Space–Time Discretization of a Wave Equation with Fractional Kelvin–Voigt Damping

This work is concerned with the numerical treatment of a wave equation with fractional Kelvin–Voigt damping, where the viscoelastic contribution is described by a Caputo derivative in time acting on the elliptic part of the model. Such models are of interest because memory effects produce hereditary damping and reduced regularity near the initial time, which makes both the analysis and the numerical discretization more delicate than in the classical wave equation. We study the problem on a bounded convex domain under homogeneous Dirichlet boundary conditions and derive a solution representation that is suitable for regularity analysis. Based on this representation, we establish stability and smoothing estimates for both homogeneous data and forcing terms, with particular attention to the influence of nonsmooth initial data. For the spatial discretization, we employ a continuous Galerkin finite element method with piecewise linear elements and prove error estimates that are explicit in the regularity of the initial displacement, initial velocity, and source term. We show that the fully discrete approximation inherits the regularity-dependent behavior of the continuous problem and achieves optimal convergence in space together with second-order accuracy in time under appropriate assumptions on the data. Several numerical experiments are presented to illustrate the theoretical findings and to confirm the predicted convergence rates, thereby supporting the effectiveness of the proposed space–time discretization.