Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 23

The Poisson equation is a BVP of the form

(1.5.3)

Below are some common phenomena modeled by the Poisson's equation.

 Electrostatics: To describe the components of the Maxwell's equations, associating the electric‐ and potential fields and to the charge density , roughly speaking, we haveand with a Dirichlet boundary condition on .

 Fluid mechanics: The velocity field of a rotation‐free fluid flow satisfies and hence, is a, so‐called, gradient field: , with being a scalar potential field. The rotation‐free incompressible fluid flow satisfies , which yields the Laplace's equation for its potential. At a solid boundary, this problem will be associated with homogeneous Neumann boundary condition, due to the fact that in such a boundary, the normal velocity is zero.

 Statistical physics: The random motion of particles inside a container until they hit the boundary is described by the probability of a particle starting at the point winding up to stop at some point on , where means that it is certain and means that it never happens. It turns out that solves the Laplace's equation , with discontinuity at the boundary: on and on where . Poisson's equation is of vital importance in describing, the coupling of nonlinearity aspects, in the system of gas kinetic/dynamic equations.