Potential Theory / Harmonic Fields

Laplace Equation Field Studio

This simulation solves the two-dimensional Dirichlet problem for Laplace's equation, ∇²φ = 0, using red-black Gauss-Seidel relaxation with successive over-relaxation. Paint fixed-potential conductors, shape the outer boundary, and watch the charge-free region settle into a harmonic potential whose field lines follow E = -∇φ.

← Back to Mathematical Simulations

Live State

Max Residual 0

Sweeps / Sec 0

Mean |E| 0

Peak |E| 0

Probe φ 0

Probe |E| 0

How To Read It

Warm colors are positive potential, cool colors are negative potential, and pale midtones sit near zero. Thin contour curves are equipotentials, and the arrow field follows the electric field direction.

Outside the painted conductors and fixed outer boundary, the solver enforces the discrete harmonic average: each cell relaxes toward the average of its four nearest neighbors.

Use drag to paint electrodes and scroll over the canvas to change brush size.

Presets

Canonical Boundary Value Problems

These presets stay within the Laplace regime by imposing fixed potentials on the box boundary or on idealized conductors, then solving only the charge-free space between them.

Solver

Relaxation Controls

Grid Resolution 160 x 160 Higher grids resolve sharper curvature but need more sweeps to converge.

Relaxation Factor 1.78 ω = 1 gives plain Gauss-Seidel; values closer to 2 accelerate convergence until instability appears.

Sweeps Per Frame 8 Each sweep updates both checkerboard parities once.

Color Span 1.15 Sets the symmetric potential range mapped into the diverging color palette.

Electrodes

Paint Fixed Potentials

Painting adds idealized conductors with prescribed potential. The surrounding space then solves Laplace's equation again, so the resulting field is always harmonic away from those fixed cells.

Brush Radius 10 px Radius is measured in grid cells rather than screen pixels so the stencil stays physically consistent.

Electrode Voltage 1.00 The positive and negative tools use this magnitude with opposite sign.

Boundary

Outer Box Potentials

The simulation box is a Dirichlet boundary. These sliders directly fix the outer edge values and immediately re-solve the interior.

Top Edge 0.00

Right Edge 0.00

Bottom Edge 0.00

Left Edge 0.00

Rendering

Visual Layers

Contour Levels 10 Equipotential contours are generated with a marching-squares pass over the harmonic field.

Show Contours

Show Field Arrows

Auto Color Scale

Glow Shading

Readout

Convergence And Diagnostics

RMS Residual0.000000

Free Cells0

Fixed Cells0

Max |φ|0.000

Harmonic Cells0%

FPS0.0

Residual History (log scale)

Method

Why This Is Laplace, Not Poisson

This page solves the charge-free region between fixed conductors and fixed walls. If you wanted free charge density inside the domain, the governing equation would become Poisson's equation instead.

∇²φ = ∂²φ/∂x² + ∂²φ/∂y² = 0

φ_i,j^* = (φ_i+1,j + φ_i-1,j + φ_i,j+1 + φ_i,j-1) / 4

φ_i,j ← φ_i,j + ω (φ_i,j^* - φ_i,j)

E = -∇φ

Discrete harmonic average. Every non-fixed cell relaxes toward the average of its four cardinal neighbors, which is the standard five-point finite-difference stencil for the 2D Laplacian.

Red-black ordering. Alternating checkerboard updates make Gauss-Seidel easy to parallelize conceptually while still using freshly updated values inside each sweep.

Successive over-relaxation. The ω slider intentionally exposes the classic convergence-speed tradeoff: too low is slow, too high overshoots.

Physical interpretation. Equipotential contours are orthogonal to the electric field in the continuous limit, so the arrows should cut across contour bands rather than flow along them.

Accuracy note: painted objects are fixed-potential conductors, not literal point charges. That keeps the solved region within Laplace's equation while still producing canonical electrostatic field patterns.