Finite volume/Finite Element methods for degenerate chemotaxis model

Mazen Saad

in collaboration with B. Andreianov, M. Bendahmane, C. Cancès, G. Chamoun, M. Ibrahim, R. Talhouk

Chemotaxis is the property of certain living organisms, which is a species of soil-living amoebae, to be repelled or attracted to chemical signals.
The celebrated model in chemotaxis was introduced by Keller and Segel.
We investigate the variant of the Keller-Segel chemotaxis model with a nonlinear degenerate diffusion law for the cells.
Finite volume -- Finite element (conforming or nonconforming) method for solving the degenerate chemotaxis model are presented.
For more information :
Finite volume on an admissible mesh (see [])
Combined finite volume/nonconforming finite element on general mesh (see [])
CVFE on general mesh (see [])

PLEASE waiiiiiiit

TEST 1. Chemattraction

Initial condition for the cell density u(0,x,y)=1 in the red sqaure(left) and
the chemoattractant v(0,x,y)=5 in the four red square (right).

Evolution of the cell density (u), at time t=1 with 0<= u <= 0.1626 (left), at time t=3 with 0<= u <= 0.3947
and at time t=20 with 0<= u <= 0.7420 (right).

The cell density (u), at time t=60 with 0<= u <= 0.3275$ (left) and the chemoattractant (v), at time 60 with 0<= v <= 0.0565 (right).

TEST 2 : Random initial distribution of cell density in the same domain

Cell density, randum initial condition with 0 <= u <= 1 (left), solution at time t=0.25 with 0 <= u <= 0.5570 (center)
and solution at time t=5. with 0. <= u <= 0.5254(right)

Initial condition with 0 <= v <= 5 (left) and distribution of chemoattractant at time t=5 with 0 <= v <= 2.25 (right).