Uniform \(l^1\) convergence in the Crank-Nicolson method of a linear integro-differential equation for viscoelastic rods and plates (Q2871182)

The author studies the numerical approximation of the Volterra integro-differential equation NEWLINE\[NEWLINE \frac{\partial u}{\partial t}+\int\limits_0^t A(t-s) Lu(s) ds=0, \quad t>0, \tag{1} NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0)=u_0 \tag{2} NEWLINE\]NEWLINE in a Hilbert space \(H\). These equations arise in the linear theory of isotropic viscoelastic rods and plates. The equation (1) 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 convergent in the \(l_t^1(0,\infty;H)\cap L^{\infty}_t(0,\infty;H)\) norm.