Poisson equation, solving with DFT

Primary authors of this description:V.M.Stepanenko, E.V.Mortikov, Vad.V.Voevodin (section 2.2)

1 Properties and structure of the algorithm

1.1 General description of the algorithm

The Poisson equation for the multidimensional space has the form $\sum_{i=1}^{N}\frac{\partial^2 \phi}{\partial x_i^2}=f,~\mathbf{x}\in D.$

Here, $D \in \mathbb{R}^N$ is the domain in which the solution $\phi(\mathbf{x})$ is defined, and $\mathbf{x}=(x_1,...,x_N)^T$ is the vector of independent variables. The Poisson equation is supplemented by the boundary conditions $B(\phi)=F, \mathbf{x} \in \Gamma(D),$ where $\Gamma(D)$ is the boundary of $D$ and $B(\phi)$ is the operator defining the boundary conditions. The case $B(\phi)=\phi$ corresponds to the Dirichlet boundary condition, while $B(\phi)=\partial\phi/\partial n$, where $\mathbf{n}$ is the outer normal to the boundary $\Gamma(D)$, corresponds to the Neumann boundary condition. Sometimes mixed boundary conditions $B(\phi)=C\phi+\partial\phi/\partial n$, where $C$ is a constant, are also used. The so-called "periodic boundary conditions" may also occur. In this case, the problem is posed on an unbounded domain, but the solution is assumed to be periodic with respect to a subset of variables from $\mathbf{x}$.

The Poisson equation emerges in many problems of mathematical physics, for instance, in electrostatics (in this case, $\phi$ is the potential of the electric force) and hydrodynamics ($\phi$ is the pressure of a fluid or a gas). The parameter $N$ is 2 and 3 for the plane and three-dimensional problems, respectively.

The analytical form of the solution to the Poisson equation is not known in the case where the right-hand side is arbitrary and the boundary conditions are inhomogeneous. Consequently, in most applications, this equation is solved numerically. The most common discretization of the Poisson equation has the form

$\sum_{i=1}^{N}\frac{\phi_{k_1,...,k_i+1,...,k_N}-2\phi_{k_1,...,k_i,...,k_N}+\phi_{k_1,...,k_i-1,...,k_N}}{\Delta x_i^2}=f_{k_1,...,k_N},~(k_1,...,k_N) \in D_N.$

Here, the second derivatives are replaced by second-order finite difference approximations (which creates the cross stencil for the plane problem), and the solution is sought on a discrete subset $D_N$ of the $N$-dimensional space. The boundary conditions are also approximated by finite differences.