Chebyshev Collocation Method for 2D Boundary Value Problems

Consider the 2D boundary value problem given by , with boundary conditions and . You can set the values of and . Using this Demonstration, you can solve the PDE using the Chebyshev collocation method adapted for 2D problems. The solution is shown as either a 3D plot or a contour plot.

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomial of the first kind, .

The Chebyshev derivative matrix at the quadrature points is an matrix given by

, , for

and

for and ,

where for and .

The discrete Laplacian is given by , where is the identity matrix, is the Kronecker product operator, , and is without the first row and first column.

Reference

[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.