Uniform 𝑙¹ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates

Xu, Da

doi:10.1090/S0025-5718-2013-02756-1

Uniform $l^{1}$ convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates
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Math. Comp. 83 (2014), 735-769 Request permission

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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