Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

• Published in: Advances in computational mathematics Vol. 45; no. 2; pp. 611 - 630
• Main Authors: Chen, Chuanjun, Li, Kang, Chen, Yanping, Huang, Yunqing
• Format: Journal Article
• Language: English
• Published: New York Springer US 02.04.2019; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Crank-Nicholson method; Estimates; Finite element method; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinear systems; Visualization; Workload; Nonlinear Sobolev equations; 65N15; Error estimates; Crank-Nicolson scheme; 65N08; Two-grid finite element method
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-018-9628-2

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.