Methodological Analysis of an ALE–FEM Framework for a Nutrient–Pressure Tumor Moving-Boundary Model

The paper considers a two-dimensional quasi-stationary problem of tumor growth in a domain with evolving geometry and a moving interface separating tumor and healthy tissues. The model is formulated as a coupled boundary value problem for the scalar fields of nutrient concentration and pressure in the subdomains corresponding to tumor and healthy tissues. Ideal contact conditions are imposed on the internal interface, ensuring the continuity of the corresponding fields and the compatibility of fluxes. The motion of the interface is described by a normal velocity law depending on the normal derivatives of the nutrient concentration and pressure. To transform the problem from the moving physical domain to a problem posed on a fixed domain, an arbitrary Lagrangian–Eulerian mapping is used. Based on this mapping, a weak formulation of the problem is derived. For each fixed time and prescribed domain geometry, existence and uniqueness of the quasi-stationary nutrient–pressure elliptic subproblem are established using the Lax–Milgram theorem. This result does not establish well-posedness of the fully coupled moving-boundary evolution problem. A finite element approximation of the solution to the problem on a moving mesh is constructed. For quadratic finite elements, a priori error estimates are obtained in the corresponding energy norms under assumptions on the regularity of the solution and the geometric regularity of the ALE mapping. It is shown that the constants in the estimates can be chosen independently of time, provided that the family of domains is uniformly regular. The theoretical error analysis predicts second-order spatial convergence for conforming quadratic finite elements under the stated regularity assumptions. No numerical convergence experiment is presented in the current paper; computational validation is left for future work.