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 that there exists 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.

翻译：在这项工作中,我们考虑在肿瘤消耗的养分存在的情况下,肿瘤生长的分散界面模型。方程式系统包括一个Cahn-Hilliard方程式,其中含有肿瘤细胞的来源条件和营养元素的反扩散方程式。我们引入了模型的完全分解的有限元素近似值,并证明离散方案具有稳定性。此外,我们显示离散解决方案存在,并持续依赖初始和边界数据。然后我们通过离散参数的极限,证明模型存在全球时空薄弱的解决方案。根据其他假设,这种薄弱的解决方案是独特的。最后,我们提出了一些数字结果,包括一个空间层面的数字错误调查,以及两个和三个空间层面的一些长期模拟。