Virtual Element Method for Quasilinear Elliptic Problems

Cangiani, A.; Chatzipantelidis, P.; Diwan, G.; Georgoulis, E.H.

Abstract

A Virtual Element Method (VEM) for the quasilinear equation ?div(? ? ?(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 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.

Citation

Cangiani, A., Chatzipantelidis, P., Diwan, G., & Georgoulis, E. (2020). Virtual Element Method for Quasilinear Elliptic Problems. IMA Journal of Numerical Analysis, 40(4), 2450–2472. https://doi.org/10.1093/imanum/drz035

• Journal Article Type: Article
• Acceptance Date: Jul 1, 2019
• Online Publication Date: Oct 14, 2019
• Publication Date: 2020-10
• Deposit Date: Aug 9, 2019
• Publicly Available Date: Oct 15, 2020
• Journal: IMA Journal of Numerical Analysis
• Print ISSN: 0272-4979
• Electronic ISSN: 1464-3642
• Publisher: Oxford University Press (OUP)
• Peer Reviewed: Peer Reviewed
• Volume: 40
• Issue: 4
• Pages: 2450–2472
• DOI: https://doi.org/10.1093/imanum/drz035
• Keywords: Applied Mathematics; General Mathematics; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/2411161
• Publisher URL: https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drz035/5586239