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43
A posteriori error control for the AllenCahn problem: circumventing Gronwall’s inequality
 M2AN Math. Model. Numer. Anal
"... Abstract. Phasefield models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually dea ..."
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Cited by 39 (0 self)
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Abstract. Phasefield models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying nonmonotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε−2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε−1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.
A posteriori error estimates for the Crank– Nicolson method for parabolic equations
 Math. Comp
"... Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates a ..."
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Cited by 37 (8 self)
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Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are secondorder Crank–Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank–Nicolson method. 1.
A POSTERIORI ERROR ESTIMATION BASED ON POTENTIAL AND FLUX RECONSTRUCTION FOR THE HEAT EQUATION ∗
"... Abstract. 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 sch ..."
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Cited by 16 (2 self)
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Abstract. 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.
OPTIMAL ORDER A POSTERIORI ERROR ESTIMATES FOR A CLASS OF RUNGE–KUTTA AND GALERKIN METHODS
"... Abstract. We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RKC) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredie ..."
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Cited by 10 (2 self)
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Abstract. We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RKC) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate onedegree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the timestep; they do not hinge on asymptotics. 1.
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.
Optimal and Robust A Posteriori Error Estimates IN L∞(L²) FOR THE APPROXIMATION OF ALLENCAHN EQUATIONS PAST SINGULARITIES
, 2009
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A posteriori error estimates in the maximum norm for parabolic problems
, 2007
"... We derive a posteriori error estimates in the L∞((0, T]; L∞(Ω)) norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori ..."
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Cited by 5 (1 self)
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We derive a posteriori error estimates in the L∞((0, T]; L∞(Ω)) norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.
A posteriori error estimates by recovered gradients in parabolic finite element equations
, 2006
"... Abstract. This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at ..."
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Cited by 4 (1 self)
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Abstract. This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the timestep error needs to be smaller than the spacediscretization error. Numerical illustrations are given. 1.
Adaptive Finite Element Methods For Variational Inequalities: Theory And Applications In Finance
, 2007
"... We consider variational inequalities (VIs) in a bounded open domain Ω ⊂ Rd with a piecewise smooth obstacle constraint. To solve VIs, we formulate a fullydiscrete adaptive algorithm by using the backward Euler method for time discretization and the continuous piecewise linear finite element method ..."
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Cited by 3 (0 self)
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We consider variational inequalities (VIs) in a bounded open domain Ω ⊂ Rd with a piecewise smooth obstacle constraint. To solve VIs, we formulate a fullydiscrete adaptive algorithm by using the backward Euler method for time discretization and the continuous piecewise linear finite element method for space discretization. The outline of this thesis is the following. Firstly, we introduce the elliptic and parabolic variational inequalities in Hilbert spaces and briefly review general existence and uniqueness results (Chapter 1). Then we focus on a simple but important example of VI, namely the obstacle problem (Chapter 2). One interesting application of the obstacle problem is the Americantype option pricing problem in finance. We review the classical model as well as some recent advances in option pricing (Chapter 3). These models result in VIs with integrodifferential operators. Secondly, we introduce two classical numerical methods in scientific computing: the finite element method for elliptic partial differential equations (PDEs) and the Euler method for ordinary different equations (ODEs). Then we combine these two
A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS USING ELLIPTIC RECONSTRUCTIONS. II: A THIRDORDER DISCONTINUOUS GALERKIN METHOD∗
, 2011
"... Abstract. A semilinear secondorder parabolic equation is considered in a regular and a singularlyperturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the discontinuous Galerkin method dG(1) are address ..."
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Cited by 2 (2 self)
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Abstract. A semilinear secondorder parabolic equation is considered in a regular and a singularlyperturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the discontinuous Galerkin method dG(1) are addressed; for the latter we employ elliptic reconstructions that are piecewisequadratic in time. We also use certain bounds for the Green’s function of the parabolic operator. Key words. a posteriori error estimate, maximum norm, singular perturbation, elliptic reconstruction, discontinuous Galerkin method dG(1), parabolic equations, reactiondiffusion. AMS subject classifications. 65M15, 65M60. 1. Introduction. Consider