Convergence of finite elements for the Eyles-King-Styles model of tumour growth

This paper presents a convergence analysis of the evolving surface finite element method (ESFEM) applied to the original Eyles-King-Styles model of tumour growth. The model consists of a Poisson equation in the bulk, a forced mean curvature flow on the surface, and a coupled velocity law between bulk and surface. Due to the non-trivial bulk-surface coupling, all previous analyses required an additional regularization term. By introducing a $H^{1/2}(Γ)$ energy estimates theory, we develop an essentially new theoretical framework that addresses the intrinsic bulk-surface coupling. Based on this framework, we provide the first rigorous convergence proof for the original model without regularization.