We develop a linear fully discrete structure-preserving finite element method for a diffuse-interface model of tumour growth. The system couples a Cahn--Hilliard type equation with a nonlinear reaction-diffusion equation for nutrient concentration and admits a dissipative energy law at the continuous level. For the discretisation, we employ a scalar auxiliary variable (SAV) formulation together with a mixed finite element method for the Cahn--Hilliard part and standard conforming finite elements for the reaction-diffusion equation in space, combined with a first-order Euler time-stepping scheme. The resulting method is unconditionally energy-stable, mass-preserving, and inherits a discrete energy dissipation law associated with the SAV-based approximate energy functional, while requiring the solution of only linear systems at each time step. Under suitable regularity assumptions on the exact solution, we derive rigorous error estimates in $L^2$, $H^1$, and $L^\infty$ norms, establishing first-order accuracy in time and optimal-order accuracy in space. A key step in this analysis is the proof of boundedness of the numerical solutions in $L^\infty$. Numerical experiments validate the theoretical convergence rates and demonstrate the robustness of the method in capturing characteristic phenomena such as aggregation and chemotactic tumour growth.

翻译：本文针对肿瘤生长的扩散界面模型，提出了一种线性全离散保结构有限元方法。该模型将Cahn--Hilliard型方程与描述营养物浓度的非线性反应-扩散方程相耦合，并在连续层面满足耗散能量定律。在离散化过程中，我们对Cahn--Hilliard部分采用标量辅助变量（SAV）公式结合混合有限元法，对空间上的反应-扩散方程采用标准协调有限元法，并结合一阶欧拉时间步进格式。所得方法无条件能量稳定且保持质量守恒，继承了基于SAV的近似能量泛函的离散能量耗散律，同时每个时间步仅需求解线性系统。在精确解满足适当正则性假设的条件下，我们推导了$L^2$、$H^1$和$L^\infty$范数下的严格误差估计，证明了时间一阶精度与空间最优阶精度。分析的关键步骤是证明数值解在$L^\infty$范数下的有界性。数值实验验证了理论收敛率，并证明了该方法在捕捉聚集和趋化性肿瘤生长等特征现象方面的鲁棒性。