The parabolic p-Laplacian with fractional differentiability

Dominic Breit; Lars Diening; Johannes Storn; Jörn Wichmann

doi:10.1093/imanum/draa081

The parabolic p-Laplacian with fractional differentiability

Dominic Breit, Lars Diening, Johannes Storn, Jörn Wichmann

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Abstract

We study the parabolic p-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions h and τ⁠. For this we show that the L²-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

Original language	English
Pages (from-to)	2110–2138
Number of pages	29
Journal	IMA Journal of Numerical Analysis
Volume	41
Issue number	3
DOIs	https://doi.org/10.1093/imanum/draa081
Publication status	Published - 9 Dec 2020

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This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record Dominic Breit, Lars Diening, Johannes Storn, Jörn Wichmann, The parabolic p-Laplacian with fractional differentiability, IMA Journal of Numerical Analysis, draa081 is available online at: https://doi.org/10.1093/imanum/draa081
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AB - We study the parabolic p-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions h and τ⁠. For this we show that the L2-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

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The parabolic p-Laplacian with fractional differentiability

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