Solution of the Laplace Equation for Temperature Distribution in a Square

collocation points

11

Consider the Laplace equation

2

∂

u

∂

2

x

+

2

∂

u

∂

2

y

=0

in a square region, where

0⩽x,y⩽1

. We wish to solve for the temperature distribution

u(x,y)

, subject to the following Dirichlet boundary conditions:

BC1:

u(x=0,y)=1

,

BC2:

u(x=1,y)=0

,

BC3:

u(x,y=0)=0

,

BC4:

u(x,y=1)=0

.

This Demonstration solves the problem using a spectral method [1] for a user-set value of the number of the Chebyshev–Gauss–Lobatto collocation points. The colored dots reflect the intensity of the scalar field

u

at the various nodes in the

[0,1]×[0,1]

square domain. You can clearly recognize the form of a contour plot of

u

in the third snapshot.