Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equationsusing an expanded mixed finite element method. We use two Newtoniterations on the fine grid in our methods. Firstly, we solve anoriginal nonlinear problem on the coarse nonlinear grid, then we useNewton iterations on the fine grid twice. The two-grid idea is fromXu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] onstandard finite method. We also obtain the error estimates for thealgorithms of the two-grid method. It is shown that the algorithmachieves asymptotically optimal approximation rate with the two-gridmethods as long as the mesh sizes satisfy$h=\mathcal{O}(H^{(4k+1)/(k+1)})$.