An Online Generalized Multiscale finite element method for heat and mass transfer problem with artificial ground freezing

In this paper, we present an Online Generalized Multiscale Finite Element Method(Online GMsFEM) for heat and mass transfer problem in heterogeneous media with artificial ground freezing pipes. The mathematical model of the process is based on the classical Stefan model, which describes heat transfer with a phase transition and takes into account filtration in a porous medium. The model is described by a system of equations for temperature and pressure. For fine grid solution, we use a finite element method using the fictitious domain method. To derive a solution on the coarse grid, we use a model reduction procedure based on Online GMsFEM. Online version of GMsFEM allows to us to take less number of offline multiscale basis functions. In our approach, we use decoupled offline basis functions constructed with snapshot space and based on spectral problems. This is the standard approach of basis construction. To take into account artificial ground freezing pipes, we compute an additional basis functions on the offline stage. For the accurate approximation of phase change we add online multiscale basis functions. We construct online basis that minimizes error by values of local residuals. Online procedure is significantly improves the accuracy of standard GMsFEM. We present numerical results in two-dimensional domain with layered heterogeneity. To investigate accuracy of the method, we present results with different number of offline and online basis functions. The presented results show that Online GMsFEM can produce solution with high accuracy and requires small computational resources.