Finite element approximation of a nonlinear cross-diffusion population model

where j = i and gi(u1, u2) := (μi − γii ui − γij uj) ui. In the above, the given data is as follows: v is an environmental potential, c i ∈ R≥0, ai ∈ R>0 are diffusion coefficients, bi ∈ R are transport coefficients, μi ∈ R≥0 are the intrinsic growth rates, and γii ∈ R≥0 are intra-specific, whereas γij , i = j, ∈ R≥0 are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ≤ 3. Finally some numerical experiments in one space dimension are presented.