Fully discrete approximation of general nonlinear Sobolev equations

Bekkouche, Fatiha; Chikouche, Wided; Nicaise, Serge

doi:10.1007/s13370-018-0626-9

Cited by 7 publications

(5 citation statements)

References 31 publications

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“…(3.6b) and (3.7b). Results for those problems are found in [6,30]. For the following error analysis, we provide two results.…”

Section: Well-posedness Of Microscopic and Macroscopic Problemsmentioning

confidence: 99%

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

Freese

2022

Multiscale Model. Simul.

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In this work, we apply the finite element heterogeneous multiscale method to a class of dispersive first-order time-dependent Maxwell systems. For this purpose, we use an analytic homogenization result, which shows that the effective system contains additional dispersive effects. We provide a careful study of the (timedependent) micro problems, including H 2 and micro errors estimates. Eventually, we prove a semi-discrete error estimate for the method.

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“…(3.6b) and (3.7b). Results for those problems are found in [6,30]. For the following error analysis, we provide two results.…”

Section: Well-posedness Of Microscopic and Macroscopic Problemsmentioning

confidence: 99%

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

Freese

2022

Multiscale Model. Simul.

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“…(Ω) and a.e. t ∈ (0, T ) (2.4) has a unique (weak) solution in C([0, T ]; H 1 0 (Ω)), see [3]. This system is a linear Sobolev equation in Ω, where some a priori error analyses were performed in [1,13,17,19,20,23] in some particular situations or with a kind of Neumann boundary conditions.…”

Section: The Continuous Time Semi-discrete and Full Discrete Problemsmentioning

confidence: 99%

“…was considered in [3] since it is more appropriate for Sobolev equations. The a posteriori error analysis that we perform below for system (2.8) is immediately applicable to (2.9) since it can be re-written as…”

Section: The Continuous Time Semi-discrete and Full Discrete Problemsmentioning

confidence: 99%

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A posteriori error estimates for a fully discrete approximation of Sobolev equations

Nicaise¹,

Bekkouche²

2019

Confluentes Mathematici

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“…This methods are based on the shifted Standard and shifted Chebyshev Tau method. In [7] the abstract quasilinear evolution equations of Sobolev type in a Hilbert setting are considered. Authors proposed two fully discrete schemes and proved some error estimates under minimal assumptions.…”

Section: Introductionmentioning

confidence: 99%

Analysis of higher order difference method for a pseudo-parabolic equation with delay

Amirali¹

2019

MMN

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In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method.

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Fully discrete approximation of general nonlinear Sobolev equations

Cited by 7 publications

References 31 publications

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

A posteriori error estimates for a fully discrete approximation of Sobolev equations

Analysis of higher order difference method for a pseudo-parabolic equation with delay

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