I am looking at a heat initial value problem\begin{align}\frac{\partial u}{\partial t}-\nabla^2u = f\quad&\text{in}\quad \Omega\times(0,T)\\u = g \quad&\text{on}\quad \partial\Omega\times(0,T)\\u = u_0\quad&\text{in}\quad \Omega\times\{0\}\end{align}which I have succesfully solved with the finite element method along with a time-stepping scheme, so that is not my question. My question is however, it seems to work to just solve the boundary value problem\begin{align}-\nabla^2u = f\quad&\text{in}\quad \Omega\\u = g \quad&\text{on}\quad \partial\Omega\\\end{align}for various fixed times, e.g. $f(x,y,0),f(x,y,0.1),$ etc. This seems to give a good approximation of the solution to the initial value problem, and I don't understand why. This simply ignores the time-derivative it seems to me (of course, the functions $f$ are the same for both problems so the information of the time-derivative is not lost, but it still confuses me). Is this simply the method of lines?Thanks
Solving Initial Value problem ignoring the time-derivative
finite element;time integration