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.

中文翻译：

---

具有趋化性和主动转运的 Cahn-Hilliard 系统模拟肿瘤生长的数值分析

在这项工作中，我们考虑了在存在被肿瘤消耗的营养物的情况下肿瘤生长的扩散界面模型。方程组由一个 Cahn-Hilliard 方程组成，其中包含肿瘤细胞的源项和营养物质的反应扩散方程。我们引入了模型的全离散有限元近似，并证明了离散方案的稳定性界限。此外，我们表明存在离散解决方案并持续依赖于初始和边界数据。然后我们传递到离散化参数的极限，并证明收敛到模型的全局时间弱解。在额外的假设下，这个弱解是唯一的。最后，