Poisson's Equation · aiwiki.page

Poisson's equation is a linear, second-order partial differential equation that relates the spatial variation of an unknown scalar function to a prescribed source. Commonly written as $-\Delta u=f$ , it is a fundamental model for potential fields and a standard boundary-value problem in mathematics. Here $u$ is the unknown function, $f$ is the source, and $\Delta$ is the Laplace operator. A complete problem also specifies the spatial domain and appropriate boundary conditions. (github.com)

Mathematical form

On a domain $\Omega$ in Euclidean space $\mathbb R^n$ , the equation is

-\Delta u(\mathbf x)=f(\mathbf x), \qquad \mathbf x\in\Omega.

The Laplace operator, or Laplacian, is the sum of the second partial derivatives in Cartesian coordinates:

\Delta u=\sum_{i=1}^{n}\frac{\partial^2u}{\partial x_i^2}.

In three dimensions this becomes $u_{xx}+u_{yy}+u_{zz}$ . The alternative convention $\Delta u=f$ is equally common; changing conventions requires changing the sign of the source. (github.com)

When $f=0$ , the equation reduces to Laplace's equation, and its solutions are called harmonic functions. Thus Poisson's equation is the inhomogeneous counterpart of Laplace's equation: it includes a source rather than describing only a source-free region. (web.stanford.edu)

Historical origin

The equation is named after Siméon-Denis Poisson (1781–1840). In 1813, Poisson investigated the potential inside attracting masses, obtaining results that subsequently found applications in electrostatics. The historical problem concerned extending potential theory from regions outside a mass distribution to regions containing the sources themselves. (mathshistory.st-andrews.ac.uk)

Boundary conditions and uniqueness

The source alone generally does not determine a unique solution. Boundary conditions specify how the unknown field interacts with the boundary $\partial\Omega$ . Two principal types are:

Dirichlet conditions: prescribe the value, $u=g$ .
Neumann conditions: prescribe the outward normal derivative, $\partial u/\partial n=h$ .

A mixed problem imposes Dirichlet conditions on one part of the boundary and Neumann conditions on another. The normal derivative is $\nabla u\cdot\mathbf n$ , where $\mathbf n$ is the outward unit normal and $\nabla u$ is the gradient. (docs.fenicsproject.org)

For the pure Neumann problem

-\Delta u=f,\qquad \frac{\partial u}{\partial n}=h,

the data must satisfy the compatibility condition

\int_\Omega f\,dx+ \int_{\partial\Omega}h\,dS=0.

This follows by integrating the equation and converting the integral of the Laplacian into a boundary flux. It expresses a balance between the interior source and the boundary data. On a connected domain, a Neumann solution is determined only up to an additive constant; a normalization such as $\int_\Omega u\,dx=0$ removes this freedom. (web.stanford.edu)

For homogeneous Dirichlet conditions on a bounded domain, uniqueness follows from an energy argument. The difference $w$ between two solutions satisfies the homogeneous equation and zero boundary data. Multiplication by $w$ and integration by parts give

\int_\Omega|\nabla w|^2\,dx=0,

which forces $w=0$ . This argument also works in the weak formulation. (math.stanford.edu)

Weak and variational formulations

A classical solution has the second derivatives needed to satisfy the equation pointwise. A weak solution instead satisfies an integral identity, permitting less regular functions and source data.

For zero Dirichlet conditions, multiply $-\Delta u=f$ by a test function $v$ and integrate by parts. The resulting problem is

\int_\Omega\nabla u\cdot\nabla v\,dx = \int_\Omega fv\,dx \quad\text{for every }v\in H_0^1(\Omega).

Here $H_0^1(\Omega)$ is a Sobolev space of functions with square-integrable weak first derivatives and zero boundary trace. Only first derivatives appear in this formulation. With Neumann conditions on part of the boundary, an additional boundary integral enters the right-hand side. (docs.fenicsproject.org)

For a bounded Lipschitz domain and $f\in L^2(\Omega)$ , the homogeneous Dirichlet problem has a unique weak solution in $H_0^1(\Omega)$ . The same formulation identifies the solution as the minimizer of

J(v)= \frac12\int_\Omega|\nabla v|^2\,dx -\int_\Omega fv\,dx.

This connects the equation to the calculus of variations. Existence and uniqueness at this level do not automatically imply that all second derivatives exist classically: additional regularity depends on the source and the boundary. (math.stanford.edu)

Fundamental solutions and Green's functions

A fundamental solution describes the response to a unit point source. For the convention $-\Delta u=f$ , it satisfies

-\Delta\Gamma=\delta_0,

where $\delta_0$ is the Dirac delta distribution concentrated at the origin. In two and three dimensions,

\Gamma(\mathbf x)= \begin{cases} -\dfrac{1}{2\pi}\log|\mathbf x|,&n=2,\\[6pt] \dfrac{1}{4\pi|\mathbf x|},&n=3. \end{cases}

For sufficiently regular, compactly supported sources in the whole space, a solution is obtained by convolution:

u(\mathbf x)= \int_{\mathbb R^n} \Gamma(\mathbf x-\mathbf y)f(\mathbf y)\,d\mathbf y.

