An a Posteriori Error Estimate and Adaptive Timestep Control for a Backward Euler Discretization of a Parabolic Problem

Johnson, Claes; Nie, Yaguang; Thomée, Vidar

doi:10.1137/0727019

Cited by 77 publications

(60 citation statements)

References 4 publications

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“…The discontinuous Galerkin method dG(q) of order q ≥ 1 is defined as follows [7,8,9,10,11,14,15,16,17,21,14,30]:…”

Section: 3)û + πF(u ) = Fmentioning

confidence: 99%

“…In contrast to the approach of [7,8,9,10,11,15,16,17,28], which is based on the strong stability of suitable dual problems, the key novel ingredient of our approach to a posteriori error analysis is a higher order reconstruction U , of degree q + 1, which yields the differential equation (…”

Section: Introductionmentioning

confidence: 99%

“…Most of the work has been done for linear or nonlinear dissipative problems by considering time discretizations based on the backward Euler method or on higher order discontinuous Galerkin methods, cf. e.g., [9,7,8,17,27,28] and [15,16,18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A posteriori error analysis for higher order dissipative methods for evolution problems

Makridakis

Nochetto

2006

Numer. Math.

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Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit Runge-KuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction U of the discrete solution U , which restores continuity and leads to the differential equation U +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 convection-diffusion equation and the wave equation, and nonlinear PDE corresponding to subgradient operators such as the p-Laplacian and minimal surfaces, as well as Lipschitz and noncoercive operators.

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“…The discontinuous Galerkin method dG(q) of order q ≥ 1 is defined as follows [7,8,9,10,11,14,15,16,17,21,14,30]:…”

Section: 3)û + πF(u ) = Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A posteriori error analysis for higher order dissipative methods for evolution problems

Makridakis

Nochetto

2006

Numer. Math.

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“…In analogy to [1] (see also [11] for the basic idea in a different context and [12] for analogous results for the heat equation), we define for each n, 1 ≤ n ≤ N , the error indicator…”

Section: A Posteriori Analysis Of the Time Discretizationmentioning

confidence: 99%

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

Bernardi¹,

Verfürth²

2004

ESAIM: M2AN

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Abstract. The time-dependent Stokes equations in two-or three-dimensional 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.Mathematics Subject Classification. 65N30, 65N15, 65J15.

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“…Most of these algorithms are based on a posteriori error estimators which provide appropriate tools for adaptive mesh refinements. The theory of a posteriori analysis of finite element methods for parabolic problems is well-developed (see, e.g., [3,4,19,22,27,29,36,40]). Surprisingly, there has been considerably less work on the error control of finite element methods for second order hyperbolic problems, despite the substantial amount of research in the design of finite element methods for the wave problem (see, e.g., [6,7,8,11,20]).…”

Section: Introductionmentioning

confidence: 99%

A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

Hou¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ≥ 0). Using mixed elliptic reconstruction method, a posteriori L ∞ (L 2 )-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

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An a Posteriori Error Estimate and Adaptive Timestep Control for a Backward Euler Discretization of a Parabolic Problem

Cited by 77 publications

References 4 publications

A posteriori error analysis for higher order dissipative methods for evolution problems

A posteriori error analysis for higher order dissipative methods for evolution problems

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

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An a Posteriori Error Estimate and Adaptive Timestep Control for a Backward Euler Discretization of a Parabolic Problem

Cited by 77 publications

References 4 publications

A posteriori error analysis for higher order dissipative methods for evolution problems

A posteriori error analysis for higher order dissipative methods for evolution problems

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

A POSTERIORI L∞(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

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Product

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A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS