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 [math] \sum_{i=1}^{N}\frac{\partial^2 \phi}{\partial x_i^2}=f,~\mathbf{x}\in D. [/math]

Here, [math]D \in \mathbb{R}^N[/math] is the domain in which the solution [math]\phi(\mathbf{x})[/math] is defined, and [math]\mathbf{x}=(x_1,...,x_N)^T[/math] is the vector of independent variables. The Poisson equation is supplemented by the boundary conditions [math] B(\phi)=F, \mathbf{x} \in \Gamma(D), [/math] where [math]\Gamma(D)[/math] is the boundary of [math]D[/math] and [math]B(\phi)[/math] is the operator defining the boundary conditions. The case [math]B(\phi)=\phi[/math] corresponds to the Dirichlet boundary condition, while [math]B(\phi)=\partial\phi/\partial n[/math], where [math]\mathbf{n}[/math] is the outer normal to the boundary [math]\Gamma(D)[/math], corresponds to the Neumann boundary condition. Sometimes mixed boundary conditions [math]B(\phi)=C\phi+\partial\phi/\partial n[/math], where [math]C[/math] 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 [math]\mathbf{x}[/math].

The Poisson equation emerges in many problems of mathematical physics, for instance, in Уравнение Пуассона возникает во многих задачах математической физики, например, в электростатике (в этом случае [math]\phi[/math] - потенциал электрической силы) и гидродинамике ([math]\phi[/math] - давление жидкости или газа); при этом [math]N=2,3[/math] для плоской и трехмерной задач, соответственно.