Solution of the Laplace Equation for Temperature Distribution in a Square
Consider the Laplace equation in a square region, where . We wish to solve for the temperature distribution , subject to the following Dirichlet boundary conditions:
This Demonstration solves the problem using a spectral method  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 at the various nodes in the square domain. You can clearly recognize the form of a contour plot of in the third snapshot.