Numerical analysis of Volterra integro-differential equations for viscoelastic rods and membranes.

Abstract We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm L t 1 (0 , ∞ ; L x 2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds. [ABSTRACT FROM AUTHOR]