 Solution

Let $K_N=\{k\in {\mathbb{Z}}^2:|k_1|,|k_2|\le N\}$, use $u_N={\sum }_{k\in K_N}{u}_k{e}^{i\pi k\cdot x}$ and impose $u_0=0$. The coefficient has $a_0=3$ and $a_{\pm e_1}=a_{\pm e_2}=1/2$, with all other coefficients zero. Fourier differentiation and convolution giveConsequently the explicit Galerkin system for each $k\in K_N\setminus \{0\}$ iswhere coefficients outside $K_N$ and $u_0$ are zero. The forcing is $f_{e_1}=f_{e_2}=1/(2i)$, $f_{-e_1}=f_{-e_2}=-1/(2i)$ and zero elsewhere. The $k=0$ equation is $0=0$ and is replaced by normalization. For a real solution $u_{-k}=\overline{u_k}$, so the system can equivalently be written in real sine/cosine coordinates.

Since $a\ge 1$, the negative of this operator has quadratic form $\int a|\nabla u_N{|}^2$, strictly positive on nonzero zero-mean trigonometric polynomials. Thus the finite linear system is invertible. This construction multiplies the coefficient and derivative before projecting; it retains the explicit nearest-neighbour mode coupling rather than dividing the forcing by $a$ pointwise.