A POSTERIORI ERROR ESTIMATES FOR LOCAL CO DISCONTINUOUS GALERKIN METHODS FOR KIRCHHOFF PLATE BENDING PROBLEMS

We derive some residual-type a posteriori error estimates for the local CO discontinuous Galerkin （LCDG） approximations （[31]） of the Kirchhoff bending plate clamped on the boundary. The estimator is both reliable and efficient with respect to the moment-field approximation error in an energy norm. Some numerical experiments are reported to demonstrate theoretical results.     

一阶双曲问题间断有限元的后验误差分析

一阶双曲问题的有限元后验误差估计至今没有得到很好的解决.本文对d维区域上一阶双曲问题的k次间断有限元逼近提出了一种新的后验误差分析方法, 进而建立了间断有限元解在DG范数下(强于L₂范数)基于误差余量型的后验误差估计. 数值计算验证了本文理论分析的有效性. 本文方法也适用于其他变分问题有限元逼近的后验误差分析.     

A UNIFIED A POSTERIORI ERROR ANALYSIS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF REACTIVE TRANSPORT EQUATIONS

Four primal discontinuous Galerkin methods are applied to solve reactive transportproblems, namely, Oden-Babuska-Baumann DG (OBB-DG), non-symmetric interior penaltyGalerkin (NIPG), symmetric interior penalty Galerkin (SIPG), and incomplete interiorpenalty Galerkin (IIPG). A unified a posteriori residual-type error estimation is derivedexplicitly for these methods. From the computed solution and given data, explicit esti-mators can be computed efficiently and directly, which can be used as error indicators foradaptation. Unlike in the reference [10], we obtain the error estimators in L~2 (L~2) norm byusing duality techniques instead of in L~2 (H~1) norm.     

MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL FULLY NONLINEAR SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （PDEs）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.     

ERROR ANALYSIS OF NONLINEAR GALERKIN METHODS FOR KURAMOTO-SIVASHINSKY EQUATIONS

Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular, by an illustrative example, we show that nonlinear Galerkin methods converge faster than the usual Galerkin method.     

A NEW DIRECT DISCONTINUOUS GALERKIN METHOD WITH SYMMETRIC STRUCTURE FOR NONLINEAR DIFFUSION EQUATIONS

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin （DDG） meth- ods for diffusion problems [17] and the direct discontinuous Galerkin （DDG） methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test func- tion, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a sym- metric property and an optimal L2 （L2） error estimate is obtained. Numerical examples are carried out to demonstrate the optimal （k ＋ 1）th order of accuracy for the method with pk polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.     

SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHOD FOR NONSTATIONARY HYPERBOLIC EQUATION

AbstractFor the first order nonstationary hyperbolic equation taking the piecewise linear discontinuous Galerkin solver, we prove that under the uniform rectangular partition, such a discontinuous solver, after postprossesing, can have two and half approximative order which is half order higher than the optimal estimate by Lesaint and Raviart under the rectangular partition.     

求解刚性Volterra延迟积分微分方程的隐显单支方法的稳定性与误差分析

本文主要研究用隐显单支方法求解一类刚性Volterra延迟积分微分方程初值问题时的稳定性与误差分析.我们获得并证明了结论:若隐显单支方法满足2阶相容条件,且其中的隐式单支方法是A-稳定的,则隐显单支方法是2阶收敛且关于初值扰动是稳定的.最后,由数值算例验证了相关结论.     

SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

In this paper,we theoretically and numerically verify that the discontinuous Galerkin(DG)methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L2-norm for even degree polynomial approximations.On uniform meshes,the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions,improving previous results.The theoretical findings are found to be sharp and consistent with numerical results.     

A NEW APPROACH TO RECOVERY OF DISCONTINUOUS GALERKIN

A new recovery operator P ：Qn^disc（T）→Qn＋1^disc（M） for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh T into a higher order polynomial space on a macro mesh M. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local L2-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.     

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO- AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.     

AN EFFICIENT ADER DISCONTINUOUS GALERKIN SCHEME FOR DIRECTLY SOLVING HAMILTON-JACOBI EQUATION

This paper proposes an efficient ADER(Arbitrary DERivatives in space and time)discontinuous Galerkin(DG)scheme to directly solve the Hamilton-Jacobi equation.Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG(RKDG)schemes,the ADER scheme is one-stage in time discretization,which is desirable in many applications.The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time.In such predictor step,a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime.The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution.The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written.The explicit formulae of the scheme at third order is provided on two-dimensional structured meshes.The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme.Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

SUPERCONVERGENCE OF A DISCONTINUOUS GALERKIN METHOD FOR FIRST-ORDER LINEAR DELAY DIFFERENTIAL EQUATIONS

This paper deals with the discontinuous Galerkin (DG) methods for delay differential equations.By an orthogonal analysis in each element,the superconvergence re...     

REFLECTION/TRANSMISSION CHARACTERISTICS OF A DISCONTINUOUS GALERKIN METHOD FOR MAXWELL＇S EQUATIONS IN DISPERSIVE INHOMOGENEOUS MEDIA

In this paper, we analyze the transmission and reflection properties of a high order discontinuous Galerkin method for dispersive Maxwell＇s equations, originally proposed by Lu et al. [J. Comput. Phys. 200 （2004）, pp. 549-580]. We study the reflection and transmission properties of the numerical method for up to second-order polynomial elements for one- and two-dimensional Maxwell＇s equations with rectangular meshes. High order accuracy has been shown for reflection and transmission coefficients near material interfaces.     

奇异摄动问题内罚间断有限方法的最优阶一致收敛性分析

本文在Bakhvalov-Shishkin网格上分析了采用高次元的内罚间断有限元方法求解一维对流扩散型奇异摄动问题的最优阶一致收敛性. 取k(k≥1)次分片多项式和网格剖分单元数为N时,在能量范数度量下, Bakhvalov-Shishkin网格上可获得O(N^-k)的一致误差估计. 在数值算例部分对理论分析结果进行了验证.     

一类加权半隐格式增量未知元方法的稳定性和误差估计

本文提出一类基于一维热传导方程数值求解的增量未知元方法加权半隐格式,并由此给出分析稳定性和整体截断误差的新方法．我们引入源于Laplace算子的两组基底,使得放大矩阵易于分析;我们利用IU性质和矩阵运算技巧,严格证明了所述加权格式的稳定性充分条件和全局误差估计,这些结果本质上优于1/4≤θ≤3/4条件下的常见情形．所得结论为恢复初始误差带来可能,为选择最优加权半隐格式提供了理论依据．