When the false transient method could make an elliptic PDE easier to solve numerically? I think that I do not fully understand the false transient method. This method consists on introducing a time derivative to an elliptic PDE to convert it to a parabolic PDE. However, it does not make sense for me, I do not get why it could be easier to solve a "transient" problem compared to a steady problem. Note that I do not know too much about numerically solving PDEs but intuitivelly I guess that transient problems are usually harder to solve.
Thanks!
 A: Solving an elliptic PDE numerically is actually solvin some set of linear equations. For example, consider this scheme for Poissons equation $\Delta u = f$ in 2D:
$$
\frac{u_{m,n-1} + u_{m-1,n} - 4 u_{m,n} + u_{m,n+1} + u_{m+1,n}}{\Delta x^2} = f_{m,n}\\
u_{0,n} = u_{M,n} = u_{m,0} = u_{m,N} = 0.
$$
It is basically a (sparse) system of linear equations with $(N-1)(M-1)$ unknowns $u_{m,n}$.
Though for this particular case there's a direct method of discrete sine transform, generally this kind of systems of equations are solved iteratively.
Consider simple Jacobi method for this equation:
$$
\frac{u^{(p+1)}_{m,n} - u^{(p)}_{m,n}}{\omega} = \frac{u_{m,n-1}^{(p)} + u_{m-1,n}^{(p)} - 4 u_{m,n}^{(p)} + u_{m,n+1}^{(p)} + u_{m+1,n}^{(p)}}{\Delta x^2} - f_{m,n}
$$
Since it is iterative, I've added $(p)$ to denote iteration number. Note this scheme is exactly the same as explicit method for solving heat equation $u_t = \Delta u - f$:
$$
\frac{u^{p+1}_{m,n} - u^{p}_{m,n}}{\Delta t} = \frac{u_{m,n-1}^{p} + u_{m-1,n}^{p} - 4 u_{m,n}^{p} + u_{m,n+1}^{p} + u_{m+1,n}^{p}}{\Delta x^2} - f_{m,n}
$$
But now $\omega$ became the timestep $\Delta t$ and $p$ denotes the current number, not the iteration number. But the numerical scheme left the same.
It appears that specific numerical schemes for parabolic equations (alternating directions, for example) perform better than general iterative methods for sparse linear systems, because they can exploit some physical properties of the system of equations.
So you're not quite right saying that "transient problems are usually harder to solve", they are not. Usually, boundary problems are harder than initial value problems. But for any elliptical problem can be easily converted into heat-like equation by introducing some kind of artificial time that is nothing but a notation for some iterative process.
In general if you have a parabolic problem, and solve it for long enouth its solution can be considered as a solition to corresponding problem with all time derivatives being canceled out.
$$
u_t = \Delta u - f \qquad\rightarrow\qquad 0 = \Delta u\big|_{t \rightarrow \infty} - f.
$$
That's due to dissipative nature of parabolic equations.
