We investigate a global-in-time variational approach to abstract evolution
by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz [ESAIM: COCV
14 (2008) 494–516]. In particular, we focus on
gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow.
Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from [S. Conti and M. Ortiz, J. Mech. Phys. Solids
56 (2008) 1885–1904.].