Pressure-Robust Finite Element Method for Stokes Interface Problem in Two-Phase Flow

This project is dedicated to the study of the Stokes interface problem in two-phase flow, a critical research area in fluid mechanics with diverse applications in physics, chemical engineering, biomedicine, microfluidic devices, and other fields. The primary objective is to enhance existing unfitted methods to ensure pressure-robustness and to develop an efficient and accurate approach to solve this problem.

To achieve this goal, we propose a new divergence-free reconstruction operator that is compatible with Nitsche's method and small cut cell processing techniques to construct the unfitted pressure-robust finite element method. Additionally, we construct auxiliary subspaces satisfying LBB stability conditions and quasi-interpolation properties for velocity and pressure in unfitted meshes. By solving the error problem based on this subspace, we obtain a reliable and effective pressure-robust a posteriori error estimator.

Furthermore, we develop a novel diagonalization technique to overcome the challenge of directly diagonalizing the saddle point problem. Based on this technique, we can efficiently calculate the a posteriori error estimator and design an adaptive algorithm. The research in this project can advance the numerical and theoretical understanding of the Stokes interface problem, provide valuable technical guidance for fluid simulation, and facilitate instrument design and improvement in the engineering field.