Virtual Element Method for Quasilinear Elliptic Problems

TL;DR

This paper develops a Virtual Element Method for solving quasilinear elliptic problems, providing theoretical analysis and numerical validation of convergence and error estimates.

Contribution

It introduces a VEM approach for quasilinear elliptic equations, including well-posedness, error estimates, and convergence of fixed point iterations.

Findings

Optimal order error estimates in H^1 and L^2 norms

Convergence of fixed point iterations for nonlinear systems

Numerical examples confirming theoretical results

Abstract

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.