A Robust Finite Element Method for Elastic Vibration Problems

Yuling Guo; Jianguo Huang

doi:10.1515/cmam-2018-0197

Article

A Robust Finite Element Method for Elastic Vibration Problems

Yuling Guo and Jianguo Huang

Published/Copyright: August 28, 2019

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From the journal Computational Methods in Applied Mathematics Volume 20 Issue 3

Abstract

A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the $P_{1}$ $P 1$ -continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. $L^{2}$ $L 2$ norm) is bounded by $O (h + k)$ $O ⁢ ( h + k )$ (resp. $O (h^{2} + k)$ $O ⁢ ( h 2 + k )$ ), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison.

Keywords: Elastic Vibration Problems; CDG Method; CR Element; Robust Method; Error Analysis

MSC 2010: 65M60; 65M12; 74B05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11571237

Funding statement: The work was supported by NSFC (Grant No. 11571237).

Acknowledgements

The authors are grateful to the referees for valuable and constructive comments and suggestions, which greatly improved an early version of the paper.

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Received: 2018-08-05

Revised: 2019-03-07

Accepted: 2019-07-08

Published Online: 2019-08-28

Published in Print: 2020-07-01

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Articles in the same Issue

https://doi.org/10.1515/cmam-2018-0197

Keywords for this article

Elastic Vibration Problems; CDG Method; CR Element; Robust Method; Error Analysis