A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM

In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method,though they have essentially different bilinear forms.More precisely,we prove its reliability and efficiency for the actual error measured in the standard DG norm.We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution.Numerical results are presented to verify the theoretical analysis.     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

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.     

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.     

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.     

A discontinuous finite volume element method for second‐order elliptic problems

In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L² ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012     

A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated.A symmetric interior penalty discontinuous Galerkin (SIPG) method is applied to the coupled system of flow and transport.Using the induction hypotheses instead of the cut-off operator and the interpolation projection properties,a priori hp error estimates are presented.The error bounds in L2(H1) norm for concentration and in L∞(L2) norm for velocity are optimal in h and suboptimal in p with a loss of power 1/2.     

Interior estimates for the wavelet Galerkin method

Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems. We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems. Received May 17, 1995 / Revised version received April 26, 1996     

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the(2k + 1)-th-order superconvergence for the cell averages, and the numerical traces in the discrete L~2 norm. In addition, superconvergence of orders k + 2 and k + 1 is obtained for the error and its derivative at generalized Radau points. All the theoretical findings are confirmed by numerical experiments.     

A partially penalty immersed interface finite element method for anisotropic elliptic interface problems

In this article, we develop a partially penalty immersed interface finite element (PIFE) method for a kind of anisotropy diffusion models governed by the elliptic interface problems with discontinuous tensor‐coefficients. This method is based on linear immersed interface finite elements (IIFE) and applies the discontinuous Galerkin formulation around the interface. We add two penalty terms to the general IIFE formulation along the sides intersected with the interface. The flux jump condition is weakly enforced on the smooth interface. By proving that the piecewise linear function on an interface element is uniquely determined by its values at the three vertices under some conditions, we construct the finite element spaces. Therefore, a PIFE procedure is proposed, which is based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. Then we prove the consistency and the solvability of the procedure. Theoretical analysis and numerical experiments show that the PIFE solution possesses optimal‐order error estimates in the energy norm and norm.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1984–2028, 2014     

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.     

A supercloseness result for the discontinuous Galerkin stabilization of convection–diffusion problems on Shishkin meshes

We consider a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h–version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer–adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

拟线性拋物方程半离散变网格有限元方法的误差估计

§1.导言近年来,变网格方法正日益为人们所重视与应用,但理论性分析文献仍不多见。文献[1]讨论了某些发展型方程变网格方法的误差估计,但未给出收敛阶估计;文献[2,3]仅对全离散方法讨论了收敛阶问题。本文对一类拟线性抛物问题,于第二节中给出了半离散Galerkin变网格计算格式及其可解性定理;第三节中建立了对称误差估计;第四节给     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

THE FULL DISCRETE DISCONTINUOUS FINITE ELEMENTANALYSIS FOR FIRST-ORDER LINEAR HYPERBOLICEQUATION

1.IntroductionLetfibeaboundeddomaininRZwithpiecewisesmoothboundaryOff,[0,T]beatimeinterval.Considerthefirst-orderhyperbolicproblemasfollowingwhereac~(%,%),p(x,t)~(gi(x,t),pZ(x,t)),Off--(t)~{xEOff:fi(x,t)'ac<0},700istheoutwardunitnormaltoOff;fi(t)~fi\Ofl--(t).Asusual,Off--(t)isreferedtoasinflowboundaryattimet,andOff (t)~Off\Ofl--(t)iscalledoutflowboundaryattimet.FOrsimplicityinfiniteelementanalysis,supposethatboundaryOff--(t)isindependentoft.ThusforalltE(0,T]wecanwriteandproblem(1.0)can…     

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn–Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori error estimates. Our analysis is valid for both symmetric and nonsymmetric versions of the discontinuous Galerkin formulation.     

Multigrid methods for the symmetric interior penalty method on graded meshes

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi‐optimal error estimates in both the energy norm and the L₂ norm for the SIP method, and prove uniform convergence of the W‐cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of hp-discontinuous Galerkin finite element methods to the rotating disk electrode problems in electrochemistry

This paper presents the interior penalty discontinuous Galerkin finite element methods (DGFEM) for solving the rotating disk electrode problems in electrochemistry. We present results for the simple E reaction mechanism (convection-diffusion equations), the EC’ reaction mechanism (reaction-convection-diffusion equation) and the ECE and EC₂E reaction mechanisms (linear and nonlinear systems of reaction-convection-diffusion equations, respectively). All problems will be in one dimension.