Elliptic equations

1 Elliptic equations

Consider a region $R$ (for example, a rectangle) in the $x y$ -plane. We might pose the following boundary value problem

\begin{array}{rcl} u_{x x} + u_{y y} & = & f (x, y) a given function, in R \\ u & = & g a given function, on the boundary of R \end{array}

if $f = 0$ everywhere, then the PDE is called Laplace’s equation
if $f$ is non-zero somewhere in $R$ then the PDE is called Poisson’s equation

Laplace’s equation models a huge range of physical situations. It is used by coastal engineers to approximate the motion of the sea; it is used to model electric potential; it can give an approximation to heat distribution in certain steady state problems. The list goes on and on. The generalisation to Poisson’s equation opens up further application areas, but for our purposes in this Section we will concentrate on how to solve the equation, rather than on how it is applied.