A Green's function incorporates the domain's boundary conditions as well as the point-source response. For zero Dirichlet data, it gives a representation of the form

u(\mathbf x)=\int_\Omega G(\mathbf x,\mathbf y)f(\mathbf y)\,d\mathbf y.

The kernel is therefore sensitive to the domain and boundary conditions, not merely to the local differential operator. (web.stanford.edu)

Physical applications

Electrostatics

In electrostatics, the electric field is expressed through the electric potential $\phi$ as

\mathbf E=-\nabla\phi.

In vacuum, a prescribed electric charge density $\rho_e$ produces

\Delta\phi=-\frac{\rho_e}{\varepsilon_0},

where $\varepsilon_0$ is the vacuum permittivity. Solving the equation determines the potential; taking its negative gradient then determines the electric field. Boundary values can represent potentials imposed on surrounding conductors. (farside.ph.utexas.edu)

Newtonian gravity

In Newtonian gravity, the gravitational potential $\Phi$ satisfies

\Delta\Phi=4\pi G\rho_m, \qquad \mathbf g=-\nabla\Phi,

where $G$ is the gravitational constant, $\rho_m$ is mass density, and $\mathbf g$ is gravitational acceleration. For a localized mass distribution with the potential chosen to vanish at infinity,

\Phi(\mathbf x)= -G\int_{\mathbb R^3} \frac{\rho_m(\mathbf y)} {|\mathbf x-\mathbf y|}\,d\mathbf y.

This integral and the differential equation are complementary descriptions of the same potential field. (farside.ph.utexas.edu)

Steady-state thermal models

A constant-coefficient thermal model can take the form

-\kappa\Delta T=q,

where $T$ is temperature, $\kappa$ is thermal conductivity, and $q$ is a prescribed heat-source density. Such a model is used, for example, in optimizing a heating pattern to obtain a desired temperature distribution. (scientificcomputing.github.io)

Numerical solution

The finite difference method replaces derivatives by grid-based approximations. For $-\Delta u=f$ on a uniform two-dimensional grid of spacing $h$ , the standard five-point stencil is

\frac{ 4u_{i,j}-u_{i+1,j}-u_{i-1,j} -u_{i,j+1}-u_{i,j-1} }{h^2}=f_{i,j}.

With Dirichlet conditions, this produces a sparse, symmetric positive-definite system of linear equations. Pure Neumann conditions instead leave a constant null space and require compatible data. (web.stanford.edu)

The finite element method discretizes the weak formulation using a finite-dimensional function space. It is particularly useful for meshes adapted to the domain's geometry. The resulting algebraic problem can be solved with direct or iterative linear solvers. (docs.fenicsproject.org)

Multigrid methods accelerate solution by combining relaxation on fine grids with corrections on coarser grids. They are widely studied for the discrete Poisson problem because errors that are difficult to remove on one grid can be addressed on another scale. (math.uci.edu)

On uniform rectangular grids with suitable boundary conditions, solvers based on the fast Fourier transform provide another efficient approach. Fourier, sine, or cosine transforms can diagonalize the discrete operator, reducing the problem to separate equations for transformed coefficients. Their straightforward application depends on the grid structure and boundary conditions. (arxiv.org)

Scope and limitations

Poisson's equation is a spatial boundary-value model, not a time-evolution equation. In physical applications, using it therefore requires an appropriate static or steady-state description. Its standard Laplacian form also assumes constant coefficients; spatially varying material properties or density can require a divergence-form operator such as $\nabla\cdot(a(\mathbf x)\nabla u)$ . (farside.ph.utexas.edu)

Smoothness should not be presumed merely because the equation is linear. Point sources lead to singular fundamental solutions, while the regularity of boundary-value solutions depends on the data and geometry. Weak or distributional formulations distinguish meaningful solutions from functions that satisfy the equation classically everywhere. (web.stanford.edu)

References

Solving the Poisson equationgithub.com
Laplace's Equation — Math 220B Lecture Notesweb.stanford.edu
Siméon-Denis Poisson (1781–1840) — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
Poisson equation — DOLFINxdocs.fenicsproject.org
CS205b/CME306 — Lecture 16web.stanford.edu
Lectures on PDE — Leon Simonmath.stanford.edu
Introduction — Computational Physicsfarside.ph.utexas.edu
Gravitational Potentialfarside.ph.utexas.edu
Optimal Control of the Poisson equationscientificcomputing.github.io
Project: Multigrid Methodsmath.uci.edu
PoisFFT — A Free Parallel Fast Poisson Solverarxiv.org