Poisson Equation

The equation

Siméon-Denis Poisson generalised Laplace's equation by allowing a source term on the right:

∇²φ = f                            (general form)

∇²φ = −ρ/ε₀                        (electrostatics; f = −ρ/ε₀)
∇²φ_grav = 4πGρ                    (Newtonian gravity)
∇²p = ρ ∇·(... advective terms ...) (fluid pressure, incompressible Navier–Stokes)

Where Laplace's equation ∇²φ = 0 governs source-free equilibrium (the potential in a vacuum), Poisson's equation ∇²φ = f governs equilibrium with a known source distribution. The right-hand side f tells you what density of "stuff" — charge, mass, fluid divergence — sits at each point. The equation asks for the potential that responds to that density.

Like Laplace, Poisson is linear and second-order elliptic. The classification matters: solutions are smooth wherever the source is smooth, propagate "instantly" rather than along characteristics, and admit a unique Dirichlet solution on bounded domains.

Laplace versus Poisson

	Laplace	Poisson
Equation	∇²φ = 0	∇²φ = f
Type	Homogeneous linear elliptic	Inhomogeneous linear elliptic
Source	None (vacuum)	f ≠ 0 somewhere
Solutions	Harmonic functions	Particular + harmonic
Mean value	Holds for solutions	Fails generically (depends on source)
Maximum principle	Strict (interior extrema forbidden)	Modified (depends on sign of f)
Physical example	Potential in vacuum	Potential with charges present

Crucially, the general solution of Poisson's equation is a particular solution (a specific function with ∇²φ_p = f) plus any harmonic function:

φ = φ_particular + φ_homogeneous,    where ∇²φ_homogeneous = 0.

The harmonic part is free to absorb the boundary conditions. Pick φ_particular = G ∗ f (convolution with the Green's function), then adjust the harmonic part to match the prescribed boundary values.

The Green's function

The trick to solving Poisson's equation is the Green's function — the response to a point source at r':

∇² G(r, r') = −δ³(r − r')

3D free space:  G(r, r') = 1 / (4π |r − r'|)

2D free space:  G(r, r') = −(1/(2π)) log |r − r'|

1D free space:  G(x, x') = −(1/2) |x − x'|

In 3D, the Green's function is the familiar inverse-distance kernel — the potential of a unit point charge sits at 1/(4π|r − r'|), a fact known since Coulomb (1785). Once you have G, the solution for any source distribution f follows by superposition:

φ(r) = −∫ G(r, r') f(r') d³r'

(The minus sign comes from the convention ∇²G = −δ; under different sign conventions the minus moves around — always sanity-check the sign of the answer against a known case like a point charge.) Each infinitesimal volume f(r') d³r' contributes a 1/|r − r'| tail, and the integral adds up the contributions.

For bounded domains with prescribed boundary conditions, the free-space G must be modified. The method of images (a mirror source on the other side of the wall) gives the Dirichlet Green's function for half-spaces and disks; eigenfunction expansions give it for general boxes.

Electrostatics — the canonical example

Maxwell's first equation in differential form is

∇·E = ρ / ε₀.

Since ∇×E = 0 in electrostatics, E is a gradient: E = −∇φ. Substituting,

∇·(−∇φ) = ρ/ε₀     ⇒    ∇²φ = −ρ/ε₀.

That is Poisson's equation. The Green's function solution gives the potential due to an arbitrary charge density:

φ(r) = (1/(4πε₀)) ∫ ρ(r') / |r − r'| d³r'

For a point charge q at the origin, ρ(r') = q δ³(r'), and the integral collapses to

φ(r) = q / (4πε₀ |r|).

That is Coulomb's potential. For a continuous distribution, you simply integrate — every volume element contributes a 1/r tail weighted by its charge.

Worked example — uniformly charged ball

Take a ball of radius R with uniform charge density ρ_0. Compute the potential.

By spherical symmetry, φ(r) = φ(|r|) depends only on radial distance r. In spherical coordinates, ∇²φ = (1/r²)(d/dr)(r² dφ/dr). Inside the ball (r < R), Poisson's equation reads

(1/r²) d/dr (r² dφ/dr) = −ρ_0/ε₀.

Integrate:  r² dφ/dr = −(ρ_0 r³)/(3 ε₀) + C₁ → dφ/dr = −ρ_0 r /(3ε₀) + C₁/r²
At r=0, dφ/dr must be finite → C₁ = 0.
Integrate again: φ_inside = −(ρ_0 r²)/(6 ε₀) + C₂

Outside (r > R), ∇²φ = 0 (Laplace's equation), so by symmetry φ_outside = A/r + B. As r → ∞, φ → 0, so B = 0. And the total enclosed charge Q = ρ_0 · (4πR³/3) generates a Coulomb potential, so A = Q/(4πε₀ R³) × ... → φ_outside = Q/(4πε₀ r).

Match φ and dφ/dr at r = R (continuity of potential and field). The result:

φ(r) = (ρ_0 / (6 ε₀)) (3R² − r²)    for r ≤ R
φ(r) = (ρ_0 R³) / (3 ε₀ r)          for r ≥ R

At the centre, φ(0) = ρ_0 R² / (2 ε₀) — exactly 1.5 times the surface potential ρ_0 R² / (3 ε₀). Compute the electric field as E = −dφ/dr: inside, E = ρ_0 r / (3ε₀); outside, E = Q/(4πε₀ r²) = Coulomb's law for the total enclosed charge. Familiar Gauss-law results, all derived from the differential Poisson equation.

