Local discontinuous Galerkin approximation of non-Fickian diffusion model in viscoelastic polymers

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings.     

Error estimate of fully discrete discontinuous Galerkin approximation for the viscoelasticity type equation

In this paper we develop fully discrete discontinuous Galerkin approximation using symmetric interior penalty method. We construct finite element spaces which consist of piecewise continuous polynomials. We introduce an appropriate elliptic-type projection of u and prove its optimal convergence. We develop fully discrete discontinuous Galerkin approximations and prove the optimal convergence in ? ^∞(L ²) normed space.     

Attractor information for discrete dynamical systems by means of optimal discrete Galerkin bases

We introduce local adaptive discrete Galerkin bases as a basis set in order to obtain geometrical and topological information about attractors of discrete dynamical systems. The asymptotic behavior of these systems is described by the reconstruction of their attractors in a finite dimensional Euclidean space and by the attractor topological characteristics including the minimal embedding dimension and its local dimension. We evaluate numerically the applicability of our geometrical and topological results by examining two examples: a dissipative discrete system and a nonlinear discrete predator–prey model that includes several types of self-limitation on the prey.     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

1．引言本文的工作主要是讨论非定常的热传导一对流问题的向后一步的Euler全离散化的非线性Galerkin混合元解的存在性及其误差估计．该工作是对山中的同一问题研究的第二部分．在第一部分［1］，我们已经讨论了此问题的半离散化的情形．由于所研究的目标都是非定常的热传导一对流问题，其背景是相同的，在此将不重复了，请参考［1］．本文的安排如下，52先回顾非定常的热传导一对流问题的混合元解的经典性质．53回顾半离散化的非线性Galerkin混合元解的性质，并导出后续讨论需要的一些关于时间导数的估计．54讨论向后一步的Euler全离散化…     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

In this paper, a fully discrete format of nonlinear Galerkin mixed element method with backward one-step Euler discretization of time for the non stationary conduction-convection problems is presented. The scheme is based on two finite element spaces XH and Xh for the approximation of the velocity, defined respectively on a coarse grid with grids size H and another fine grid with grid size h<< H, a finite element space Mh for the approximation of the pressure and two finite element spaces AH and Wh, for the approximation of the temperature,also defined respectivply on the coarse grid with grid size H and another fine grid with grid size h. The existence and the convergence of the fully discrete mixed element solution are shown. The scheme consists in using standard backward one step Euler-Galerkin fully discrete format at first L0 steps (L0 2) on fine grid with grid size h, but using nonlinear Galerkin mixed element method of backward one step Euler-Galerkin fully discrete format through L0 + 1 step to end step. We have proved that the fully discrete nonlinear Galerkin mixed element procedure with respect to the coarse grid spaces with grid size H holds superconvergence.     

An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems

We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two-level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions.

     

一类更广泛的Захаров方程组的有限元分析

其中n=n(x,i)为离子的扰动量(实函数,ε为场量(复函数)。该方程组具有一系列重要性质,如具有一维孤立子解,即Langmuir孤立子,它的形成、发展和相互作用不同于KDV方程的孤立子,因而引起人们的兴趣和关注.[2]研究了这个方程组的周期初值问     

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.     

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L^∞(L²) for velocity and concentration and the super convergence in L^∞(H¹) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

求解Brinkman方程的修正弱有限元方法

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.     

A numerical study of variable depth KdV equations and generalizations of Camassa–Holm-like equations

In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method.     

Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

In this paper, we propose and analyze a fully discrete local discontinuous Galerkin (LDG) finite element method for time-fractional fourth-order problems. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Stability is ensured by a careful choice of interface numerical fluxes. We prove that our scheme is unconditional stable and convergent. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems

We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method.     

Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. To define the necessary coarse-level solver required for the definition of the proposed preconditioner, we exploit ideas from composite finite element methods, which allow for the definition of finite element schemes on general meshes consisting of polygonal (agglomerated) elements. The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two- and three-dimensions.     

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.     

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Superconvergence of a combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem

A combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is studied. That is to say, the mixed finite element method is applied to the flow equation, and the transport equation is solved by an interior penalty discontinuous Galerkin method. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is obtained. Numerical experimental results are presented to verify the theoretical analysis.