Given \(f\in L^2\) on the torus in \(2\)-space, a solution \(Y\) of \(\operatorname{div}Y=F\) which is stable in the \(L^\infty\) norm cannot be constructed by a linear operator. This was proven by J. Bourgain and H. Brezis [J. Am. Math. Soc. 16, 393–426 (2002; Zbl 1075.35006)]. Therefore an approximation is considered here which is based on truncated Fourier series and a nonlinear optimization procedure. The essential tools are relations between \(L^\infty\)-norms, \(H^1\)-norms, and \(L^p\)-norms \((p=p(N))\) of trigonometric polynomials.