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

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