Virtual element method for nonlinear Sobolev equation on polygonal meshes

In this work, the virtual element method (VEM) on convex polygonal meshes for the nonlinear Sobolev equations is developed, where the semi-discrete and fully discrete formulations are presented and analyzed. To overcome the complexity of nonlinear terms, the nonlinear coefficient is approximated by employing the orthogonal L 2 projection operator, which is directly computable from the degrees of freedom. Under some assumptions about the nonlinear coefficient, the existence and uniqueness of the semi-discrete solution are analyzed. Furthermore, a priori error estimate showing optimal order of convergence with respect to the H 1 semi-norm was derived. Finally, some numerical experiments are conducted to illustrate the theoretical convergence rate.