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GUARANTEED AND ROBUST DISCONTINUOUS GALERKIN A POSTERIORI ERROR ESTIMATES FOR CONVECTION–DIFFUSION–REACTION PROBLEMS
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2008
"... We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to anal ..."
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Cited by 21 (5 self)
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We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty 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 H(div, Ω)conforming diffusive and convective flux reconstructions, thereby extending previous work on pure diffusion problems. The resulting estimates are semirobust 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 skewsymmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical
hpDISCONTINUOUS GALERKIN METHODS FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER
, 2009
"... This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form ..."
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Cited by 20 (3 self)
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This paper develops some interior penalty hpdiscontinuous Galerkin (hpDG) methods for the Helmholtz equation in two and three dimensions. The proposed hpDG methods are defined using a sesquilinear form which is not only meshdependent but also degreedependent. In addition, the sesquilinear form contains penalty terms which not only penalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts. It is proved that the proposed hpdiscontinuous Galerkin methods are absolutely stable (hence, wellposed). For each fixed wave number k, suboptimal order error estimates in the broken H 1norm and the L 2norm are derived without any mesh constraint. The error estimates and the stability estimates are improved to optimal order under the mesh condition k 3 h 2 p −1 ≤ C0 by utilizing these stability and error estimates and using a stabilityerror iterative procedure To overcome the difficulty caused by strong indefiniteness of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [19, 20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h, the polynomial degree p, the wave number k, as well as all the penalty parameters for the numerical solutions.
A posteriori error estimation based on potential and flux reconstruction for the heat equation
, 2010
"... We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in tim ..."
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Cited by 15 (1 self)
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We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H 1conforming reconstruction of the potential, continuous and piecewise affine in time, and a locally conservative H(div)conforming reconstruction of the flux, piecewise constant in time. They yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative. Localintime lower bounds are also derived; for nonconforming methods on timevarying meshes, the lower bounds require a mild parabolictype constraint on the meshsize.
Preasymptotic error analysis of CIPFEM and FEM for the Helmholtz equation with high wave number. Part I: linear version
, 2014
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A posteriori energynorm error estimates for advectiondiffusion equations approximated by weighted interior penalty methods
 J. Comput. Math
"... Abstract We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations of advectiondiffusionreaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diff ..."
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Cited by 8 (2 self)
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Abstract We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations of advectiondiffusionreaction 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 nonconforming 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 Verfürth on continuous finite elements, that the local lower error bounds can be written with constants involving a cutoff for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor. Mathematics subject classification: 65N30, 65N15, 76R99
A posteriori error control for discontinuous Galerkin methods for parabolic problems
 SIAM J. Numer. Anal
"... Abstract. We derive energynorm a posteriori error bounds for an Euler timestepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by in ..."
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Cited by 8 (5 self)
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Abstract. We derive energynorm a posteriori error bounds for an Euler timestepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler timestepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energynorm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which prompts the derivation of new a posteriori error bounds. For the analysis of the timedependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it. 1.
Weighted error estimates of the continuous interior penalty method for singularly perturbed problems
 IMA J. of Num. Anal
"... Abstract. In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convectiondominated singularly perturbed convectiondiffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain o ..."
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Cited by 7 (2 self)
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Abstract. In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convectiondominated singularly perturbed convectiondiffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods. 1.
Explicit RungeKutta schemes and finite elements with symmetric stabilization for firstorder linear PDE systems
 SIAM J. NUMER. ANAL
, 2009
"... We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a firstorder linear differential operator in space of Friedrichstype. For the time discretization, we consider explicit second and thirdorder Runge–Kutta sch ..."
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Cited by 7 (2 self)
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We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a firstorder linear differential operator in space of Friedrichstype. For the time discretization, we consider explicit second and thirdorder Runge–Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish L²norm error estimates with (quasi)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for thirdorder Runge–Kutta schemes and any polynomial degree in space and for secondorder Runge– Kutta schemes and firstorder polynomials in space. For secondorder Runge–Kutta schemes and higher polynomial degrees in space, a tightened 4/3CFL condition is required. Numerical results are presented for the advection and wave equations.
A unified framework for a posteriori error estimation for the Stokes problem
 NUMERISCHE MATHEMATIK
, 2011
"... In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on [H10 (Ω)] dconforming velocity reconstruction and H(div, Ω)conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper boun ..."
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Cited by 4 (1 self)
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In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on [H10 (Ω)] dconforming velocity reconstruction and H(div, Ω)conforming, locally conservative flux (stress) reconstruction. It gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. Numerical experiments illustrate the theoretical developments.