A Posteriori Error Analysis of Finite Element Approximation to an Evolution Equation with a Positive-Type Memory Term

This paper develops reliable a posteriori error estimates for finite element approximation of an evolution equation with a positive-type memory kernels on bounded convex polygonal or polyhedral domains, including both smooth and weakly singular kernels. For the weakly singular kernel $t^{α-1}/Γ(α),\; 0<α<1,$ the model captures the transition from parabolic diffusion to wave propagation, recovering the corresponding limiting equations as $α\to0^+$ and $α\to1^-$, respectively, with the memory term governing the intermediate dynamics. The spatial discretization employs time-dependent finite element spaces, while the backward Euler method combined with numerical quadrature is used for time integration. A unified \emph{a posteriori} analysis is established for both semidiscrete and fully discrete approximations. Elliptic reconstruction yields optimal spatial error estimates, while a memory-lag estimate preserves the essential cancellation in the convolution term, yielding a first-order temporal estimator whose kernel dependence is characterized by its integral. The resulting estimators are robust, with constants independent of the final time, spatial mesh size, and time step. Their kernel dependence involves only computable quantities, avoiding exponential growth in time. Moreover, for the weakly singular kernel, the $α$-dependent factors remain uniformly bounded for $α\in(0,1).$ Numerical experiments on a convex domain validate the theoretical results, and a space-time adaptive algorithm is also demonstrated on an L-shaped domain featuring a corner singularity and two temporal layers.