Structure-preserving parametric finite element methods for two-phase Stokes flow based on Lagrange multiplier approaches

TL;DR

This paper introduces a new parametric finite element approach for two-phase Stokes flow that preserves energy decay and volume at the discrete level, employing Lagrange multipliers and higher-order time discretizations.

Contribution

It proposes a novel Lagrange multiplier formulation that ensures structure preservation in finite element methods for two-phase Stokes flow, with efficient nonlinear solvers.

Findings

Methods achieve desired temporal accuracy.

Exact preservation of energy-decaying property.

Volume-preserving at the discrete level.

Abstract

We present a novel formulation for parametric finite element methods to approximate two-phase Stokes flow. The new formulation is based on the classical Stokes equation in the bulk and a novel choice of interface conditions with additional Lagrange multipliers. This new Lagrange multiplier approach ensures that the numerical methods exactly preserve two physical structures of two-phase Stokes flow at the fully discrete level: (i) the energy-decaying and (ii) the volume-preserving properties. Moreover, different types of higher-order time discretization methods are employed, including the Crank--Nicolson method and the second-order backward differentiation formula approach. The resulting schemes are nonlinear and can be efficiently solved by using the Newton method with a decoupling technique. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy while preserving the two physical structures of the two-phase Stokes system.