A damped Kačanov scheme for the numerical solution of a relaxed p(x)-Poisson equation

The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a truncation of the nonlinearity from below and from above by usin...

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Bibliographic Details
Published inarXiv.org
Main Author Heid, Pascal
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.04.2023
Subjects
Convergence
Iterative methods
Poisson equation
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Summary:The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a truncation of the nonlinearity from below and from above by using a pair of positive cut-off parameters. We will then verify that, for any such pair, a damped Kačanov scheme generates a sequence converging to a solution of the relaxed equation. Subsequently, it will be shown that the solutions of the relaxed problems converge to the solution of the original problem in the discrete setting. Finally, the discrete solutions of the unrelaxed problem converge to the continuous solution. Our work will finally be rounded up with some numerical experiments that underline the analytical findings.
ISSN:2331-8422
DOI:10.48550/arxiv.2304.01566