Dimension splitting algorithm for a three-dimensional elliptic equation

This paper presents a finite-element dimension splitting algorithm (DSA) for a three-dimensional (3D) elliptic equation in a cubic domain. The main idea of DSA is that a 3D elliptic equation can be transformed into a series of two-dimensional (2D) elliptic equations in the X–Y plane along the Z-direction. The convergence speed of the DSA for a 3D elliptic equation depends mainly on the mesh scale of the Z-direction. P ₂ finite-element discretization in the Z-direction for DSA is adopted to accelerate the convergence speed of DSA. The error estimates are given for DSA applying P ₁ or P ₂ finite-element discretization in the Z-direction. Finally, some numerical examples are presented. We apply the parallel solving technology to our numerical examples and obtain good parallel efficiency. These numerical experiments test and verify theoretical results.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

Adaptive Finite Element Methods for Microstructures? Numerical Experiments for a 2-Well Benchmark

Macroscopic simulations of non-convex minimisation problems with enforced microstructures encounter oscillations on finest length scales – too fine to be fully resolved. The numerical analysis must rely on an essentially equivalent relaxed mathematical model. The paper addresses a prototype example, the scalar 2-well minimisation problem and its convexification and introduces a benchmark problem with a known (generalised) solution. For this benchmark, the stress error is studied empirically to asses the performance of adaptive finite element methods for the relaxed and the original minimisation problem. Despite the theoretical reliability-efficiency gap for the relaxed problem, numerical evidence supports that adaptive mesh-refining algorithms generate efficient triangulations and improve the experimental convergence rates optimally. Moreover, the averaging error estimators perform surprisingly accurate.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

发展型对流占优扩散方程的FD-SD法的后验误差估计及空间网格调节技术

０．引言 流线扩散法（ｓｔｒｅａｍｌｉｎｅ　ｄｉｆｆｕｓｉｏｎ ｍｅｔｈｏｄ,简称 ＳＤ法）是由Ｈｕｇｈｅｓ和 Ｂｒｏｏｋｓ在１９８０年前后提出的一种数值求解对流占优扩散问题的新型有限元算法．随后,Ｊｏｈｎｓｏｎ和 Ｎａｖｅｒｔ把ＳＤ法推广到发展型对流扩散问题．这一方法因其兼具良好的数值稳定性和高阶精度,近年来在理论与实践方面都得到了很大发展． 对于发展型对流扩散问题的ＳＤ法均采用时空有限元,即把时间、空间同等对待,这样做虽然使关于时间、空间的精度很好地统一起来,但与传统的有限元相比,由于维数增加,计…     

A family of third-order parallel splitting methods for parabolic partial differential equations

A family of numerical methods, based upon a new rational approximation to the matrix exponential function, is developed for solving parabolic partial differential equations. These methods are L-acceptable, third-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by third-order finite-difference approximations.- Parallel algorithms are developed and tested on the one-dimensional heat equation, with constant coefficients, subject to homogeneous boundary conditions and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions.     

High-order time splitting for the Stokes equations

A pseudo-spectral (or collocation) approximation of the unsteady Stokes equations is presented. Using the Uzawa algorithm the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the Pseudo-Laplacian for the pressure. In order to avoid spurious modes the pressure is approximated with lower order (two degrees lower) polynomials than the velocity. Only one grid (no staggered grids) with the standard Chebyshev Gauss-Lobatto nodes is used. Here we further compare our treatment with a Neumann boundary value problem for the pressure. The highly improved accuracy of our method becomes obvious. In the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure. It is numerically shown that a stable third order method in time can be achieved.     

Simple a posteriori error estimators for the <Emphasis Type="Italic">h</Emphasis>-version of the boundary element method

The h-h/2-strategy is one well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. One considers to estimate the error , where is a Galerkin solution with respect to a mesh and is a Galerkin solution with respect to the mesh obtained from a uniform refinement of . This error estimator is always efficient and observed to be also reliable in practice. However, for boundary element methods, the energy norm is non-local and thus the error estimator η does not provide information for a local mesh-refinement. We consider Symm’s integral equation of the first kind, where the energy space is the negative-order Sobolev space . Recent localization techniques allow to replace the energy norm in this case by some weighted L ²-norm. Then, this very basic error estimation strategy is also applicable to steer an h-adaptive algorithm. Numerical experiments in 2D and 3D show that the proposed method works well in practice. A short conclusion is concerned with other integral equations, e.g., the hypersingular case with energy space and , respectively, or a transmission problem. Dedicated to Professor Ernst P. Stephan on the occasion of his 60th birthday.     

Interval arithmetic error estimation for the solution of Fredholm integral equation

Finite element method with a posteriori estimation using interval arithmetic is discussed for a Fredholm integral equation of the second kind. This approach is general. It leads to the guaranteed L ^∞ asymptotically exact estimate without the usual overestimation when interval arithmetic is used. An algorithm is provided for determination of an approximate solution such that the computed error bound between the exact solution and its approximation in L ^∞ is less than the given tolerance ?. Numerical solution for the equation with only C ¹ kernel illustrates the approach.     

快速椭圆曲线签名验证算法

从签名方程出发,推导出一组椭圆曲线签名验证算法,这些算法在验证步骤中,采用护士Q的形式,只运用了一次标题乘运算,而目前国际上公认的椭圆曲线签名算法为ECDSA算法,也还有一些相关的签名算法,这些算法在验证步骤中都采用kP±lQ的形式,需要计算两次标题乘运算.因为关键运算为标题乘运算,其它运算时间可忽略不计,所以给出算法的验证步骤的运算时间仅为ECDSA算法所需时间的一半,接着还通过实验进行了验证.     

A robust a posteriori error estimate for the Fortin-Soulie finite-element method

The subject of a posteriori error estimation is widely studied, and a variety of such error estimates have been used for elasticity problems in recent years. Of particular interest is the work carried out in ^{1. and 2.}. In this paper, we derive a new a posteriori error estimator for the quadratic nonconforming Fortin-Soulie element for the error in an energy-like norm. Then, we illustrate the new error bound by presenting some numerical examples, and show an example of a sequence of adaptively refined meshes.     

A new low-order non-conforming mixed finite-element scheme for second-order elliptic problems

The main aim of this paper is to develop a new low-order non-conforming mixed finite-element method to solve the second-order elliptic problems on rectangular meshes. The convergence analysis is presented and the optimal error estimates are derived with the lowest regularity of the exact solution. Numerical results which coincide with our theoretical analysis show that this element indeed has very good convergence behaviour.     

Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

A multiscale mortar mixed finite element method is established to approximate non-linear second order elliptic equations. The method is based on non-overlapping domain decomposition and mortar finite element methods. The existence and uniqueness of the approximation are demonstrated, and a priori $L^2$-error estimates for the velocity and pressure are derived. An error bound for mortar pressure is proved. Convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete-pressure dependent coefficient. Optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation. The quadratic convergence of the Newton–Raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem. Numerical experiments are performed to support theoretic results.     

Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations

Poor performances can be obtained from classical domain decomposition algorithms to solve advection-diffusion equations in the case of convection dominated flows. Therefore, adaptive domain decomposition have been developed for such flows. We investigate the properties of some algorithms of this kind in the framework of a finite volume/finite element discretization.This research was carried out while the author was visiting the Group of Applied Mathematics and Simulation of CRS4, and was supported by an HCM fellowship.     

求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.