Finite element approximation of a nonlinear cross-diffusion population model |
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Authors: | Email author" target="_blank">John W?BarrettEmail author James F?Blowey |
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Institution: | (1) Department of Mathematics, Imperial College, London, SW7 2AZ, UK;(2) Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK |
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Abstract: | 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):=( i– ii ui– ij uj) ui. In the above, the given data is as follows: v is an environmental potential, ci![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) , ai![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) are diffusion coefficients, bi![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) are transport coefficients, i![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) are the intrinsic growth rates, and ii![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) are intra-specific, whereas ij, i j, ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) ![isin](/content/rv2a0ecj7fv0ucy0/xxlarge8712.gif) ![thinsp](/content/rv2a0ecj7fv0ucy0/xxlarge8201.gif) 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.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK. |
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