二阶椭圆问题的各向异性非协调Crouzeix-Raviart型有限元方法

对满足最大角条件和坐标系条件的二维区域中的各向异性一般三角形网格,研究了二阶椭圆问题的非协调Crouzeix-Raviart型线性三角形有限元逼近,得到了最优的能量模和L2-模误差估计结果．     

An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.     

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.     

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Boundary integral equations provide a powerful tool for the solution of scattering problems. However, often a singular kernel arises, in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision, thus special treatment is needed to handle the singular behavior. Especially, for inhomogeneous media, it is difficult if not impossible to find out an analytical expression for Green’s function. In this paper, an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media. This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient (FFT-PCG) solver. A remarkable point of this method is that there is no need to know analytical expressions for Green’s function. Numerical experiments are provided to demonstrate the advantage of the current approach, including its simplicity in implementation, its high accuracy and efficiency.     

Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.     

Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data

This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.     

A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

三维弹塑性有摩擦接触问题求解的一个新算法

空间弹塑性有摩擦问题是两种非线性相互耦合的边值待定问题,由于涉及到空间接触问题,切向滑动状态的确定就成为一个重要的方面,为获得高精度的结果,大量的计算工作是难免的.规划法与迭代法是处理这类问题的两种重要的方法.作者提出了将这两种方法相结合的规划-迭代算法,充分利用两种方法各自的长处来处理空间弹塑性接触问题.数值结果表明了所提出算法的良好性态.     

AN H-BASED A- φMETHOD WITH A NONMATCHING GRID FOR EDDY CURRENT PROBLEM WITH DISCONTINUOUS COEFFICIENTS

In this paper, we investigate the finite element A-φ method to approximate the eddy current equations with discontinuous coefficients in general three-dimensional Lipschitz polyhedral eddy current region. Nonmatching finite element meshes on the interface are considered and optimal error estimates are obtained.     

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method.     

基于三维弹性问题的多极边界元法核函数分解

多极边界元法已经成功地应用于大规模工程计算中.得到并且证明了基于三维弹性问题的多极边界元法核函数分解的定理(定理1),完善了多击边界元法的数学理论.     

三维泊松方程基于Richardson外推法的高阶紧致差分方法

基于Richardson外推法提出了数值求解三维泊松方程的高阶紧致差分方法.方法通过利用四阶和六阶紧致差分格式,分别在细网格和粗网格上求解,然后利用Richardson外推技术和算子插值方法,得到三维泊松方程在细网格上的六阶和八阶精度的数值解.数值实验结果验证了该方法的精确性和有效性.     

H1-superconvergence of finite difference method based on Q1-element on quasi-uniform mesh for the 3D Poisson equation

In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q₁-element on a quasi-uniform mesh. First, under the regularity assumption of , the H¹-superconvergence of the FD solution u_h based on the Q₁-element to the first-order interpolation function is obtained. Next, the H¹-superconvergence of the second-order interpolation postprocessing function based on the FD solution u_h to u is provided. Finally, numerical tests are presented to show the H¹-superconvergence result of the FD postprocessing function to u if .