Virtual element method for quasilinear elliptic problems

By Andrea Cangiani, Panagiotis Chatzipantelidis, Ganesh C. Diwan, Emmanuil H. Georgoulis. IMA Journal of Numerical Analysis, 2017

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A virtual element method for the quasilinear equation $-\textrm{div} ({\boldsymbol \kappa }(u)\operatorname{grad} u)=f$ 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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

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