In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_t+{\int }_0^t\beta (t-s)Au\left(s\right)ds=0,u\left(0\right)=v,t>0,$ where A is an elliptic partial-differential operator and $\beta \left(t\right)$ is positive, nonincreasing and log-convex on $(0,\infty )$ with $0\le \beta \left(\infty \right)<\beta \left(0^{+}\right)\le \infty$. Error estimates are derived in the norm of $L_t^1(0,\infty ;L_x^2)$, and some estimates for the first order time derivatives of the errors are also given.

EUDML-ID : urn:eudml:doc:287834
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@article{702901,
     title = {Uniform $L^1$ error bounds for semi-discrete finite element solutions of evolutionary integral equations},
     booktitle = {Applications of Mathematics 2012},
     series = {GDML\_Books},
     publisher = {Institute of Mathematics AS CR},
     address = {Prague},
     year = {2012},
     pages = {144-162},
     mrnumber = {MR3204408},
     zbl = {1313.65336},
     url = {http://dml.mathdoc.fr/item/702901}
}

Lin, Qun; Xu, Da; Zhang, Shuhua. Uniform $L^1$ error bounds for semi-discrete finite element solutions of evolutionary integral equations, dans Applications of Mathematics 2012, GDML_Books,  (2012), pp. 144-162. http://gdmltest.u-ga.fr/item/702901/