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 u _i , the population of the i^th species, i=1 and 2, such that where j≠i and g _i (u₁,u₂):=(μ _i −γ _ii  u _i −γ _ij  u _j ) u _i . In the above, the given data is as follows: v is an environmental potential, c _i  ∈ ℝ, a _i  ∈ ℝ are diffusion coefficients, b _i  ∈ ℝ are transport coefficients, μ _i  ∈ ℝ are the intrinsic growth rates, and γ _ii  ∈ ℝ are intra-specific, whereas γ _ij , 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.