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Virtual Element Method for Quasilinear Elliptic Problems

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journal contribution
posted on 23.04.2020, 14:14 by A Cangiani, P Chatzipantelidis, G Diwan, EH Georgoulis
A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = 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 H1 - and L2 -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.A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = 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 H1 - and L2 -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.

Funding

This research was initiated during the visit of PC to Leicester funded by the LMS Scheme 2 grant (Project RP201G0158). AC was partially supported by the EPSRC (Grant EP/L022745/1). EHG was supported by a Research Project Grant from The Leverhulme Trust (grant no. RPG 2015-306). All this support is gratefully acknowledged. We also express our gratitude to Martin Nolte (Albert-Ludwigs-Universit¨at Freiburg) and Andreas Dedner (University of Warwick) for supporting the implementation of the VEM within DUNE-FEM.

History

Citation

IMA Journal of Numerical Analysis, drz035, https://doi.org/10.1093/imanum/drz035 Published: 14 October 2019

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

IMA Journal of Numerical Analysis

Pagination

drz035

Publisher

Oxford University Press (OUP) for Institute of Mathematics and its Applications

eissn

1464-3642

Copyright date

2019

Available date

14/10/2019

Publisher version

https://academic.oup.com/imajna/article/doi/10.1093/imanum/drz035/5586239

Language

en

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