A stabilized Nitsche fictitious domain method for the Stokes problem

Andre Massing; Mats G. Larson; Anders Logg; Marie E. Rognes

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A stabilized Nitsche fictitious domain method for the Stokes problem

Andre Massing, Mats G. Larson, Anders Logg, Marie E. Rognes

arXiv:1206.1933v1Jun 9, 2012math.NA

TL;DR

Complex or moving geometries make conforming mesh generation difficult, while fictitious domain methods introduce cut-element stability and implementation challenges. The paper develops a stabilized Nitsche fictitious domain method for Stokes flow using low-order elements and ghost penalties, proving stability, optimal-order error estimates, and boundary-position-independent conditioning. It also demonstrates three-dimensional applicability on complex geometries, while leaving time-dependent Navier–Stokes and fluid–structure interaction for future work.

Problem
Fictitious domain methods avoid mesh generation for complex geometries but introduce non-regular cut elements and associated formulation, analysis, and implementation challenges.
Method
The paper combines a low-order stabilized Galerkin Stokes method with Nitsche boundary treatment and velocity and pressure ghost penalties near boundary-intersected elements.
Results
The method is inf-sup stable, has optimal-order a priori error estimates, and yields a stiffness-matrix condition number bounded independently of boundary position.
Takeaways & Limitations
The framework supports general three-dimensional simulations with arbitrary boundary triangulations embedded in tetrahedral meshes, including complex-geometry examples.
Takeaways & Limitations
The study is restricted to the static Stokes model; time-dependent Navier–Stokes and fluid–structure interaction are deferred to future work.

Abstract

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We develop a Nitsche fictitious domain method for the Stokes problem starting from a stabilized Galerkin finite element method with low order elements for both the velocity and the pressure. By introducing additional penalty terms for the jumps in the normal velocity and pressure gradients in the vicinity of the boundary, we show that the method is inf-sup stable. As a consequence, optimal order a priori error estimates are established. Moreover, the condition number of the resulting stiffness matrix is shown to be bounded independently of the location of the boundary. We discuss a general, flexible and freely available implementation of the method in three spatial dimensions and present numerical examples supporting the theoretical results.

1. Introduction.

Fictitious domain methods avoid remeshing complex or moving geometries by embedding the computational domain in a background mesh, but introduce geometric and formulation challenges. This paper develops a stabilized Nitsche method for Stokes flow, with three-dimensional treatment and implementation support.

Motivation: Fictitious domain methods represent the computational domain using a background mesh and an interior surface, avoiding conventional domain-conforming mesh generation.The surface–mesh intersection and resulting non-regular elements create new analysis and implementation challenges.
Problem setting: The paper considers a Nitsche fictitious domain method for the Stokes problem on bounded domains in 2D or 3D.The formulation uses velocity, pressure, body-force, and prescribed boundary-velocity data.
Method: The method uses low-order stabilized finite element spaces and penalty terms for jumps in normal gradients near boundary-intersected elements.The approach targets stability and optimal a priori error estimates for both continuous piecewise linear pressure and piecewise constant pressure cases.
Three-dimensional contribution: The work treats arbitrary boundary triangulations embedded in three-dimensional tetrahedral meshes and provides a freely available implementation.This requires integration over arbitrary polyhedral domains and computational-geometry procedures for boundary–mesh intersections.
Organization: The paper introduces the Nitsche fictitious domain formulation after defining computational domains, meshes, function spaces, and norms.The paper then analyzes stability, error, conditioning, implementation, and numerical behavior.

2. Preliminaries.

The preliminaries define the fictitious domain as a mesh-covered extension of the physical domain and distinguish cut, non-cut, boundary, and skeleton entities. They also specify geometric assumptions, discrete spaces, norms, and the stabilized-method context.

Computational domain and meshes: The physical domain Ω lies inside a larger fictitious domain Ω∗ covered by a shape-regular background tessellation T ∗.The background mesh can be constructed from a larger, easy-to-generate mesh and restricted to elements intersecting Ω.
Computational domain and meshes: Exterior facets belong to one background element, whereas interior facets are shared by two elements and form the skeleton mesh.Facets associated with boundary-intersected elements are separately identified for stabilization and geometric treatment.
Computational domain and meshes: The cut mesh T is formed from background elements intersecting Ω, with corresponding boundary and skeleton meshes obtained by intersecting facets with Ω.These sets contain both standard simplicial entities and non-standard cut entities.
Computational domain and meshes: The boundary zone consists of elements intersected by Γ and their associated facets, including elements where only a small fraction lies inside Ω.Figure 2.2 illustrates this cut-element region.
Geometric assumptions: The mesh and boundary satisfy geometric conditions controlling facet intersections, local boundary parametrization, and the number of facet crossings from cut to non-cut elements.These assumptions ensure that Γ is reasonably resolved by the background mesh.
Finite element spaces: The discrete velocity space uses continuous piecewise linear vector fields, while the pressure space uses either piecewise constants or continuous piecewise linears.The notation distinguishes the two pressure discretizations used in the method.
Norms: The paper introduces Sobolev norms, inner products, and mesh-dependent norms on the fictitious domain for finite element analysis.These norms are proper norms for discrete finite element functions defined on Ω∗.
Stabilization context: The preliminary framework reviews stabilized Stokes finite element formulations before extending them to a Nitsche-based fictitious domain method.This establishes the notation and analytical setting used in the subsequent formulation.

