# Trignometric/Fourier spectral collocation with zero Dirichlet BC in 2D

I am concerned with numerical solution to the following problem on $[0,1]\times[0,1]$.

$\dfrac{\partial\theta}{\partial t}+u(x,t).\nabla \theta(x,t)=\kappa \nabla^2\theta(t,x)$

with Dirichlet boundary condition $\theta(t,x)=0$ on the boundary of rectangle.

My question in general situation, and in particular for the problem above is:

Can I use trignometric (Fourier) spectral Galerkin approach here ? In other words, would using $\phi_{i,j}=sin(2i\pi x)sin(2j\pi y)$ as basis for galerkin give rise to Gibbs phenomenon or not ?

The initial condition I am interested in is smooth and zero at boundaries as well.

I know that for non-periodic domains, we ought to be using Chebyshev polynomials instead. But here, can we assume that solution is "periodic" since it is zero at the boundaries ?