无单元Galerkin法在热传导中的应用

对热传导问题的微分方程采用无单元Galerkin法进行数值求解.首先,将微分方程用Galerkin加权残量法转化为等效的积分形式.然后,先将时间变量看作参数,对空间变量进行离散化,得到方程的半离散形式,接着,对时间采用向后Euler—Galerkin格式进行离散,得到方程的全离散形式最后,编制MATLAB程序,上机计算.列举了两个热传导算例,通过计算说明EFG法适用于热传导问题,且其计算速度快,精确度高、前后处理也十分方便,是一种具有潜力的温度场数值计算的新方法.     

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.     

Error estimates for distributed parameter identification in parabolic problems with output least squares and Crank-Nicolson method

The identification problem of a functional coefficient in a parabolic equation is considered. For this purpose an output least squares method is introduced, and estimates of the rate of convergence for the Crank-Nicolson time discretization scheme are proved, the equation being approximated with the finite element Galerkin method with respect to space variables.     

On the Rate of Convergence of Projection-Difference Methods for Smoothly Solvable Parabolic Equations

A linear parabolic problem in a separable Hilbert space is solved approximately by the projection-difference method. The problem is discretized in space by the Galerkin method orientated towards finite-dimensional subspaces of finite-element type and in time by using the implicit Euler method and the modified Crank-Nicolson scheme. We establish uniform (with respect to the time grid) and mean-square (in space) error estimates for the approximate solutions. These estimates characterize the rate of convergence of errors to zero with respect to both the time and space variables.     

A Sparse Grid Stochastic Collocation Discontinuous Galerkin Method for Constrained Optimal Control Problem Governed by Random Convection Dominated Diffusion Equations

A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients. By the optimal control theory, an optimality system is obtained for the problem, which consists of a state equation, a co-state equation and an inequality. Based on finite dimensional noise assumption of random field, the random coefficients are assumed to have finite term expansions depending on a finite number of mutually independent random variables in the probability space. An approximation scheme is established by using a discontinuous Galerkin method for the physical space and a sparse grid stochastic collocation method based on the Smolyak construction for the probability space, which leads to the solution of uncoupled deterministic problems. A priori error estimates are derived for the state, co-state and control variables. Numerical experiments are presented to illustrate the theoretical results.     

Stabilization method for continuous approximation of transient convection problem

The aim of this work is to study a new finite element (FE) formulation for the approximation of nonsteady convection equation. Our approximation scheme is based on the Streamline Upwind Petrov Galerkin (SUPG) method for space variable, x, and a modified of the Euler implicit method for time variable, t. The most interest for this scheme lies in its application to resolve by continuous (FE) method the complex of viscoelastic fluid flow obeying an Oldroyd‐B differential model; this constituted our aim motivation and allows us to treat the constitutive law equation, which expresses the relation between the stress tensor and the velocity gradient and includes tensorial transport term. To make the analysis of the method more clear, we first study, in this article this modified method for the advection equation. We point out the stability of this new method and the error estimate of the approximation solution is discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A Mixed Problem with Integral Condition for the Hyperbolic Equation

In this paper, we study a mixed problem for the hyperbolic equation with a boundary Neumann condition and a nonlocal integral condition. We justify the assertion that there exists a unique generalized solution of the problem under consideration. The proof of uniqueness is based on an estimate, derived a priori, in the function space introduced in the paper, while the existence of a generalized solution is proved by the Galerkin method.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

In this paper, we investigate the numerical solution of the three-dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first-order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three-dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non-linear term. Thereafter we prove a positive-type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.     

Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

Solvability of the fractional Volterra‐Fredholm integro differential equation by hybridizable discontinuous Galerkin method

This article proves the existence and uniqueness of the solution obtained by the hybridizable discontinuous Galerkin (HDG) method of the fractional Volterra‐Fredholm integro differential equation. The method based on local solvers and transmission condition is applied to the equation using two auxiliary variables. The form of the equation is amenable for achieving the solvability criteria of the problem according to the HDG method. We also calculate a numerical solution of the problem whose exact solution is taken as a smooth or fractional function. This results in a tridiagonal, symmetric, and positive definite stiffness matrix.     

The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation

In the present paper, we analyze a second-order in time fully discrete finite element method for the BBM equation. The discretization in space is based on the standard Galerkin method, for the time discretization the Crank–Nicolson scheme is used. We also prove the convergence of a linearized Galerkin modification scheme.     

A Galerkin approximation for integro-differential equations in electromagnetic scattering from a chiral medium

We consider the scattering of time-harmonic electromagnetic waves from a chiral medium. It is known for the Drude–Born–Fedorov model that the forward scattering problem can be described by an integro-differential equation. In this paper we study a Galerkin finite element approximation for this integro-differential equation. Our Galerkin scheme, which relies on a suitable periodization of the integral equation, enables the use of the fast Fourier transform and a simple numerical implementation. We establish a quasi-optimal convergence analysis for the Galerkin method. Explicit formulas for the discrete scheme are also provided.     

A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

This work concerns with the discontinuous Galerkin (DG) method for the time‐dependent linear elasticity problem. We derive the a posteriori error bounds for semidiscrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time‐dependent problem through the error estimation of the associated stationary elasticity problem. For fully discrete scheme, we make use of the backward‐Euler scheme and an appropriate space‐time reconstruction. The technique here can be applicable for a variety of DG methods as well.     

Error estimate for a symmetric scheme of the projection-difference method for an abstract hyperbolic-parabolic system of the type of systems of thermoelasticity equations

We study the convergence of the projection-difference method for a system of abstract differential equations in Hilbert spaces generalizing a number of linear systems of coupled thermoelasticity equations. We consider a computational scheme that combines a scheme of the Galerkin method with respect to space and a symmetric three-layer difference scheme with weights in time. We impose no special conditions on the projection subspaces of the Galerkin method. For the error, we obtain an energy estimate of second order in time.     

Limit Galerkin-Petrov schemes for a nonlinear convection-diffusion equation

In the present paper, we suggest a version of the nonconformal finite-element method (a perturbed Galerkin method) for approximating a quasilinear convection-diffusion equation in divergence form. A grid scheme is constructed with the use of an approach based on the Galerkin-Petrov approximation to the mixed statement of the original problem. The separated coordinate approximation of the solution components for the mixed problem permits one to take into account the direction of convective transport and preserve the main properties of the spatial operator of the original problem. We prove the stability of the line method scheme and a two-layer weighted scheme for the original problem.     

Superconvergence of a full‐discrete combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem

An efficient time‐stepping procedure is investigated for a two‐dimensional compressible miscible displacement problem in porous media in which the mixed finite element method with Raviart‐Thomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin approximation on Cartesian meshes. Based on the projection interpolations and the induction hypotheses, a superconvergence error estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrödinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.     

广义Burgers方程的Legendre-Galerkin Chebyshev-配置方法

本文对广义Burgers方程的Neumann和Robin型边值问题构造了LegendreGalerkinChebyshev-配置方法.Legendre-GalerkinChebyshev-配置方法整体上按LegendreGalerkin方法形成,但对非线性项采用在Chebyshev-Gauss-Lobatto点上的配置法处理.文中给出了方法的稳定性和收敛性分析,获得了按H1-模的最佳误差估计.数值实验证实了方法的有效性.     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.