We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^\infty(0,T;H^1(\Omega))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of $\gamma$.

翻译：我们研究了逼近具有幂律扩散（指数$\gamma \in [1,3]$）且满足周期边界条件的抛物-椭圆型Keller-Segel系统的有限体积格式。针对趋化剂，我们推导了$L^\infty(0,T;H^1(\Omega))$范数下误差的条件后验界，并对密度采用拟范数类量建立了类似结果。这些结论基于稳定性估计及数值解合适的保形重构。通过数值实验，我们展示了误差界与网格步长呈线性关系，并阐明了误差估计量随$\gamma$变化的规律。