Uniform approximation of non-autonomous evolution equations

We study $L^2$-maximal regularity for non-autonomous evolution equations of the form \begin{equation}\label{Abstract equation} \dot u(t)+\mathcal A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation} where $\mathcal A(t),\ t\in [0,T]$ arise from a non-autonomous sesquilinear forms $a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ $L^2-$maximal regularity result is proved recently in \cite{Ar-Mo15} when $a$ is H\"older continuous of type $\alpha>1/2.$ In this paper we recover the same results by an approximation method developed in \cite{ELLA13}, \cite{LASA14} and \cite{ELLA15}. The method uses an appropriate approximation $\mathcal A_\Lambda(\cdot)$ of $\mathcal A(\cdot)$ for which \begin{equation}\label{Abstract equation approx} \dot u_{\Lambda}(t)+\mathcal A_{\Lambda}(t)u_{\Lambda}(t)=f(t)\ \ t\in[0,T],\ \ u_{\Lambda}(0)=u_0 \end{equation} has $L^2$-maximal regularity where $\Lambda$ is a subdivision of $[0,T].$ Moreover, under a little more assumptions on the modulus of continuity we show that the solutions of (\ref{Abstract equation approx}) converges in $L^2(0,T,V)\cap H^1(0,T,H)\cap C(0,T,V)$ uniformly on the initial datas $(u_0,f)$ to the solution of (\ref{Abstract equation}) as $|\Lambda| \rightarrow 0.$