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13
AN EXAMPLE
IN TWO DIMENSIONS
To illustrate the construction of polynomial approximations of the solution of a partial
differential equation, we examine a simple boundary-value problem in two dimensions.
13.1 Poisson’s equation
Let Ω be the open square ] − 1,1[×] − 1,1[ in R 2 . We denote by ∂Ω the boundary
of Ω. Given the function f : Ω → R, we want to find the solution U : ¯ Ω → R of the
boundary-value problem
⎧ 2 ∂ U
⎪⎨ −∆U := −
∂x
(13.1.1)
⎪⎩
2 + ∂2U ∂y2
U ≡ 0 on ∂Ω,
= f in Ω.
The partial differential equation in (13.1.1) is known as Poisson’s equation. The symbol
∆ is called the Laplace operator. There are many applications of equation (13.1.1) in
the field of physics. For example, if Ω is a plane region of dielectric material surrounded
by a grounded conductor, and f is the charge density in Ω, then the solution U of
(13.1.1) is proportional to the electrostatic potential on Ω.