# 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 ?

## 1 Answer

Yes, you can use Fourier expansions. It will lead to Gibb's phenomenon, but since you're on the unit square you can likely use enough Fourier modes that that doesn't matter.

If the boundary values are equal on opposing boundaries, you can use periodic boundary conditions.

• Thanks for you reply. As I mention in my post, the boundary value is 0 on all boundaries. Would it still lead to Gibb's phenomenon, or is it then equivalent to periodic BC ? I would appreciate a reference. The number of Fourier modes I use is a concern, since my aim is to use this code for control design, and I would like to keep the problem size small. Thanks – H.Poincare Nov 5 '14 at 4:40
• You get Gibbs phenomena when you don't have enough resolution to resolve any layers that might be present. It has nothing to do with whether you have Dirichlet BCs or periodic BCs or any particular values at the boundaries. – Bill Barth Nov 5 '14 at 14:09
• Correct. In particular, you get Gibbs phenomenon any time you use any kind of basis that is continuous when you want to resolve discontinuous solutions. The Fourier basis is one example. Continuous finite element shape functions are another example. – Wolfgang Bangerth Nov 5 '14 at 19:05
• One comment: Oscar Bruno has a way to address Gibbs issues via Fourier continuation - add a fictitious domain, construct a periodic continuation of the solution over that domain and then use Fourier. At least in the eyeball norm, the results are much improved. math.ksu.edu/pas/2011/albin_pas_2011.pdf sciencedirect.com/science/article/pii/S0021999109006391 – Jesse Chan Nov 18 '14 at 19:55