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Virtual element method for quasilinear elliptic problems

TitleVirtual element method for quasilinear elliptic problems
Publication TypeJournal Article
Year of Publication2019
AuthorsCangiani, A, Chatzipantelidis, P, Diwan, G, Georgoulis, EH
JournalIMA Journal of Numerical Analysis
Volume40
Pagination2450-2472
Date Published07
ISSN0272-4979
Abstract

A virtual element method for the quasilinear equation \\$-\\textrm\{div\} (\{\\boldsymbol \ąppa \}(u)\\operatorname\{grad\} u)=f\\$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the \\$H^1\\$- and \\$L^2\\$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

URLhttps://doi.org/10.1093/imanum/drz035
DOI10.1093/imanum/drz035

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