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This dissertation studies novel computational methodologies for multi-phase problems. The first part of the thesis focus on computational models for complex interface coupled flow problems. The problem is studied both numerically and analytically. More specifically, an unfitted finite ele- ment approach for the simulation of a two-phase flow with an immersed material viscous interface is studied in the first and second chapters. The interaction between the bulk and surface flows is characterized by no-penetration and slip with friction interface conditions. The system is shown to be dissipative and a model stationary problem is proved to be well-posed. The finite element method applied in this thesis belongs to a family of unfitted discretization. For the unfitted gen- eralized Taylor–Hood finite element pair, an inf-sup stability property is shown with a stability constant that is independent of the viscosity ratio, slip coefficient, the position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. To study numerically the coupled problem, we introduce an iterative procedure based on the splitting of the system into bulk and surface problems. Numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach. Solving strongly-coupled nonlinear partial differential equations which characterize multi-scale, multi-physics processes with high dimensional chaotic dynamics is computationally expensive for many practical reasons. The necessity for developing a simulation and prediction strategy with high fidelity by only utilizing observational data highly increased over the last decades. In the second part of the thesis, the performance of two deep learning methods for reproducing short-term and long-term statistics of spatio-temporal data from the surface Cahn–Hilliard phase-field model is examined. The deep learning methods are echo state network (ESN) and long short-term mem- ory (LSTM). The numerical discretization scheme of the Cahn-Hilliard system is briefly discussed. Then we present architectures of the ESN and the LSTM. We show that LSTM substantially out- performs ESN in short-term and long-term prediction, and give accurate forecasting trajectories for numerical solver’s time steps. This dissertation is based in part on the previously published articles listed below. I have permis- sion from my co-authors/publishers to use the works listed below in my dissertation. Olshanskii, Maxim, Quaini, Annalisa and Sun, Qi. ”An unfitted finite element method for two-phase Stokes problems with slip between phases”, Journal of Scientific Computing, V. 89 (2021), Article 41; Olshanskii, Maxim, Quaini, Annalisa and Sun, Qi. ”A Finite Element Method for Two-Phase Flow with Material Viscous Interface” , Computational Methods in Applied Mathematics, vol. 22, no. 2, 2022, pp. 443-464.



Finite element method, Machine learning


Portions of this document appear in: Olshanskii, Maxim, Annalisa Quaini, and Qi Sun. "An unfitted finite element method for two-phase Stokes problems with slip between phases." Journal of Scientific Computing 89, no. 2 (2021): 1-23; and in: Olshanskii, Maxim, Annalisa Quaini, and Qi Sun. "A finite element method for two-phase flow with material viscous interface." Computational Methods in Applied Mathematics 22, no. 2 (2022): 443-464.