I'm solving a simple 1D heat diffusion problem $$u_t=u_{xx},\quad \Omega\times[0,T]$$ $$u=0\quad, \partial\Omega\times [0,T]$$ $$u(x,0)=f$$using a fully discrete galerkin finite element method.  This effectively reduces solution of the parabolic problem into solving a recursive sequence of elliptic problems $\{U_h\}^t$, each using the solution from the previous timestep $\{U_h\}^{t-1}$.  I have been told that it is not a good idea to use the initial condition $f$ as $\{U_h\}^0$.  Instead, it has been recommended to me to use the projection of $f$ onto the variational space chosen for the weak formulation as $\{U_h\}^0$.  I have run some numerical experiments using $f$ instead of its projection and I observe that The errors still approach zero as the spatial and temporal step sizes approach zero.The orders of convergence are still preserved.Heuristically, it seems to me that projecting the initial condition would introduce more error into the problem than using the $f$ as is.  These numerical experiments seem to indicate that it really isn't necessary to obtain consistent, stable solutions (at least, for this problem).  Are there situations in which it is crucial to project the initial condition, or is more a matter of taste/mathematical formality?
Is it necessary to project the initial condition onto the variational space in a fully discrete galerkin method?
pde;finite element;parabolic pde