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= O(H2) 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= O(H3). 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= O(H4). Furthermore, even the choice h= O(H5) 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.
Bibliographical noteFunding Information:
The authors would like to express sincere thanks to the anonymous referee whose invaluable comments led to an improved version of the paper. Dongho Kim was supported by NRF-2013R1A1A2007462 and NRF-2018R1D1A1B07050583 and Eun-Jae Park was supported by NRF-2015R1A5A1009350 and NRF-2016R1A2B4014358.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Computational Mathematics