Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel

We study the second-order continuous time Galerkin approximation of the solution ut + ∫0t β(t — s)Au(s) ds = 0, u(0) = v, t > 0, where A is an elliptic partial-differential operator and β(t) is completely monotonic and locally integrable, but not constant. Error estimates are derived in the norm of Lt1(0, ∞; Lx2).