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.     

An iterative algorithm for elliptic variational inequalities of the second kind

In this paper, we propose an iterative algorithm for a simplified friction problem which is formulated as an elliptic variational inequality of the second kind. We approximate the simplified friction problem by a discrete system with the finite element method. Based on the use of the linearized technique and by constructing a particular function, we put forward the new algorithm to get the discrete solution. This algorithm is attractive due to its simple proof of convergence and easy implementation. A linear equation is solved in each iteration. Numerical results confirm that our algorithm is efficient and mesh independent.     

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.     

A posteriori error analysis of nonconforming finite-element discretization for the Stokes equations

In this paper, we extend the unifying theory for a posteriori error analysis of the nonconforming finite-element methods to the Stokes problems. We present explicit residual-based computable error indicators, we prove its reliability and efficiency based on two assumptions concerning both the weak continuity and the weak orthogonality of the nonconforming finite-element spaces, respectively, and we apply the unified framework to various nonconforming finite elements from the literature.     

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L²- and H¹-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.     

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 Newton-like method for solving a non-smooth elliptic equation

In this paper, we use finite element method to discrete a non-smooth elliptic equation and present some error estimates. Non-smooth Newton-like method is applied to solve the discrete problem. Since Newton's equations have a very bad conditioner when the mesh-size is finer, multigrid technique is used to solve the subproblems. It is shown that if we use V-cycle or cascadic multigrid as an inner iterator, an (nearly) optimal property can be obtained. Numerical results are illustrated to confirm the error estimates we obtained and the efficiency of the non-smooth Newton-like method combining with multigrid technique. Especially, if the mesh-size h becomes much smaller, the method can save substantial computational work.     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

New local discontinuous Galerkin method for a fractional time diffusion wave equation

ABSTRACT

This paper is devoted to the study of a new discontinuous finite element idea for the time fractional diffusion-wave equation defined in bounded domain. The time fractional derivatives are described in the Caputo's sense. By applying the sine transform on the time fractional diffusion-wave equation, we make the equation depend on time. Then we use definition of Caputo's derivative and by defining l-degree discontinuous finite element with interpolated coefficients we solve the mentioned equation. Error estimate, existence and uniqueness are proved. Finally, the theoretical results are tested by some numerical examples.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

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.     

A posteriori error analysis for a fractional differential equation

Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.     

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.     

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.