A posteriori error estimation for a new stabilized discontinuous Galerkin method

A posteriori error estimates are derived for a stabilized discontinuous Galerkin method (DGM) [1]. Equivalence between the error norm and the norm of the residual functional is proved, and consequently, global error estimates are obtained by estimating the norm of the residual. One- and two-dimensional numerical experiments are shown for a reaction-diffusion type model problem.     

Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems

In this article, we develop functional a posteriori error estimates for discontinuous Galerkin (DG) approximations of elliptic boundary‐value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations developed by S. Repin (see e.g., Math Comp 69 (2000) 481–500). On these grounds, we derive two‐sided guaranteed and computable bounds for the errors in “broken” energy norms. A series of numerical examples presented confirm the efficiency of the estimates. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes. C. Carstensen and M. Jensen supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the Hausdorff Institute of Mathematics in Bonn, Germany. C. Carstensen, T. Gudi, and M. Jensen supported by DST-DAAD (PPP-05) project no. 32307481.     

Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems

We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using -conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.     

A posteriori error estimates for discontinuous Galerkin methods of obstacle problems

We present a posteriori error analysis of discontinuous Galerkin methods for solving the obstacle problem, which is a representative elliptic variational inequality of the first kind. We derive reliable error estimators of the residual type. Efficiency of the estimators is theoretically explored and numerically confirmed.     

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.     

A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

This work concerns with the discontinuous Galerkin (DG) method for the time‐dependent linear elasticity problem. We derive the a posteriori error bounds for semidiscrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time‐dependent problem through the error estimation of the associated stationary elasticity problem. For fully discrete scheme, we make use of the backward‐Euler scheme and an appropriate space‐time reconstruction. The technique here can be applicable for a variety of DG methods as well.     

An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations

^** Email: paul.houston{at}nottingham.ac.uk^*** Corresponding author. Email: ilaria.perugia{at}unipv.it^**** Email: schoetzau{at}math.ubc.ca We introduce a residual-based a posteriori error indicator fordiscontinuous Galerkin discretizations of H(curl; )-ellipticboundary value problems that arise in eddy current models. Weshow that the indicator is both reliable and efficient withrespect to the approximation error measured in terms of a naturalenergy norm. We validate the performance of the indicator withinan adaptive mesh refinement procedure and show its asymptoticexactness for a range of test problems.     

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem on arbitrary (convex or nonconvex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization technique based on bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.     

A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations

^** Email: jingtang{at}lsec.cc.ac.cn^*** Email: hermann{at}math.mun.ca In this paper we establish a posteriori error estimates forthe discontinuous Galerkin (DG) method applied to linear, semilinearand non-standard (non-linear) Volterra integro-differentialequations. We also present an analysis of the DG method withquadrature for the memory term. Numerical experiments basedon three integro-differential equations are used to illustratevarious aspects of the error analysis.     

A unified a posteriori error estimate of local discontinuous Galerkin approximations for reactive transport problems

To solve reactive transport problems in porous media, local discontinuous Galerkin (LDG) approximations are investigated. Based on the duality technique and the residual error notations, a unified a posteriori error estimate in L ²(L ²) norm is obtained, which is usually used for guiding anisotropic and dynamic mesh adaptivity.     

Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem

This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e., elements with very large aspect ratio. Our analysis covers two‐ and three‐dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh, which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

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.     

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d‐dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017     

A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

A posteriori error estimation for a fully discrete discontinuous Galerkin approximation to a kind of singularly perturbed problems

The time variable in the semi-discrete problem is still continuous. In order to obtain an expected numerical solution, discretization the time variable from the semi-discrete form (full discretization) is needed. For a kind of non-stationarily singular perturbation problem in 1D, a fully discrete discontinuous Galerkin (DG) method is considered. That is to say, space variable is discretized with a primal DG method with penalty, and time variable is done using the backward Euler method. By virtue of duality arguments, inverse estimation of finite element method and interpolation theory, we present a residual-type a posteriori error indicator, which is usually used for adaptivity.     

An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods

In this paper we analyze the coupling of local discontinuous Galerkin (LDG) and boundary element methods as applied to linear exterior boundary value problems in the plane. As a model problem we consider a Poisson equation in an annular polygonal domain coupled with a Laplace equation in the surrounding unbounded exterior region. The technique resembles the usual coupling of finite elements and boundary elements, but the corresponding analysis becomes quite different. In particular, in order to deal with the weak continuity of the traces at the interface boundary, we need to define a mortar-type auxiliary unknown representing an interior approximation of the normal derivative. We prove the stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Finally, we apply local and global approximation properties of the subspaces involved to obtain the a priori error estimate in the energy norm.

     

Robustness in a posteriori error analysis for FEM flow models

Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint. Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001