Uniform <i>L</i> 1 error estimates for the semidiscrete finite element solutions of a parabolic integro-differential equation

Abstract We analyze the semidiscrete piecewise linear finite element methods for parabolic integro-differential equations with some nonempty spectral sets that arise in the modeling of viscoelasticity fluid in porous media. The long time error estimates are obtained with smooth initial data. More precisely, by introducing piecewise linear finite element residual resolvent solution and its spectral resolution we derive optimal order $ L^{1}(0,\infty ;L^{2}(\varOmega )) $ error estimates with smooth initial data. Numerical experiments confirm the convergence rates of the proposed estimates.