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A cut finite element method for a Stokes interface problem
Peter Hansbo, Mats G. Larson, Sara Zahedi
TL;DR
The paper addresses Stokes flow for two immiscible fluids whose discontinuous interface need not align with a mesh. It proposes a weighted Nitsche formulation with consistent velocity-pressure stabilization, proving optimal accuracy and interface-position-independent conditioning. Numerical examples show accurate discontinuous-pressure representation and machine-epsilon spurious currents for a static drop.
Problem
Unfitted, evolving interfaces with viscosity and pressure discontinuities are difficult for standard methods, while fitted meshes require remeshing and complicate topological changes.
Method
The method weakly enforces interface jump conditions through weighted Nitsche fluxes and adds consistent least-squares stabilization controlling normal-gradient jumps near the interface.
Results
The method has optimal-order accuracy and condition number O(h^-2) independent of interface location; for a static drop, spurious currents and pressure error are of machine-epsilon order.
Takeaways & Limitations
The formulation accurately represents discontinuous pressure on coarse meshes and is robust for interfaces positioned arbitrarily relative to a fixed background mesh.
Takeaways & Limitations
The formulation still requires stabilization terms for a well-conditioned linear system, and the stated analysis uses particular element and mesh constructions.
Abstract
from arXiv · showhide
We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.
1 Introduction
Unfitted interfaces avoid remeshing but challenge standard discretizations and conditioning. The proposed stabilized Nitsche method handles discontinuities in two-phase Stokes flow while retaining optimal convergence and conditioning.
- 1 Introduction: Interfaces may move arbitrarily across a fixed mesh, making standard finite element and finite difference schemes inaccurate unless the mesh is fitted.Mesh fitting requires remeshing and complicates topological changes such as breakup or coalescence.
- 1 Introduction: Regularizing discontinuities reduces accuracy near interfaces and therefore requires very fine local meshes.Unfitted methods aim to avoid both regularization and remeshing.
- 1 Introduction: Small interface-cut supports can make XFEM systems ill-conditioned, especially when moving interfaces enter new elements.Neglecting such basis functions requires careful criteria to avoid losing accuracy.
- 1 Introduction: Nitsche interface methods weakly enforce jump conditions with locally weighted flux coefficients, achieving optimal convergence for discontinuities internal to elements.Earlier Nitsche-based approaches still suffered from ill-conditioning, motivating stabilization.
- 1 Introduction: The proposed method targets two-phase Stokes flow with viscosity jumps and surface tension using weighted Nitsche fluxes and velocity-pressure stabilization.The stabilization controls normal-gradient jumps near the interface, supports an inf-sup condition, and yields optimal conditioning.
- 1 Introduction: The paper proves optimal convergence and shows that the condition number is O(h^-2) independently of interface position relative to the mesh.Numerical examples compare the method with existing techniques.
2 The interface problem and the finite element method
The paper formulates a two-fluid Stokes interface problem with discontinuous viscosity and surface tension, then constructs a mesh-independent CutFEM discretization using weak interface enforcement and stabilization.
- 2.1 The two-fluid incompressible Stokes equations: The model describes two immiscible incompressible fluids separated by a smooth interface, with viscosity jumps and surface-tension-induced interface conditions.
- 2.2 Mesh and assumptions: The background triangulation is generated independently of the interface, which may intersect elements and places intersected elements in both subdomain meshes.
- 2.2 Mesh and assumptions: Mesh and interface assumptions require quasi-uniform elements, controlled interface intersections, neighboring elements in both subdomains, and boundary conformity.
- 2.3 The finite element method: The discrete pressure and velocity spaces use separate subdomain-wise functions, allowing pressure discontinuities at the interface and employing a refined velocity mesh.
- 2.4 Penalty parameters and averaging operators: As the second viscosity tends to zero with the first fixed, the interface condition converges to a Neumann condition because the interface terms vanish.
3 Analysis
The analysis establishes consistency, continuity, coercivity, inf-sup stability, and optimal convergence properties for the stabilized Nitsche formulation.
