Elliptic PDE · Iterative solver

Laplace
Equation

∇²u = 0 · u(x,y) on boundary fixed

The Laplace equation describes steady-state heat, electrostatics, and fluid potential. The Gauss–Seidel method turns the PDE into a local average — each interior point becomes the mean of its four neighbours.

Heat diffusion Electrostatics Potential flow Image processing

Gauss–Seidel update

u_i,j = u_i+1,j + u_i-1,j + u_i,j+1 + u_i,j-1 4

Sweep left→right, bottom→top · use newest values immediately

            ⬆ top = 1 ⬇ bottom = 0 ⬅ left = 0 ➡ right = 0
          

Algorithm · Gauss–Seidel

How it works

Discrete Laplace equation

∇²u = 0 → u_i+1,j − 2u_i,j + u_i-1,jh_x² + u_i,j+1 − 2u_i,j + u_i,j-1h_y² = 0

With uniform grid (h_x = h_y), the update is simply the average of four neighbours.

u_i,j = u_i+1,j + u_i-1,j + u_i,j+1 + u_i,j-14

Convergence check

Stop when max(|u^k+1 − u^k|) < ε

Typical ε = 1e-6 · converges in ~1800 iterations for 50×50 grid

Set up the domain

Define [xmin,xmax]×[ymin,ymax] and fix boundary values.

Discretise

Create a grid with nx × ny points. Initialise interior to 0 (or any guess).

Gauss–Seidel sweep

Loop over every interior point, update using the newest neighbour values (in-place).

Check convergence

After each full sweep, compute the maximum change. If below tolerance, stop.

Visualise

Plot the solution as a contour or 3D surface — smooth harmonic function.

Interactive solver

Run Gauss–Seidel

Grid dimension N (N×N)

Tolerance (ε)

Top boundary

Bottom boundary

Left boundary

Right boundary

contour (heatmap) 3D surface

Ready

LaplaceEquation