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A posteriori analysis of the finite element discretization of some parabolic equations
 MR2136996 RECONSTRUCTION FOR DISCRETE PARABOLIC PROBLEMS 1657
, 2005
"... Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respe ..."
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Cited by 14 (4 self)
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Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes. Résumé. Nous considérons la discrétisation d’équations paraboliques, soit linéaires soit semilinéaires, par un schéma d’Euler implicite en temps et par éléments finis en espace. L’idée de cet article est de construire des indicateurs d’erreur liés à l’approximation en temps et en espace et de prouver leur équivalence avec l’erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d’éléments finis adaptés à la solution. 1.
Elliptic reconstruction and a posteriori error estimates for parabolic problems
 SIAM J. Numer. Anal
"... Abstract. We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, ..."
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Cited by 10 (4 self)
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Abstract. We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0,T;L2(Ω)) and the higher order spaces, L∞(0,T;H1 (Ω)) and H1 (0,T;L2(Ω)), with optimal orders of convergence. 1.
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 9 (2 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.
Finite Element Approximation of an AllenCahn/CahnHilliard System
"... We consider an AllenCahn/CahnHilliard system with a nondegenerate mobility and (i) a logarithmic free energy and (ii) a nonsmooth free energy (the deep quench limit). This system arises in the modelling of phase separation and ordering in binary alloys. In particular we prove in each case that t ..."
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Cited by 7 (2 self)
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We consider an AllenCahn/CahnHilliard system with a nondegenerate mobility and (i) a logarithmic free energy and (ii) a nonsmooth free energy (the deep quench limit). This system arises in the modelling of phase separation and ordering in binary alloys. In particular we prove in each case that there exists a unique solution for suciently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear nite element approximation of (i) and (ii) in one and two space dimensions (and three space dimensions for constant mobility). The error bound being optimal in the deep quench limit. In addition an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments are presented. 1 Introduction Let be a bounded domain in R d ; d 3, with a Lipschitz boundary @ We consider the AllenCahn/CahnHilliard system with varying mobility and logarithmic free energy: (P ) Find fu (x; t); v (x; t); w (x...
Gradient flows of non convex functionals in Hilbert spaces and applications
 ESAIM CONTROL OPTIM. CALC. VAR
, 2006
"... This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space H ( u ′ (t) + ∂ℓφ(u(t)) ∋ f(t) a.e. in (0, T), u(0) = u0, where φ: H → (−∞, +∞] is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ∂ℓφ ..."
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Cited by 6 (2 self)
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This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space H ( u ′ (t) + ∂ℓφ(u(t)) ∋ f(t) a.e. in (0, T), u(0) = u0, where φ: H → (−∞, +∞] is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ∂ℓφ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures. Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.
Error estimates for semidiscrete gauge methods for the Navier–Stokes equations
 Math. Comp
"... Abstract. The gauge formulation of the NavierStokes equations for incompressible fluids is a new projection method. It splits the velocity u = a+∇φ in terms of auxiliary (nonphysical) variables a and φ and replaces the momentum equation by a heatlike equation for a and the incompressibility constr ..."
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Cited by 5 (2 self)
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Abstract. The gauge formulation of the NavierStokes equations for incompressible fluids is a new projection method. It splits the velocity u = a+∇φ in terms of auxiliary (nonphysical) variables a and φ and replaces the momentum equation by a heatlike equation for a and the incompressibility constraint by a diffusion equation for φ. This paper studies two timediscrete algorithms based on this splitting and the backward Euler method for a with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure. The analyses are variational and hinge on realistic regularity requirements on the exact solution and data. Both Neumann and Dirichlet boundary conditions are, in principle, admissible for φ but a compatibility restriction for the latter is uncovered which limits its applicability. 1. The gauge or impulse formulation Given an open bounded polyhedral domain Ω in Rd,withd = 2 or 3, we consider the timedependent NavierStokes equations of incompressible fluids: ut +(u ·∇)u + ∇p − µ∆u = f, in Ω, (1.1) div u =0, in Ω,
A posteriori error analysis for higher order dissipative methods for evolution problems
"... Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit RungeKuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction Û of th ..."
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Cited by 5 (0 self)
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Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit RungeKuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction Û of the discrete solution U, which restores continuity and leads to the differential equation Û ′ +ΠF(U) = F for a suitable interpolation operator Π. The error analysis hinges on careful energy arguments and the monotonicity of the operator F, in particular its angle bounded structure. We discuss applications to linear PDE such as the convectiondiffusion equation and the wave equation, and nonlinear PDE corresponding to subgradient operators such as the pLaplacian and minimal surfaces, as well as Lipschitz and noncoercive operators. 1.
An Improved Error Bound for a Finite Element Approximation of a Model for Phase Separation of a MultiComponent Alloy with a Concentration Dependent Mobility Matrix
 IMA J. Numer. Anal
"... this paper we consider only the case of a nondegenerate mobility matrix L; that is, L min (i) in (1.3b) satisfies the assumption in (D) above. Elliott and Garcke (1997) have proved existence of a solution to (P ` ) and its deep quench limit (P) for the physically interesting case of mobility matric ..."
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Cited by 4 (0 self)
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this paper we consider only the case of a nondegenerate mobility matrix L; that is, L min (i) in (1.3b) satisfies the assumption in (D) above. Elliott and Garcke (1997) have proved existence of a solution to (P ` ) and its deep quench limit (P) for the physically interesting case of mobility matrices L which degenerate in the pure phases; that is, L min (i) in (1.3b) satisfies L min (i) ? 0 for all i lying in the interior of the Gibbs simplex Q
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 3 (0 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 Estimates and Adaptive Finite Elements for a Nonlinear Parabolic Problem Related to Solidification
"... A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L² in time H¹ in space error is bounded above and below by an e ..."
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Cited by 2 (0 self)
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A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L² in time H¹ in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the effectivity index is close to one. An adaptive finite element algorithm is proposed and a solutal dendrite is computed.