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

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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.## References

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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
- 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.
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3143690