肿瘤生长的演化体-表面模型的数值分析
Numerical analysis of an evolving bulk--surface model of tumour growth
AI总结：
本文针对肿瘤生长中耦合 Poisson 方程与受迫平均曲率流的体-表面模型，提出演化体-表面有限元方法，通过调和延拓处理表面速度，并结合稳定性与相容性估计获得最优阶 $H^1$ 误差界。
摘要：
本文研究一种用于组织生长模型的演化体-表面有限元方法，该模型是对 Eyles、King 和 Styles（2019）模型的改进。该模型将区域上的 Poisson 方程与自由边界的受迫平均曲率流耦合，在演化表面的速度律和 Poisson 方程的边界条件中均存在非平凡的体-表面耦合。数值方法对平均曲率和外法向的演化方程进行离散，并利用表面速度的调和延拓将其延拓到体内。该离散格式在多项式次数至少为二的连续有限元情形下可进行收敛性分析。离散体-表面耦合的稳定性是一个主要关注点。误差分析结合稳定性估计与相容性估计，为计算得到的组织压力以及表面位置、速度、法向量和平均曲率导出了最优阶 $H^1$-范数误差界。数值实验说明并补充了理论结果。
英文摘要：
This paper studies an evolving bulk--surface finite element method for a model of tissue growth, which is a modification of the model of Eyles, King and Styles (2019). The model couples a Poisson equation on the domain with a forced mean curvature flow of the free boundary, with nontrivial bulk--surface coupling in both the velocity law of the evolving surface and the boundary condition of the Poisson equation. The numerical method discretizes evolution equations for the mean curvature and the outer normal and it uses a harmonic extension of the surface velocity into the bulk. The discretization admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The stability of the discretized bulk--surface coupling is a major concern. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed tissue pressure and for the surface position, velocity, normal vector and mean curvature. Numerical experiments illustrate and complement the theoretical results.
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