标题： 基于趋化性和主动运输的Cahn-Hilliard系统肿瘤生长的数值分析 显示英文标题

标题： Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport

摘要： 在本工作中，我们考虑了一种扩散界面模型，用于描述在存在营养物质的情况下肿瘤的生长。该方程组包括一个带有源项的Cahn--Hilliard方程，用于肿瘤细胞，以及一个反应-扩散方程用于营养物质。我们引入了该模型的全离散有限元近似，并证明了离散方案的稳定性边界。此外，我们证明了离散解的存在性，并且离散解连续依赖于初始和边界数据。然后，我们对离散化参数取极限，并证明了离散方案收敛到模型的全局时间弱解。在额外假设下，该弱解是唯一的。最后，我们展示了一些数值结果，包括一维空间中的数值误差分析以及二维和三维空间中的一些长时间模拟。 显示英文摘要

摘要： In this work, we consider a diffuse interface model for tumour growth in the presence of a nutrient which is consumed by the tumour. The system of equations consists of a Cahn--Hilliard equation with source terms for the tumour cells and a reaction-diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.