Any differential operator can be written as a finite difference matrix acting on a vector of values. I recently used it to solve $\Delta A = j$ by writing the Laplacian as a finite difference tridiagonal (second order approximation) matrix. I then used SciPy's
spsolve, which solves matrix equations of the form $Ax=b$ by not explicitly inverting the sparse matrix (which blows up really fast).
It seems to me, that because of the direct accuracy estimate you get from the finite difference scheme, and the extreme parallellization that can be achieved with methods like
spsolve, this would be an optimal way to solve this equation.
Is this the case (or is it at least a competitive method)? Why are all classes and lectures on numerical methods focusing on all other methods and I can only find sparse mention of the above described method, although it is both intuitive, simple to implement, and at least relatively fast.
I only saw finite difference matrices in light of eigenvalue calculations (in which I believe it is also quite optimal due to extreme optimization of the sparse eigenvalue calculation procedures), and thought of this "integration" technique myself. I might have just overlooked all books and other literature on the subject, hence this question directed to the experts.