Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Makridakis, Charalambos; Nochetto, Ricardo H.

doi:10.1137/s0036142902406314

Cited by 150 publications

(178 citation statements)

References 35 publications

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“…The continuous Galerkin reconstruction U ∈ H q+1 'solves' problem (3.4) that is in a sense 'dual' to (3.5). Note the similarity to the relation between the elliptic projection and the elliptic reconstruction of [22] in the a posteriori error analysis of space discrete finite element methods for parabolic equations.…”

Section: Remark 32 (A Priori Projection and Elliptic Reconstruction)mentioning

confidence: 81%

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Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

Akrivis

Makridakis²,

Nochetto³

2009

Numer. Math.

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Summary 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 (RK-C) 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 one-degree 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 time-step; they do not hinge on asymptotics. 1

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Section: Remark 32 (A Priori Projection and Elliptic Reconstruction)mentioning

confidence: 81%

“…We stress that our reconstruction U is not a higher order approximation of u than U , which is another important difference with these two rather popular techniques. We also mention the related technique of elliptic reconstruction, introduced for a posteriori error analysis of space discrete finite element approximations in [22].…”

Section: Introductionmentioning

confidence: 99%

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

Akrivis

Makridakis²,

Nochetto³

2009

Numer. Math.

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“…Its origins can be traced back at least to the elliptic projection of Wheeler [53]. In some aspects, it is close to the elliptic reconstruction of Makridakis and Nochetto [31]; however, in [31] it is used to restore optimal order of the a posteriori estimate in L ∞ (0, T ; L 2 (Ω)), whereas here we employ it to obtain a bound on an energy-like norm.…”

Section: A2 Proof Of Theorem 52mentioning

confidence: 99%

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

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“…Elliptic reconstruction. It will be convenient to employ the elliptic reconstruction defined for semidiscrete problems in [MN03] and for fully discrete schemes in [LM06]. Analogous to our definition of u h , we first define the reconstruction at time nodes and then interpolate linearly between them.…”

Section: We Finally Compute That If T(0) ≥ H(x 0 )mentioning

confidence: 99%

Sharply local pointwise a posteriori error estimates for parabolic problems

Demlow

Makridakis

2010

Math. Comp.

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Abstract. We prove pointwise a posteriori error estimates for semi-and fullydiscrete finite element methods for approximating the solution u to a parabolic model problem. Our estimates may be used to bound the finite element error, where D is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from D. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from D in order to ensure that local solution quality is not polluted by global effects.

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Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Cited by 150 publications

References 35 publications

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Sharply local pointwise a posteriori error estimates for parabolic problems

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