Linear System: Direct vs Iterative

What you are seeing: the discrete 1D Poisson problem $-u'' = \sin(\pi x)$ on $N$ grid points. The same right-hand side is fed to a Thomas direct solver (exact, $O(N)$ ), Jacobi, Gauss-Seidel, and conjugate gradient. The lower panel tracks the residual norm vs iteration count; CG converges in at most $N$ steps in exact arithmetic.

Figure 1. Solution and residual history for four solvers on

-u'' = f

N (grid)32

solver

WHAT TO TRY

Switch the iterative solver between Jacobi, Gauss-Seidel and conjugate gradient: each drives the residual down at a different rate, and the log-residual plot shows conjugate gradient winning by far.
Raise the grid size N: the iterative methods slow down (their iteration count grows with N) while the Thomas direct solver stays exact in O(N). That scaling is why solver choice matters.
Watch the solution converge toward the exact parabola: the iterate creeps up from below as the residual shrinks, the visible meaning of an iterative method homing in on the answer.