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We prove $L^2$ $$\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)\text{for}\text{a.e.}t\in [0,T],u\left(0\right)=u_0,$$ where the operator $A\left(t\right)$ arises from a time depending sesquilinear form $𝔞(t,·,·)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of...