Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems

Leykekhman, Dmitriy; Vexler, Boris

doi:10.1137/17m114100x

Cited by 9 publications

(4 citation statements)

References 34 publications

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“…Logarithmically quasi-maximal parabolic regularity results for dG methods were earlier established in [9] in general Banach spaces for autonomous equations and in [10] in Hilbert spaces for nonautonomous equations. These works cover also the cases of variable time steps as well as the critical exponents p = 1, ∞.…”

Section: 2mentioning

confidence: 97%

“…In this section, we extend the maximal parabolic regularity stability estimates for dG methods to nonautonomous parabolic equations by a perturbation argument. For similar ideas and results, we refer to [10] and [7, §3.6], [3] for the dG method with piecewise constant elements and for Radau IIA methods, respectively. Furthermore, we establish optimal order a priori and a posteriori error estimates.…”

Section: Extension To Nonautonomous Equationsmentioning

confidence: 99%

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On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

Akrivis¹,

Makridakis²

2022

SIAM J. Numer. Anal.

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The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.

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Section: 2mentioning

confidence: 97%

Section: Extension To Nonautonomous Equationsmentioning

confidence: 99%

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

Akrivis¹,

Makridakis²

2022

SIAM J. Numer. Anal.

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“…To proceed, we will first prove the boundedness of z kh in L ∞ (I; L 2 (Ω)). To this end, we employ the discrete transformation argument from [18]. For µ > 0 a sufficient large number to be chosen later let y kh,m be defined as…”

Section: Discrete Maximal Parabolic Estimates For a Linear Auxiliary mentioning

confidence: 99%

Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations

Meidner

Vexler

2018

ESAIM: M2AN

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We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme) and with conforming finite elements in space. The main contribution of this paper is the proof of the uniform boundedness of the discrete solution. This allows us to obtain optimal error estimates with respect to various norms.

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“…The linear inhomogeneous version of (1) (F (t, u) = f (t)) was investigated in [6,5,7], [1, Chapter III, Section 12] and the references therein. To the best of our knowledge, the nonlinear case is not yet investigated in the scientific literature.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

Tambue¹,

Mukam²

2020

Preprint

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In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method.To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the scheme toward the mild solution of the PDE under polynomial growth condition of the nonlinearity. Our convergence rate are obtain for smooth and nonsmooth initial data and is similar to that of the autonomous case. Our convergence result for smooth initial data is very important in numerical analysis. For instance, it is one step forward in approximating non-autonomous stochastic partial differential equations by the finite element method. In addition, we provide realistic conditions on the nonlinearity, appropriated to achieve optimal convergence rate without logarithmic reduction by exploiting the smooth properties of the two parameters evolution operator.

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Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems

Cited by 9 publications

References 34 publications

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

On Maximal Regularity Estimates for Discontinuous Galerkin Time-Discrete Methods

Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations

Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

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