Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations - Chen, Chuanjun; Li, Kang; Chen, Yanping; Huang, Yunqing (Advances in computational mathematics, 2019)
In this paper, we present a second-order accurate Crank-Nicolson scheme for the two-grid finite element methods of the nonlinear Sobolev equations. This method involves solving a small nonlinear system on a coarse mesh with mesh size H and a linear system on a fine mesh with mesh size h , which can still maintain the asymptotically optimal accuracy compared with the standard finite element method. However, the two-grid scheme can reduce workload and save a lot of CPU time. The optimal error estimates in H 1 -norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O ( H 2 ). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.