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 electrostatics (in this case, [math]\phi[/math] is the potential of the electric force) and hydrodynamics ([math]\phi[/math] is the pressure of a fluid or a gas). The parameter [math]N[/math] 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
[math] \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. [/math]
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 [math]D_N[/math] of the [math]N[/math]-dimensional space. The boundary conditions are also approximated by finite differences.
1.2 Mathematical description of the algorithm
Here, we examine a finite difference scheme for the most common problem related to the Poisson equation in the three-dimensional space:
[math] \frac{\phi_{i+1,j,k}-2\phi_{i,j,k}+\phi_{i-1,j,k}}{ \Delta x^2}\,+\, \frac{\phi_{i,j+1,k}-2\phi_{i,j,k}+\phi_{i,j-1,k}}{ \Delta y^2}\,+\, \frac{\phi_{i,j,k+1}-2\phi_{i,j,k}+\phi_{i,j,k-1}}{ \Delta z^2} = f_{i,j,k},~(i,j,k) \in D_h, [/math]
where [math]D_h=\{0:N_x-1\}\times\{0:N_y-1\}\times\{0:N_z-1\} [/math] is a parallelepiped in the grid domain. For simplicity, we impose the so-called 3-D periodic boundary conditions
[math] \begin{align} \phi_{0,j,k} &= \phi_{N_x,j,k},\\ \phi_{i,0,k} &= \phi_{i,N_y,k},\\ \phi_{i,j,0} &= \phi_{i,j,N_z}. \end{align} [/math]
