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Uniform $ l^{1} $ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates


Author: Da Xu
Journal: Math. Comp. 83 (2014), 735-769
MSC (2010): Primary 65J08, 65D32; Secondary 45K05
DOI: https://doi.org/10.1090/S0025-5718-2013-02756-1
Published electronically: September 5, 2013
MathSciNet review: 3143690
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Abstract: We study the numerical approximation of a certain Volterra
integro-differential equation in Hilbert space which arises in the linear theory of isotropic viscoelastic rods and plates. The equation is discretized in time using a method based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step method, a convolution quadrature rule, constructed again from the trapezoidal rule, is used to approximate the integral term. The resulting scheme is shown to be convergence in the $ l_{t}^{1}(0,\infty ;H)\bigcap \,l_{t}^{\infty }(0,\infty ;\,H) $ norm.


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Additional Information

Da Xu
Affiliation: Department of Mathematics, Hunan Normal University, Changsha 410081, Hunan, People’s Republic of China
Email: daxu@hunnu.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-2013-02756-1
Keywords: Volterra integro-differential equation, time discretization, trapezoidal rule, $l_{t}^{1}$ convergence
Received by editor(s): January 22, 2011
Received by editor(s) in revised form: June 5, 2012
Published electronically: September 5, 2013
Additional Notes: This work was supported in part by the National Natural Science Foundation of China, contract grant numbers 11271123, 10971062.
Article copyright: © Copyright 2013 American Mathematical Society