Variants and special cases

Screened Poisson (Yukawa)

Adding a mass-like term k² gives the screened Poisson equation:

(∇² − k²) φ = f

Green's function (3D):  G(r, r') = e^{−k|r − r'|} / (4π |r − r'|)

The exponential decay tames the 1/r tail. This is the Yukawa potential, used to model the nuclear strong force (Yukawa, 1935, with k = m_π c/ℏ), Debye screening in plasmas (1/k_D is the Debye length), and Thomas–Fermi screening of charges in metals. It is also the static (zero-frequency) limit of the Klein–Gordon equation.

Poisson on a 2D disk — log kernel

In 2D, the Green's function is logarithmic: G = −(1/2π) log|r − r'|. This means potentials of line charges (2D point sources) grow without bound at infinity — a 2D Coulomb world has no "potential at infinity" reference. Different physics, same equation.

Discrete Poisson — graph Laplacian

On a discrete graph or grid, the analogue of Poisson is L φ = f, where L is the graph Laplacian (degree matrix minus adjacency matrix on a graph; the 5-point stencil on a 2D grid). This shows up in spectral graph theory, network analysis, semi-supervised learning, and image processing.

Solving Poisson numerically

Poisson solves are the workhorse of computational physics — they appear in every incompressible-flow timestep, every electrostatic field calculation, every multigrid library.

Discretise on a grid with spacing h. The 5-point (2D) or 7-point (3D) Laplacian stencil reads

∇²φ_{i,j} ≈ (φ_{i+1,j} + φ_{i−1,j} + φ_{i,j+1} + φ_{i,j−1} − 4 φ_{i,j}) / h²

Setting this equal to f_{i,j} gives a large sparse linear system Aφ = b. Choices:

• Direct sparse solvers (e.g., Cholesky on the SPD matrix A) — exact but O(N^{3/2}) memory in 2D, more in 3D. Practical only for small grids.
• Jacobi / Gauss–Seidel / SOR iterations — simple but slow. Convergence rate ∝ 1 − O(1/N).
• Conjugate gradient — Krylov iteration. A is SPD so CG works. Roughly O(N^{1.5}) on Poisson; faster with a preconditioner.
• FFT — on periodic domains or boxes with simple boundary conditions, the FFT diagonalises the Laplacian. O(N log N) — extremely fast.
• Multigrid — O(N) cost via nested grid hierarchies. The standard "good" Poisson solver for general domains.

Stable-fluids algorithms in computer graphics run a Poisson solve every frame to enforce ∇·v = 0 via pressure projection. Real-time fluid effects in games and films rest on multigrid/FFT Poisson solvers that hit interactive speeds.

Where Poisson's equation appears

• Electrostatic potential. The textbook example. Given any charge distribution in vacuum (or in a dielectric, with adjusted ε), the potential satisfies ∇²φ = −ρ/ε. Solved by Green's function convolution or numerically.
• Newtonian gravity. ∇²Φ = 4πGρ_mass. The same structure as electrostatics with G/(−1/(4πε₀)) replaced by 4πG. The Green's function gives Newton's 1/r law for the potential.
• Incompressible flow pressure. The pressure in incompressible Navier–Stokes satisfies a Poisson equation derived by taking the divergence of the momentum equation and using ∇·v = 0. Every CFD code does this solve to project velocity onto the divergence-free subspace.
• Steady-state diffusion with sources. Heat with a heat source: ∇²T = −q/k. Solute with a source: ∇²C = −σ/D. The steady-state version of the diffusion-source equation.
• Image processing. Poisson image editing (Pérez–Gangnet–Blake 2003) for seamless cloning. Poisson surface reconstruction (Kazhdan et al. 2006) for converting point clouds to triangle meshes. Gradient-domain HDR rendering and tone mapping.
• Computational electromagnetics. The first step of every full-wave Maxwell solver in low-frequency regimes; capacitance/inductance extraction in circuit-level chip design.
• Plasma physics. Particle-in-cell (PIC) codes solve Poisson on a grid each timestep to compute the self-consistent electrostatic potential of millions of plasma particles.

Common mistakes

• Sign confusion in ∇²φ = -ρ/ε₀. The minus sign matters. A positive charge density makes ∇²φ negative — meaning φ has a local maximum at the charge, consistent with positive potential near positive charges. Wrong sign and you get the field pointing inward at a positive charge.
• Using the free-space Green's function on a bounded domain. 1/(4π|r − r'|) gives the right answer in infinite space, but on a bounded domain you need the Green's function consistent with the boundary conditions — typically obtained by method of images or eigenfunction expansion.
• Forgetting that Poisson + boundary determines uniqueness, not the equation alone. Without boundary conditions, the general solution is any particular Poisson solution plus an arbitrary harmonic function — infinitely many answers. Boundary data picks one.
• Missing the source-induced break of the mean value property. Harmonic (Laplace) solutions have the mean-value property. Poisson solutions do not — the mean of φ over a sphere around x exceeds (or falls below) φ(x) by an amount proportional to the integrated source in the ball.
• Treating ∇²φ in spherical coordinates as a sum of unweighted second partials. The metric factors in spherical/cylindrical coordinates matter — see Laplace's equation, same issue.
• Confusing Poisson's equation with the Poisson distribution. Different Poissons. The equation comes from the same Pierre-Simon, but the discrete probability distribution is unrelated to the PDE.