Issue: 10 , Volume: 66 , Уear of publication: 2026

Using the Stefan problem as an example, the validity of a modified variational formulation of the discontinuous Galerkin method in a three-dimensional domain with a moving boundary is investigated. The h-adaptivity strategy is applied to solve this class of problems, which leads to a large size of the discrete analog. The new idea in the finite element approximation is to apply the principle of decomposing the numerical solution into a continuous macroscale component and a discontinuous microscale component near the phase transition front, introducing a special operator linking the two scales into the variational formulation. This strategy addresses two main problems arising in the discretization of problems in domains with moving boundaries. The first problem is related to constructing a weak solution in the space of L₂-class functions and preserving the global regularity property of the mathematical model at the discrete level. For this purpose, a locally conservative computational scheme of the discontinuous Galerkin method is used, which makes it possible to obtain physically relevant results in domains with a rapidly varying solution gradient. The second problem is the explosive growth in the number of degrees of freedom when using the discontinuous Galerkin computational scheme throughout the entire computational domain. Since the microscale discontinuous component of the solution is introduced locally, the discrete analog of the Stefan problem has a significantly smaller size. The article presents a numerical analysis of the spectral properties of discrete analogs, the projection operator linking the large-scale (continuous) and microscale (discontinuous) function spaces, as well as the results of numerical simulation of the ice melting process in a three-dimensional medium with a moving boundary.