A unified framework for two-grid methods for a class of nonlinear problems

In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence results indicate that the correct scaling between fine and coarse meshes is given by \(h={{\mathcal {O}}}(H^2)\) for all the nonlinear problems which can be written in and applied to the BRR framework. Next, a correction step can be added to the two-grid algorithm, which allows the choice \(h={\mathcal O}(H^3)\). On the other hand, the particular BRR framework with duality pairing, if it is applied to a semilinear problem, allows a higher order relation \(h={{\mathcal {O}}}(H^4)\). Furthermore, even the choice \(h={{\mathcal {O}}}(H^5)\) is possible with the correction step either on fine mesh or coarse mesh. In addition, elliptic problems with gradient nonlinearities and the Naiver–Stokes equations are considered to illustrate our unified theory. Finally, numerical experiments are conducted to confirm our theoretical findings. Numerical results indicate that the correction step used as a simple postprocessing enhances the solution accuracy, particularly for problems with layers.