We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model.

中文翻译：

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抛物线-抛物线 Keller-Segel 模型的标量辅助变量有限元方案

我们描述和分析抛物线​​-抛物线 Keller-Segel 模型的有限元数值方案。标量辅助变量方法用于在离散水平上检索与系统相关的能量的单调衰减。这种方法依赖于将 Keller-Segel 模型解释为梯度流。由此产生的数值方案高效且易于实现。我们展示了独特的非负解的存在，并且由于使用了 SAV 方法，获得了修改后的离散能量。我们还证明了离散解对连续 Keller-Segel 模型弱形式解的收敛性。