Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations
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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 H1-norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O(H2). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.
KeywordsNonlinear Sobolev equations Two-grid finite element method Error estimates Crank-Nicolson scheme
Mathematics Subject Classification (2010)65N08 65N15
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The authors thank the referees for valuable constructive comments and suggestions, which led to a significant improvement of this paper.
The work is supported by the National Natural Science Foundation of China (Grant No. 11771375, 11671157, 11571297, 91430213), Shandong Province Natural Science Foundation(Grant No. ZR2018MAQ008), and China Postdoctoral Science Foundation funded project (Grant No. 2017M610501).