A Mini Immersed Finite Element Method for Two-Phase Stokes Problems on Cartesian Meshes

TL;DR

This paper introduces a mini immersed finite element method for efficiently solving two-phase Stokes problems on Cartesian meshes, accounting for interface conditions, discontinuous viscosities, and surface forces, with proven stability and optimal error estimates.

Contribution

The paper develops a novel IFE method with explicit basis functions for two-phase Stokes problems, ensuring stability and optimal accuracy regardless of interface position.

Findings

Method achieves optimal error estimates.

Stability constants are independent of mesh size.

Numerical results confirm theoretical predictions.

Abstract

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.