A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin(DG)method using the interior penalty flux and discontinuous basis functions,both in space and in time,is pre-sented and fully analyzed for the second-order scalar wave equation.Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method.The theoretical analysis shows that the DG discre-tization is stable and converges in a DG-norm on general unstructured and locally refined meshes,including local refinement in time.The space-time interior penalty DG discre-tization does not have a CFL-type restriction for stability.Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time stepΔt satisfy h≅CΔt,with C a positive constant.The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems.These calculations also show that for pth-order tensor product basis functions the convergence rate in the L∞and L2-norms is order p+1 for polynomial orders p=1 and p=3 and order p for polynomial order p=2.

A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems

There have been many theoretical studies and numerical investigations of nonlocal diffusion(ND)problems in recent years.In this paper,we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems,based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense.We show that the proposed discontinuous Galerkin scheme is stable and convergent.Moreover,the local limit of such DG scheme recovers classical DG scheme for the corresponding local diff usion problem,which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization.Numerical tests are also presented to demonstrate the eff ectiveness and the robustness of the proposed method.

Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem

The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.

The improved element-free Galerkin method forthree-dimensional wave equation

The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.

daptive Hybridized Interior Penalty Discontinuous Galerkin Methods for H(curl)-Elliptic Problems

We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations.The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain.It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method.The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals.Within a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency error.The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.

A Discontinuous Galerkin Finite Element Method without Interior Penalty Terms

A conforming discontinuous Galerkinfinite element method was introduced by Ye and Zhang,on simplicial meshes and on polytopal meshes,which has theflexibility of using discontinuous approximation and an ultra simple formulation.The main goal of this paper is to improve the above discontinuous Galerkinfinite element method so that it can handle nonhomogeneous Dirichlet boundary conditions effectively.In addition,the method has been generalized in terms of approximation of the weak gradient.Error estimates of optimal order are established for the correspond-ing discontinuousfinite element approximation in both a discrete H1 norm and the L2 norm.Numerical results are presented to confirm the theory.

Multilevel Preconditioners for the Interior Penalty Discontinuous Galerkin Method II-Quantitative Studies

This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.Several concrete realizations of splitting the nonconforming trial spaces into a conforming and(remaining)nonconforming part are identified and shown to give rise to uniformly bounded condition numbers.These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.

A POSTERIORI ENERGY-NORM ERROR ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.

Two-Level Schwarz Preconditioners for Super Penalty Discontinuous Galerkin Methods

We extend the construction and analysis of the non-overlapping Schwarz preconditioners proposed in[2,3]to the(non-consistent)super penalty discontinuous Galerkin methods introduced in[5]and[8].We show that the resulting preconditioners are scalable,and we provide the convergence estimates.We also present numerical experiments confirming the sharpness of the theoretical results.

Weighted Interior Penalty Methodwith Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh.There are three weights including the arithmetic,the harmonic,and the geometric weight in the weighted discontinuous Galerkin scheme.For the time discretization,we treat the nonlinear diffusion coefficients explicitly,and apply the semiimplicit integration factormethod to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization.The semi-implicit integration factor method can not only avoid severe timestep limits,but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method.Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

多尺度椭圆问题的非对称内罚Galerkin超样本多尺度有限元方法

本文针对多尺度椭圆问题,构造了非对称内罚Galerkin超样本多尺度有限元方法,并且给出了系数周期情况下的最优误差估计.

高精度IPDG湍流模拟及通量格式数值特性

为更高精度数值求解复杂湍流问题,在内罚间断伽辽金(internal penalty discontinuous Galerkin,IPDG)方法框架内发展了基于SST k-ω模型的湍流模拟方法,通过对亚/跨/超声速工况下流场的数值计算与湍流特征捕捉,验证了方法的适用性,进而系统分析了AUSM、Lax-F、HLL和Roe 4种通量格式在IPDG湍流模拟中的数值特性。结果表明:AUSM格式在超声速工况下脱体激波面“褶皱”现象明显;Lax-F格式和HLL格式数值耗散大,在激波解析方面精度较低,且Lax-F格式在激波脚后诱导产生流动分离;Roe格式具备宽速域适用性,计算精度较高且能精确解析激波结构。

二维低波速Helmholtz方程的异质多尺度-内部惩罚间断有限元方法

将异质多尺度方法和内部惩罚间断有限元方法相结合,构造了求解二维低波速Helmholtz方程的异质多尺度-内部惩罚间断有限元方法,并在局部周期条件下给出了算法的最佳误差估计。

Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws

In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.

Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration

This work pertains to numerical aspects of a finite element method based discontinuous functions.Our study focuses on the Interior Penalty Discontinuous Galerkin method(IPDGM)because of its high-level of flexibility for solving the full wave equation in heterogeneousmedia.We assess the performance of IPDGMthrough a comparison study with a spectral element method(SEM).We show that IPDGM is as accurate as SEM.In addition,we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

Local Discontinuous Galerkin Method with Reduced Stabilization for Diffusion Equations

We extend the results on minimal stabilization of Burman and Stamm[J.Sci.Comp.,33(2007),pp.183-208]to the case of the local discontinuous Galerkin methods on mixed form.The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum.Stability in the form of a discrete inf-sup condition is proved and optimal convergence follows.Some numerical examples using high order approximation spaces illustrate the theory.

H(curl)-椭圆问题自适应内罚间断有限元方法的收敛性分析

针对H(curl)-椭圆问题的自适应内罚间断有限元方法,给出了相应的收敛性证明:把间断有限元空间分解成棱有限元空间及其正交补空间,然后结合误差的整体上界估计、关于加密网格之间的网格尺寸的2个条件以及后验误差指示子的单调性等性质,证明了在连续迭代过程中,关于误差函数的能量范数与尺度化的误差指示子之和是压缩的,即自适应内罚间断有限元方法是收敛的.

一维奇异摄动问题的间断有限元(DG)方法

对于一维奇异摄动对流扩散方程,采用一种非对称的间断有限元(DG)方法进行求解.在线性有限元上进行了误差估计并给出数值例子.

加权梯度限制器在节点DG格式中的应用

在使用节点间断Galerkin(DG)方法求解存在激波的流动问题时,为抑制Gibbs效应导致的非物理振荡,对一种加权梯度限制器进行研究,在二维非结构网格上对定常欧拉方程求解。限制器以原始变量作为限制对象,对其梯度进行限制,以加权形式进行重构。对翼型的跨音速流场、超燃冲压发动机内流场和前台阶模型流场进行模拟,结果表明限制器能很好抑制间断解附近的非物理振荡,有效避免计算中断,体现了DG格式精度高、对间断捕捉敏感的特点。