A Crank-Nicolson ADI Galerkin approach for numerical simulation of nonlinear Volterra integrodifferential models arising in viscoelastic plates

This work develops an efficient and powerful approach for nonlinear Volterra integrodifferential equations arising, for example, in the investigation of plates composed of isotropic viscoelastic materials. We focus on convolution kernels that lack explicit analytic representations and are instead characterized through their Laplace transforms. The proposed scheme combines a Crank-Nicolson approach together with the second-order quadrature of convolution types for temporal localization and employs a finite element Galerkin approximation in spatial variables. To alleviate computational cost in multi-dimensional settings, the ADI (i.e., alternating direction implicit) approach is incorporated. Convergence and stability of the full localization approach are established via a discrete energy argument, yielding the optimal error estimate O(τ2+hr+1) under appropriate assumptions, where h and τ denote the spatial and temporal mesh sizes. Numerical experiments substantiate the robustness of the approach and confirm the predicted convergence rates.