This visualization isolates the core multigrid idea on a one-dimensional Poisson problem −u″=f, with u(0)=u(1)=0 and a known exact solution made from low- and high-frequency sine modes. Starting from zero, ordinary Jacobi and Gauss–Seidel relaxation quickly damp the high-frequency error, which is why they are called smoothers. The remaining smooth, low-frequency error decays very slowly on the fine grid. A V-cycle fixes that by computing the residual r=f−Au, restricting it to coarser grids where the smooth fine-grid error becomes more oscillatory and cheaper to solve, then prolonging the correction back to the fine grid. The top panel plots the current error u−u* for Jacobi, Gauss–Seidel, and multigrid; the lower panel shows logarithmic L2-error histories.
Who it's for: Students in numerical PDEs and scientific computing learning relaxation, smoothers, residual correction, restriction/prolongation, and why multigrid can be grid-size independent.
Key terms
Multigrid
Jacobi relaxation
Gauss-Seidel
V-cycle
Coarse-grid correction
Residual
Restriction
Prolongation
How it works
A 1D Poisson multigrid visualization: Jacobi and Gauss-Seidel relaxations smooth high-frequency error, while a V-cycle uses residual restriction and coarse-grid correction to remove low-frequency error rapidly.
Key equations
Solve −u″=f, u(0)=u(1)=0. Relaxation damps oscillatory error; multigrid computes residual r=f−Au, restricts it to coarse grids, solves Ae=r, and prolongs the correction u←u+e.
Frequently asked questions
Why do Jacobi and Gauss-Seidel remove high-frequency error first?
Local relaxation updates each point from nearby values. Oscillatory error creates large local mismatches, so it is damped rapidly. Smooth low-frequency error looks locally almost consistent, so fine-grid relaxation barely changes it.
What does the coarse grid do?
A smooth error on the fine grid appears less smooth relative to a coarser grid. Solving the residual equation there captures a global correction cheaply, then interpolation brings that correction back to the fine grid.
Is this a full production multigrid solver?
No. It is a compact geometric V-cycle for a 1D Poisson operator. Real solvers add better transfer operators, boundary-condition handling, adaptive stopping, pre/post-smoother choices, and extensions to complex 2D/3D meshes.