Virtual Element Method for Quasilinear Elliptic Problems
It extends VEM to a class of nonlinear problems, providing theoretical guarantees for practitioners in computational PDEs.
The paper develops a Virtual Element Method for quasilinear elliptic Dirichlet problems, proving well-posedness, optimal error estimates, and convergence of fixed-point iterations, with numerical validation.
We present a Virtual Element Method (VEM) for the solution of Dirichlet problems for the quasilinear equation $-\text{div} (k(u)\text{grad} u)=f$ with essential boundary conditions. Within the VEM 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 $L^2$ norms are proven. In addition, the convergence of fixed point iterations for the solution of the resulting nonlinear system is established. Numerical examples confirm the convergence analysis.