Finite element approximation of a nonlinear cross-diffusion population model

Summary.We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find ui, the population of the ith species, i=1 and 2, such that 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, ci ∈ ℝ, ai ∈ ℝ are diffusion coefficients, bi ∈ ℝ are transport coefficients, μi ∈ ℝ are the intrinsic growth rates, and γii ∈ ℝ are intra-specific, whereas γij, i≠j,  ∈ ℝ 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.