# Article

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Keywords:
evolutionary integral equation; semi-discrete finite element solution; uniform error bound; Galerkin approximation; elliptic partial-differential operator
Summary:
In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t}+\int_{0}^{t}\beta (t-s) Au(s)ds=0,u(0)=v,t>0,$ where A is an elliptic partial-differential operator and $\beta(t)$ is positive, nonincreasing and log-convex on $(0,\infty)$ with $0\leq\beta(\infty)<\beta(0^{+})\leq\infty$. Error estimates are derived in the norm of $L^{1}_{t}(0,\infty;L^{2}_{x})$, and some estimates for the first order time derivatives of the errors are also given.

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