求解间断系数椭圆型问题的一种改进的DG方法

本文考虑对间断系数椭圆型问题的普通DG方法进行改进,提出了一种综合了DG方法及区域分解方法的优点的新方法.对此法进行了先验误差分析并给出其残量型后验误差估计,且通过数值实验验证了该方法及其自适应方法的有效性.     

A posteriori error estimate for a modified weak Galerkin method solving elliptic problems

A residual‐type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving second‐order elliptic problems. This estimator is proven to be both reliable and efficient because it provides computable upper and lower bounds on the actual error in a discrete H¹‐norm. Numerical experiments are given to illustrate the effectiveness of the this error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 381–398, 2017     

A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems

We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.     

The a Posteriori Error Estimator of SDG Method for Variable Coefficients Time-Harmonic Maxwell's Equations

In this paper, we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations. We propose two a posteriori error estimators, one is the recovery-type estimator, and the other is the residual-type estimator. We first propose the curl-recovery method for the staggered discontinuous Galerkin method (SDGM), and based on the super-convergence result of the postprocessed solution, an asymptotically exact error estimator is constructed. The residual-type a posteriori error estimator is also proposed, and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations. The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.     

简化摩擦接触问题的对称弱超内罚间断Galerkin方法的先验和后验误差估计

本文讨论了简化摩擦接触问题的一类对称弱超内罚间断Galerkin方法.首先,在能量范数意义下得到最优先验误差估计.进一步,我们推导了一类残量型后验误差估计子,并证明了它的可靠性和有效性.     

定常的Navier-Stokes方程的非线性Galerkin混合元法及其后验估计

提出了定常的Navier-Stokes方程的一种非线性Galerkin混合元法,并导出非线性Galerkin混合元解的存在性和误差估计及其后验误差估计.     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

A Posteriori Error Estimation for the Dual Mixed Finite Element Method of the Stokes Problem

The paper presents an a posteriori error estimator of residual type for the stationary Stokes problem in a possibly multiply-connected polygonal domain of the plane, using the dual mixed finite element method (FEM). We prove lower and upper error bounds with the explicit dependence of the viscosity parameter and without any regularity assumption on the solution.     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.     

A posteriori error estimators for the steady incompressible Navier–Stokes equations

A residual-based a posteriori error estimator for finite element discretizations of the steady incompressible Navier–Stokes equations in the primitive variable formulation is discussed. Though the estimator is similar to existing ones, an alternate derivation is presented, involving an abstract estimate that may prove of some intrinsic value. The estimator is particularized to Hood–Taylor and modified Hood–Taylor finite elements and proved to be a global upper bound (up to a positive multiplicative constant) of the true error. Numerical examples are provided to illustrate the performance of the resulting mesh adaptation process. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 561–574, 1997     

A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS

In this paper, we discuss a posteriori error estimates of the eigenvalue $\lambda_h$ given by Adini nonconforming finite element. We give an asymptotically exact error estimator of the $\lambda_h$. We prove that the order of convergence of the $\lambda_h$ is just 2 and the $\lambda_h$ converge from below for sufficiently small $h$.     

Adaptive 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.     

体积约束的非局部扩散问题的后验误差分析

本文针对体积约束的非局部扩散问题构造了新的后验误差指示器,证明了后验误差指示器的可靠性以及有效性.数值算例验证了理论结果.     

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.     

On the coupled NBEM and FEM for a class of nonlinear exterior Dirichlet problem in R~(2*)

In this paper, based on the natural boundary reduction advanced by Feng and Yu, we couple the finite element approach with the natural boundary element method to study the weak solvability and Galerkin approximation of a class of nonlinear exterior boundary value problems. The analysis is mainly based on the variational formulation with constraints. We prove the error estimate of the finite element solution and obtain     

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.     

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.