3. Finite element formulation.

The finite element formulation starts from stabilized low-order Stokes elements and augments the Nitsche formulation with velocity and pressure ghost penalties on facets near the embedded boundary. These terms address instability associated with cut elements and support the resulting fictitious-domain scheme.

3.1. Stabilized Stokes elements: The standard low-order velocity–pressure spaces do not satisfy the inf-sup condition, so the formulation uses a consistently stabilized discrete Stokes problem.The stabilized problem is written as a variational equation involving bilinear and linear forms.
3.1. Stabilized Stokes elements: The stabilized bilinear form combines velocity, pressure–velocity coupling, and pressure stabilization through the forms ah, bh, and ch.The form ch contains element- and facet-scaled stabilization contributions.
3.1. Stabilized Stokes elements: For piecewise linear velocity functions, the formulation includes a pressure-Poisson stabilization term whose Laplacian contribution vanishes elementwise.The resulting form remains consistent while providing stabilization for the discrete pressure treatment.
3.2. A stabilized Nitsche fictitious domain method: The velocity ghost penalty controls jumps across facets associated with elements intersecting the boundary, and the pressure stabilization is augmented correspondingly.The pressure treatment depends on whether Qh uses piecewise constants or continuous piecewise linears.
3.2. A stabilized Nitsche fictitious domain method: The ghost-penalty construction uses jumps across facets with an arbitrary fixed unit normal and introduces additional positive penalty parameters.The combined pressure form separates contributions for the two pressure discretizations.
3.2. A stabilized Nitsche fictitious domain method: The fictitious-domain method augments the stabilized Stokes form with ghost-penalty terms and solves Ah(uh, ph; vh, qh) + Jh(uh, ph; vh, qh) = Lh(vh, qh).The finite element spaces are defined relative to the background mesh T ∗, while the standard forms are evaluated on the cut mesh.
3.2. A stabilized Nitsche fictitious domain method: The method’s ghost penalty accounts for small intersections |T ∩ Ω| ≪ |T| near the boundary Γ.This is the stabilization mechanism introduced for fictitious-domain elements with very small physical portions.
3.2. A stabilized Nitsche fictitious domain method: For continuous piecewise linear pressure, the pressure ghost penalty is motivated by locally scaled Poisson stabilization and generalized in the analysis to higher-order elements.The pressure-gradient jump terms are part of the augmented fictitious-domain formulation.

4. Approximation properties.

The paper establishes interpolation tools for finite element functions on the fictitious domain, including extension, stability, trace, inverse, and boundary estimates. These tools support the later stability and error analyses.

Mesh and norm assumptions: The finite element framework uses shape-regular meshes and constants independent of the mesh size h.The relevant constants may depend on domain regularity, mesh shape regularity, and polynomial order, but not h.
Extension and interpolation: An extension operator maps functions from Ω to the larger fictitious domain Ω∗ while preserving Sobolev regularity.This enables interpolation on the background mesh rather than only on the physical domain.
Interpolation estimates: The extended interpolation operator satisfies elementwise and facetwise error estimates with orders h^(s−r) and h^(s−r−1/2), respectively.The estimates apply for 0 ≤ r ≤ s ≤ 2 and use neighboring element patches.
Energy-norm estimates: The paper derives energy-norm interpolation estimates for the chosen velocity and pressure finite element spaces.The proof combines trace inequalities, interpolation estimates, and stability of the interpolation operator.
Role in the analysis: The resulting interpolation framework is used in proving inf-sup stability for the stabilized formulation.The section explicitly connects the interpolation properties to the subsequent stability analysis.

5. Stability estimates.

The stability analysis shows that boundary-zone jump penalties control finite element norms throughout the fictitious domain and yield inf-sup stability of the stabilized Nitsche formulation.

