Low-dimensional numerical approximations for two boundary value problems of the complex Ginzburg-Landau equation in a chaotic regime are constructed. Spatial structures called principal interaction patterns are extracted from the system according to a nonlinear variational principle and used as basis functions in a Galerkin approximation. The dynamical description in terms of principal interaction patterns requires fewer modes than more conventional approaches using Fourier modes or Karhunen-Loève modes as basis functions.