半线性抛物方程的时空有限元方法

讨论半线性抛物方程的连续Galerkin时空有限元方法,利用有限元方法和有限差分方法相结合的技巧,证明了弱解的存在唯一性,给出了时间最大模,空间L~2模,即L~∞(L~2)模误差估计.并给出数值算例证明了连续时空有限元方法对于半线性抛物方程的有效性.     

On the convergence of boundary element methods for initial-Neumann problems for the heat equation

In this paper we study boundary element methods for initial-Neumann problems for the heat equation. Error estimates for some fully discrete methods are established. Numerical examples are presented.

     

热传导型半导体瞬态问题特征变网格有限元法及其分析

热传导型半导体器件的瞬时状态由四个方程的非线性偏微分方程组的初边值问题所决定,其中电子位势方程是椭圆型的,电子和空穴浓度方程是对流扩散型的,温度方程为热传导型的。本文提出解这类问题的特征变网格有限元法,并进行了理论分析,在一定条件下,得到了某种意义下的最佳L^2误差估计结果。     

Continuous Galerkin finite element methods for a forward‐backward heat equation

A space‐time finite element method is introduced to solve a model forward‐backward heat equation. The scheme uses the continuous Galerkin method for the time discretization. An error analysis for the method is presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 257–265, 1999     

Error estimates of H^1-Galerkin mixed finite element method for Schrodinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

A finite element method and variable transformations for a forward-backward heat equation

The Galerkin finite element method for the forward-backward heat equation is generalized to a wider class of equations with the use of a result on the existence and uniqueness of a weak solution to the problems. First, the theory for the Galerkin method is extended to forward-backward heat equations which contain additional convection and mass terms on an irregular domain. Second, variable transformations are constructed and applied to solve a wide class of forward-backward heat equations that leads to a substantial improvement. Third, Error estimates are presented. Finally, conducted numerical tests corroborate the obtained results. Received February 4, 1997 / Revised version received December 8, 1997     

NONCONFORMING FINITE ELEMENT PENALTY METHOD FOR STOKES EQUATION

a special penalty method is presented to improve the accuracy of the standard penaltymethod (or solving Stokes equation with nonconforming finite element, It is shown that thismethod with a larger penalty parameter can achieve the same accuracy as the staodaxd methodwith a smaller penalty parameter. The convergence rate of the standard method is just hall order of this penalty method when using the same penalty parameter, while the extrapolationmethod proposed by Faik et al can not yield so high accuracy of convergence. At last, we alsoget the super-convergence estimates for total flux.     

A reduced-order extrapolated approach to solution coefficient vectors in the Crank-Nicolson finite element method for the uniform transmission line equation

This work is concerned mainly with developing and testing the reduced-order extrapolated approach to the unknown coefficient vectors in the Crank-Nicolson finite element (CNFE) solutions for the uniform transmission line equation. For this objective, the CNFE functional and matrix models and the existence, stability, and errors of the CNFE solutions of the uniform transmission line equation are first derived. Then a reduced-order extrapolated CNFE (ROECNFE) matrix model is established by means of a proper orthogonal decomposition technique, and the existence, stability, and error estimates of the ROECNFE solutions are demonstrated by matrix analysis, leading to an elegant theoretical development. Especially, our work shows that the basis functions and accuracy of the ROECNFE matrix model are the same as those of the CNFE matrix model. Finally, some numerical tests are illustrated to computationally experimentally confirm the validity and sharpness of the ROECNFE method.     

多孔介质中可压缩可混溶驱动问题的有限体积元法

有界区域上多孔介质中可压缩可混溶驱动问题由两个非线性抛物型方程耦合而成：压力方程和饱和度方程均是抛物型方程．运用有限体积元法对两个方程进行数值分析,给出了全离散有限体积元格式,并通过详细的理论分析,得到了近似解与原问题真解的最优H^1模误差估计。     

非结构网格下2D Riesz分数阶方程的Galerkin有限元方法

讨论了2D Riesz分数阶扩散方程的Galerkin有限元方法.基于非结构网格,采用Lagrange线性分片多项式作为基函数,详细描述了分数阶扩散方程的有限元实现.与现有方法相比,该方法有效地降低了计算成本,提高了刚度矩阵的精度.最后,数值算例验证了所提方法的有效性.     

Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters and are sufficiently small.

     

A new a priori error estimate of nonconforming finite element methods

This paper is devoted to a new error analysis of nonconforming finite element methods.Compared with the classic error analysis in literature,only weak continuity,the F-E-M-Test for nonconforming finite element spaces,and basic Hm regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed.The analysis is motivated by ideas from a posteriori error estimates and projection average operators.One main ingredient is a novel decomposition for some key average terms on(n.1)-dimensional faces by introducing a piecewise constant projection,which defines the generalization to more general nonconforming finite elements of the results in literature.The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.     

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L∞(0,T;L2) for concentration c,in L2(0,T;L2)for cxand L∞(0,T;L2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method,the nonlinearity,and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally,we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.     

Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation

The Ciarlet–Raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi-harmonic equation. The optimality conditions consisting of the state and the co-state equations is derived. Also, the a priori error estimates are analyzed. In the analysis of the a priori error estimates, the improved convergent rate of the higher order than existed results is proved. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate.     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.     

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.     

半线性抛物型积分微分方程的间断时空有限元方法

本文讨论了一类半线性抛物型积分微分方程的间断时空有限元方法．利用有限元和有限差分方法相结合的技巧,在时间离散区间内,利用Radau点处Lagrange插值多项式的特性,去掉间断时空有限元的传统证明过程中对时空网格的限制条件,并给出了时间最大模、空间L_2模,即L_∞(L_2)模的误差估计．     

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.