- Consistency: The finite element formulation is consistent up to the stabilization terms Ju and Jp.The consistency relation identifies these terms explicitly.
- Continuity: The analysis proves continuity bounds whose constants are independent of the viscosities under the stated assumption.The continuity constant Ccont is independent of µ1 and µ2.
- Stability: The stability constants and coercivity constant are independent of µ1 and µ2 under assumption (2.4).This independence is stated for the inf-sup and coercivity results.
4 Estimate of the condition number
The condition-number analysis represents the discrete solution through coefficient vectors and derives bounds using auxiliary inequalities and stability estimates.
- Matrix representation: The stiffness matrix A is associated with the formulation and has dimension (N1 + N2) × (N1 + N2).Coefficient expansions map the discrete velocity and pressure functions to vectors used in the matrix analysis.
- Condition-number proof: The spectral condition-number estimate is obtained by separately bounding the norms of A and A−1.The proof uses a Poincare-type inequality, an inverse estimate, continuity, and inf-sup stability.
Then the following estimate holds
The condition-number theorem combines bounds on the stiffness matrix and its inverse to establish the stated spectral estimate.
- Condition-number estimate: The analysis derives an estimate for the spectral condition number of the stiffness matrix.The theorem states the condition-number bound with a positive constant C.
- Proof structure: The estimate follows by combining separate bounds for |A|N and |A−1|N.These bounds use continuity, the Poincare-type inequality, inverse estimates, and inf-sup stability.
5 Numerical examples
The numerical examples evaluate accuracy, convergence, conditioning, and stabilization for continuous, static-drop, and discontinuous Stokes interface problems. The proposed method achieves optimal convergence, represents discontinuous pressure accurately, and maintains conditioning independently of interface location when both stabilization terms are used.
- Experimental setup: The experiments use a level-set interface representation, separate velocity and pressure meshes, and stabilization parameters εp = 1 and εu = 10^-3.The velocity mesh size hx is reported, with the pressure mesh twice as coarse; the errors are not sensitive to these stabilization parameters.
- Example 1: A continuous problem: For the continuous problem, the proposed method attains second-order velocity convergence and better-than-first-order pressure convergence in the L2 norm.Its pressure errors are slightly larger than standard continuous FEM but much smaller than those reported for the method in.
- Example 1: A continuous problem: The stabilized method’s condition number grows as O(h^-2), matching standard finite elements and remaining close to the standard method at fixed mesh size.Without pressure stabilization, the condition number grows erratically as the mesh is refined, although the estimated inf-sup constant is essentially mesh-independent.
- Example 2: Static drop: For the static drop, the new method yields spurious currents and pressure errors of order machine epsilon and accurately represents discontinuous pressure on coarse meshes.Standard continuous methods produce large spurious currents with sharp surface tension and either oscillatory or smeared pressure representations.
- Example 3: A discontinuous problem: For the discontinuous problem, the method approximates discontinuous pressure and weakly discontinuous velocity on a mesh that does not coincide with the interface.The velocity shows first-order convergence in H1, while the pressure shows better-than-first-order convergence in L2.
- Example 3: A discontinuous problem: Both pressure and velocity stabilization terms are needed to obtain a well-conditioned system matrix independently of the interface’s location relative to mesh lines.This experiment varies εu and εp while keeping the mesh size fixed.
6 Conclusions
The proposed method accurately handles Stokes flow with immiscible fluids, discontinuities positioned arbitrarily relative to a fixed mesh, and surface tension. It achieves optimal-order accuracy while stabilization makes the system conditioning independent of interface location.
- The method solves Stokes equations for two immiscible fluids with different viscosities and surface tension while allowing arbitrary interface placement relative to a fixed background mesh.The interface may be represented explicitly or implicitly.
- The inf–sup stable P1–iso–P2 element gives the method optimal-order accuracy.
- The stabilization terms Ju(u_h, v_h) and Jp(p_h, q_h) ensure that the system-matrix condition number is O(h^-2), independent of interface location.
- The authors characterize the method as simple to implement and robust, with desirable properties for problems involving moving interfaces.They expect applicability in three space dimensions and describe higher-order stabilization extensions.