Two-grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data

In this paper, we propose and analyze a two-grid finite element method for a class of quasilinear elliptic problems under minimal regularity of data in a bounded convex polygonal Ω⊂R ², which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasilinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasilinear elliptic problem on a coarse space. Convergence estimates in the H ¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Moreover, we propose a natural and computationally efficient residual-based a posteriori error estimator of the two-grid finite element method for this nonmonotone quasilinear elliptic problem and derive the global upper and lower bounds on the error in the H ¹-norm. Numerical experiments are provided to confirm our theoretical findings.