second one,respectively.Finally,two numerical examples are provided to illustrate the efficiency and accuracy of the methods.