Volume 13, Issue 4
A Two-Grid Algorithm of Fully Discrete Galerkin Finite Element Methods for a Nonlinear Hyperbolic Equation

Kang Li & Zhijun Tan

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 1050-1067.

Published online: 2020-06

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  • Abstract

A two-grid finite element approximation is studied in the fully discrete scheme obtained by discretizing in both space and time for a nonlinear hyperbolic equation. The main idea of two-grid methods is to use a coarse-grid space ($S_H$) to produce a rough approximation for the solution of nonlinear hyperbolic problems and then use it as the initial guess on the fine-grid space ($S_h$). Error estimates are presented in $H^1$-norm, which show that two-grid methods can achieve the optimal convergence order as long as the two different girds satisfy $h$ = $\mathcal{O}$($H^2$). With the proposed techniques, we can obtain the same accuracy as standard finite element methods, and also save lots of time in calculation. Theoretical analyses and numerical examples are presented to confirm the methods.

  • Keywords

Nonlinear hyperbolic equation, two-grid algorithm, finite element method, fully discrete, error estimate.

  • AMS Subject Headings

65N12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-13-1050, author = {Kang Li , and Zhijun Tan , }, title = {A Two-Grid Algorithm of Fully Discrete Galerkin Finite Element Methods for a Nonlinear Hyperbolic Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {4}, pages = {1050--1067}, abstract = {

A two-grid finite element approximation is studied in the fully discrete scheme obtained by discretizing in both space and time for a nonlinear hyperbolic equation. The main idea of two-grid methods is to use a coarse-grid space ($S_H$) to produce a rough approximation for the solution of nonlinear hyperbolic problems and then use it as the initial guess on the fine-grid space ($S_h$). Error estimates are presented in $H^1$-norm, which show that two-grid methods can achieve the optimal convergence order as long as the two different girds satisfy $h$ = $\mathcal{O}$($H^2$). With the proposed techniques, we can obtain the same accuracy as standard finite element methods, and also save lots of time in calculation. Theoretical analyses and numerical examples are presented to confirm the methods.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0200}, url = {http://global-sci.org/intro/article_detail/nmtma/16966.html} }
TY - JOUR T1 - A Two-Grid Algorithm of Fully Discrete Galerkin Finite Element Methods for a Nonlinear Hyperbolic Equation AU - Kang Li , AU - Zhijun Tan , JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1050 EP - 1067 PY - 2020 DA - 2020/06 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0200 UR - https://global-sci.org/intro/article_detail/nmtma/16966.html KW - Nonlinear hyperbolic equation, two-grid algorithm, finite element method, fully discrete, error estimate. AB -

A two-grid finite element approximation is studied in the fully discrete scheme obtained by discretizing in both space and time for a nonlinear hyperbolic equation. The main idea of two-grid methods is to use a coarse-grid space ($S_H$) to produce a rough approximation for the solution of nonlinear hyperbolic problems and then use it as the initial guess on the fine-grid space ($S_h$). Error estimates are presented in $H^1$-norm, which show that two-grid methods can achieve the optimal convergence order as long as the two different girds satisfy $h$ = $\mathcal{O}$($H^2$). With the proposed techniques, we can obtain the same accuracy as standard finite element methods, and also save lots of time in calculation. Theoretical analyses and numerical examples are presented to confirm the methods.

Kang Li & Zhijun Tan. (2020). A Two-Grid Algorithm of Fully Discrete Galerkin Finite Element Methods for a Nonlinear Hyperbolic Equation. Numerical Mathematics: Theory, Methods and Applications. 13 (4). 1050-1067. doi:10.4208/nmtma.OA-2019-0200
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