The role of the boundary zone jump-penalties: Boundary-zone jump penalties control finite element norms on barely intersected fictitious elements using neighboring interior elements and penalty terms.The control is propagated across chains of neighboring elements from the cut region into the physical domain.
The role of the boundary zone jump-penalties: The jump-penalty construction provides the foundation for controlling norms on Ω∗ using quantities computed on Ω and in the intersection zone.The paper notes that this structure can extend to various norms and higher-order elements.
The role of the boundary zone jump-penalties: The propagation argument applies analogous estimates to velocity gradients and pressure across the boundary-zone facets.Shape regularity, trace estimates, and inverse estimates provide the required bounds for the summed facet terms.
Inf-sup stability: The bilinear form Ah + Jh satisfies an inf-sup stability condition with respect to the paper’s mesh-dependent norm.The proof combines continuity and coercivity estimates with a pressure-lifting argument and specially chosen test functions.
Inf-sup stability: The pressure coupling form bh is continuous in the relevant mesh-dependent norms.This continuity is one of the component estimates used in the inf-sup proof.

6. A priori error estimate.

The paper establishes weak Galerkin orthogonality and derives a priori error estimates for the stabilized Nitsche fictitious domain method. It also shows that the stiffness-matrix condition number is uniformly controlled relative to boundary location, while noting a consistency issue from pressure ghost penalties.

Consistency and orthogonality: Weak Galerkin orthogonality relates the exact and discrete solutions while accounting for the stabilization form Jh.This identity is used in the subsequent discrete-error estimate.
A priori error estimate: The stabilization’s weak consistency estimate controls the consistency contribution arising in the error analysis.The argument uses the regularity assumptions and interpolation properties of the extended operator.
A priori error estimate: Theorem 6.1 provides an a priori error estimate for velocity and pressure under u ∈ [H2(Ω)]d and p ∈ H1(Ω).The proof combines interpolation estimates, inf-sup stability, and weak consistency of the stabilization.
Condition number: The stiffness-matrix condition number is bounded by Ch^-2 independently of the location of the boundary relative to the background mesh.The estimate concerns the finite element formulation introduced for the fictitious domain method.

7. Condition number estimate.

The section establishes a condition-number estimate for the Nitsche fictitious domain stiffness matrix by bounding the matrix and its inverse using continuity, inverse, Poincaré, and inf-sup estimates.

Matrix representation: The stiffness matrix A is defined from the combined variational forms Ah and Jh on velocity-pressure coefficient vectors.The construction uses the Euclidean coefficient-space representation of the finite element functions.
Matrix representation: Because enclosed Stokes flow leaves pressure undetermined up to a constant, A is analyzed on a quotient space with its constant-pressure kernel removed.The kernel is ker(A) = span{C(0, 1)}.
Estimate ingredients: The analysis establishes continuity of Ah + Jh in the relevant norm and combines inverse and Poincaré inequalities with the inf-sup estimate.These ingredients provide bounds for both the operator and inverse-operator norms.
Final estimate: Theorem 7.1 states the condition-number estimate for the stiffness matrix associated with the Nitsche fictitious domain method.The proof derives separate estimates for |A|N and |A−1|N before combining them.

8. Numerical examples.

The numerical examples demonstrate automated assembly for fictitious domains, verify first-order convergence, examine boundary-position effects on conditioning, and apply the method to complex three-dimensional geometry.

Implementation: Automated assembly handles cut elements and facets through the open-source DOLFIN-OLM library and computational-geometry extensions.The implementation generates low-level code for cut-cell, cut-facet, surface, and standard integrals from high-level variational forms.
Convergence rates.: The manufactured-solution experiments use refined background meshes with three configurations that vary how the computational domain intersects the mesh.The velocity uses H1(Ω∗) errors and the pressure uses L2(Ω∗) errors for two low-order velocity-pressure pairs.
Convergence rates.: First-order convergence is obtained for both the velocity H1 error and pressure L2 error across the tested finite element pairs and mesh scenarios.The errors decrease monotonically toward zero in the reported cases, matching the theoretical prediction.
Convergence rates.: Background-mesh positioning changes error magnitudes and can produce non-monotone decreases, while scenario (C) exhibits pronounced superconvergence.The superconvergence is attributed to the shrinking fictitious-domain volume used in the error norms during refinement.
Influence of the boundary position on the condition number.: Positive ghost-penalty stabilization keeps the scaled condition number moderately bounded as the boundary position changes, whereas omitting it leads to dramatic growth.The condition number also grows with the penalty parameter when β > 0.025.
Stokes flow in a complex geometry.: The complex-geometry example shows that the method enforces prescribed velocity values on an aneurysm surface despite coarse fictitious-domain resolution.The example uses a Circle of Willis geometry embedded in a structured background mesh.
Conclusions.: The experiments verify optimal-order convergence and boundary-location-independent conditioning while demonstrating applicability to complex three-dimensional geometries.These numerical findings support the theoretical results and the method's intended use on embedded geometries.

9. Conclusions.

The paper is restricted to the static Stokes model, while its methodology and implementation are motivated by future treatment of time-dependent Navier–Stokes flow and fluid–structure interaction on complex, evolving geometries.

Limitations: The study addresses only the static Stokes model, leaving time-dependent Navier–Stokes equations and fluid–structure interaction for future work.The stated motivation extends toward complex and evolving geometries.