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

This paper presents a convergence analysis of evolving surface finite element methods (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 -- which exclusively relied on energy-estimate based approaches -- required an additional regularization term. By adopting the $\widehat{H}^{3/2}$ theory and the multilinear forms, we develop an essentially new theoretical framework that enables the application of PDE regularity theory to stability analysis. Based on this framework, we provide the first rigorous convergence proof for the original model without regularization.