Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 40E / b / Solution

Expand $a,u,f$ in the two-dimensional Fourier basis. Applying the derivative and product rules from part (a), the $(m,n)$ coefficient of the equation isThe normalization and compatibility conditions areThus, on ${\mathbb{Z}}^2\setminus \{(0,0)\}$, the infinite Fourier spectral method system isFor a square spectral truncation, chooseretain equations and unknowns indexed by ${\Lambda }_N$, and set omitted coefficients to zero. This is the Fourier–Galerkin method: it projects the residual onto the retained Fourier modes. Other finite mode sets may be used in the same way.