A fictitious domain approach for a mixed finite element method solving the two-phase Stokes problem with surface tension forces

TL;DR

This paper introduces a fictitious domain mixed finite element method for the two-phase Stokes problem with surface tension, employing stabilization techniques to ensure convergence and robustness in interface deformation simulations.

Contribution

It develops a simplified fictitious domain approach inspired by XFEM, incorporating an augmented Lagrangian for stable multiplier convergence in two-phase Stokes problems.

Findings

Achieves optimal convergence rates verified numerically.

Demonstrates robustness in interface deformation tests.

Ensures mesh-independent stability through theoretical analysis.

Abstract

In this article we study a mixed finite element formulation for solving the Stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor, on an interface that splits the domain into two subdomains. Equality of velocities is assumed at the interface. The interface conditions are taken into account with multipliers. A suitable Lagrangian functional is introduced for deriving a weak formulation of the problem. A specificity of this work is the consideration of the interface with a fictitious domain approach. The latter is inspired by the XFEM approach in the sense that cut-off functions are used, but it is simpler to implement since no enrichment by singular functions is provided. In that context, getting convergence for the dual variables defined on the interface is non-trivial. For that purpose, an augmented Lagrangian technique stabilizes the convergence of the multipliers, which is important because their value would determine the dynamics of the interface in an unsteady framework. Theoretical analysis is provided, where we show that a discrete inf-sup condition, independent of the mesh size, is satisfied for the stabilized formulation. This guarantees optimal convergence rates, that we observe with numerical tests. The capacity of the method is demonstrated with robustness tests, and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2.