Pressure-Robust Unfitted Finite Element Methods for Stokes Interface Problems in Two-Phase Flow

This project is focused on the Stokes interface problem in two-phase flow, which is a significant research area in fluid mechanics with extensive applications in physics, chemical engineering, biomedicine, microfluidic devices, and other related fields. The aim of this project is to enhance the existing unfitted methods to achieve pressure-robustness and develop an efficient and accurate method for this problem. Firstly, a divergence-free reconstruction operator compatible with Nitsche's method and small cut cell processing techniques is developed to construct the unfitted pressure-robust finite element method. Secondly, auxiliary subspaces satisfying LBB stability conditions and quasi-interpolation properties are constructed for velocity and pressure in unfitted meshes. Subsequently, a reliable and effective pressure-robust a posteriori error estimator is obtained by solving the error problem based on these subspaces. Finally, a novel diagonalization technique is developed to overcome the challenge of directly diagonalizing the saddle point problem. This technique enables the rapid calculation of a posteriori error estimators and the design of an adaptive algorithm. The research conducted in this project can significantly contribute to the numerical and theoretical results of the Stokes interface problem, providing technical reference for fluid simulation and instrument design and improvement in the engineering field.