Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations
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...
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| Published in | Advances in computational mathematics Vol. 45; no. 2; pp. 611 - 630 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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02.04.2019
Springer Nature B.V |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Li, Kang Huang, Yunqing Chen, Yanping Chen, Chuanjun |
| Author_xml | – sequence: 1 givenname: Chuanjun surname: Chen fullname: Chen, Chuanjun organization: School of Mathematics and Information Sciences, Yantai University – sequence: 2 givenname: Kang surname: Li fullname: Li, Kang organization: School of Mathematics and Information Sciences, Yantai University, School of Data and Computer Science, Sun Yat-sen University – sequence: 3 givenname: Yanping surname: Chen fullname: Chen, Yanping email: yanpingchen@scnu.edu.cn organization: School of Mathematical Sciences, South China Normal University – sequence: 4 givenname: Yunqing surname: Huang fullname: Huang, Yunqing organization: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University |
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| Keywords | Nonlinear Sobolev equations 65N15 Error estimates Crank-Nicolson scheme 65N08 Two-grid finite element method |
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| Snippet | 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... |
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| SubjectTerms | 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 |
| Title | Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations |
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