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32
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 43 (12 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 Finite Element Discretizations Of The Heat Equation
, 2003
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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.
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 15 (3 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 ANISOTROPIC ERROR ESTIMATOR FOR THE CRANKNICOLSON METHOD: APPLICATION TO A PARABOLIC PROBLEM.
"... Abstract. In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the CrankNicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic inte ..."
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Cited by 13 (1 self)
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Abstract. In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the CrankNicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following [AMN06], while the second one is new. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm. 1. Introduction. A
A posteriori error analysis of the fully discretized timedependent Stokes equations
"... The timedependent Stokes equations in two or threedimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indic ..."
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Cited by 11 (2 self)
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The timedependent Stokes equations in two or threedimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
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.
Time and Space Adaptivity for the SecondOrder Wave Equation
, 2004
"... The aim of this paper is to show that, for a linear secondorder hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise linear finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a ..."
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Cited by 6 (2 self)
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The aim of this paper is to show that, for a linear secondorder hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise linear finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.
A POSTERIORI ERROR ANALYSIS FOR PARABOLIC VARIATIONAL INEQUALITIES
 ESAIM: MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
, 2007
"... Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ R d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler ..."
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Cited by 4 (3 self)
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Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ R d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L 2 (0,T; H 1 (Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate noncontact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the noncontact region, and for the time error estimator. Numerical results for d =1, 2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the noncontact regions.
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