Optimal error estimate of unconditionally positivity-preserving, mass-conserving and energy stable method for the Keller-Segel chemotaxis model

In this paper, we prove the optimal error estimate of an unconditionally positivity-preserving, mass-conserving and energy stable method for the Keller-Segel chemotaxis equations. Applying a log-transformation to preserve the positivity and utilizing a recovery to ensure the mass-conversing, we consider a decoupled and linear fully discrete finite element method for the Keller-Segel chemotaxis equations. Then, supposing that the initial mass is less than certain critical threshold which guarantees that the system does not blow up at a finite time, and deriving the errors of the temporal and spatial discretizations in a proper sequence to avoid extra regularity requirements of the weak solutions, we prove that the method is unconditionally energy stable and can achieve the optimal convergence order in L 2 L^2 norm under weaker assumptions on the solutions than the existent ones. The shown numerical examples confirm the correctness of the theoretical prediction.