approximation of weakly differentriable bochner functions

Question

Given a function $u\in L^2(0,T;H^1(\Omega))$ with $u_t\in L^2(0,T;(H^1(\Omega))^*)$. Can we approximate $u$ by functions $u^k$ with $$u^k=\sum\limits_{i=1}^{n(k)}c_i^k\phi_i^k,\text{ where } c_i^k\in H^1(\Omega)\text{ and }\phi_i^k\in C_c^\infty(0,T),$$ such that $$\|u-u^k\|_{L^2(0,T;H^1)}\to 0\text{ and }\|u_t-u_t^k\|_{L^2(0,T;(H^1)^*)}\to 0$$ as $k\to\infty$?
I tried it by mollifying simple functions of the form $$f=\sum\limits_{i=1}^nc_i\chi_{E_i},\text{ where }c_i\in H^1(\Omega)\text{ and }E_i\subset (0,T),$$ which are dense in $L^2(0,T;H^1(\Omega))$, with respect to $t$. This yields a sequence of functions of the demanded form and the convergence in $L^2(0,T;H^1(\Omega))$, but I failed to show the $L^2(0,T;(H^1)^*)$-convergence.