Laplace and Poisson Equations

The Laplace equation (in 2-D)

$$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial x^2} = 0$$ (7)

and the Poisson equation (in 2-D)

$$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial x^2} = S(x,y)$$ (8)

are encountered in the solution for potential functions, for instance in electrostatic and gravitational problems, and in describing the steady state reached in heat transmission problem. The function u(x,y) is a time independent field, i.e., a potential or a temperature profile function; S(x,y) represent a source term, i.e., a mass distribution, a charge distribution, or a heat source.

These equations are said to be of Elliptic type. No time dependence is involved here. A time independent solution for u(x,y) is sought after within a domain $\Omega$. In 2-D the domain could be a simple rectangle, $0 \le x \le 1$ and $0 \le y \le 1$, or could be of arbitrary shape.

The solution, or its derivative, is specified at the boundary of the domain, $\Gamma$. The possibilities are:

1.

Dirichlet Boundary conditions, in which the function is specified on the boundary of the domain

u(x,y) = f(x,y), (9)

2.

Newman boundary conditions, in which the normal derivative is specified,

$$\frac{\partial u}{\partial n} = g(x,y),$$ (10)

3.

The Fourier case in which a mix of function values and normal derivatives is specified.