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Error estimates and blow-up analysis of a finite-element approximation for the parabolic-elliptic Keller-Segel system
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The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme, which are important properties of the original system. In this paper, we establish the error estimates of this scheme. Then, based on the error estimates, we derive the finite-time blowup of nonradial numerical solutions under some conditions on the mass and the moment of the initial data.
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Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise
A fully discrete mixed finite element method for stochastic Keller-Segel is analyzed, but the claimed O(k^{1/2}+h+k^{-1/2}h^2) rate rests on an invalid expectation step.
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