Construction of trace-preserving Fortin operators

Franziska Eickmann; Tabea Tscherpel

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Construction of trace-preserving Fortin operators

Franziska Eickmann, Tabea Tscherpel

arXiv:2609.04926v1Sep 4, 2026math.NA

TL;DR

The paper addresses the need for Fortin operators that provide more than abstract divergence preservation, including locality, stability, approximation, and trace control. It develops a unified correction framework, with velocity enrichment where required, and applies it across several conforming finite element pairs. The resulting properties support applications to nonlinear fluids, nonhomogeneous boundary conditions, and uniform inf-sup stability for domain approximation and moving domains.

Problem
Abstract Fortin operators do not generally provide the locality, W 1,p stability, approximation, and trace-preservation properties needed in several incompressible-flow applications.
Method
The paper splits Fortin corrections into trace and divergence components and constructs local operators for several conforming finite element pairs, enriching velocity spaces with facet bubbles when needed.
Results
The framework covers Bernardi–Raugel, Pd−P0, conforming Crouzeix–Raviart, modified MINI, Taylor–Hood, and higher-order elements, and establishes uniform discrete inf-sup stability for p∈(1,∞) in domain approximation and moving-domain settings.
Takeaways & Limitations
The constructed Fortin operators support applications to non-Newtonian fluids, inhomogeneous Dirichlet conditions, and uniform inf-sup analysis for changing domains.
Takeaways & Limitations
Boundary-facet bubble enrichment can require higher-degree functions and may be more difficult to implement despite potentially reducing degrees of freedom.

Abstract

from arXiv · show

We present a unifying framework to construct local Fortin operators for conforming mixed finite element pairs for the Stokes equations. The operators are constructed to satisfy the divergence-preservation property, local stability and approximation properties, and certain trace-preservation properties. For the latter, some of the finite element pairs require an enrichment of the velocity space. We present the construction for the $P_d-P_0$ element, the Bernardi-Raugel element, the conforming Crouzeix-Raviart element, a modified MINI element and generalised Taylor-Hood elements. Furthermore, we discuss implications of the existence of such a Fortin operator beyond inf-sup stability. These include applications to non-Newtonian fluid flow, problems with inhomogeneous Dirichlet boundary conditions, as well as uniform inf-sup stability relevant for domain approximation and moving domains.

1. Introduction

The paper develops a unified framework for local, stable, divergence-preserving Fortin operators that also preserve selected traces, extending constructions across conforming mixed finite element pairs. These properties support applications beyond inf-sup stability, including nonlinear fluids, nonhomogeneous boundary conditions, and domain approximation.

Motivation: Fortin operators are central to discrete inf-sup stability because they preserve the discrete divergence when tested against discrete pressures.The abstract Fortin lemma does not itself provide locality, broad W 1,p stability, approximation, or trace preservation.
Framework: The framework splits the correction into trace and divergence components, using velocity-space enrichment with boundary facet bubbles when necessary.For discontinuous pressures, local integration by parts naturally produces facet terms; continuous pressures require modified corrections.
Finite element pairs: The construction covers Bernardi–Raugel, Pd−P0, conforming Crouzeix–Raviart, modified MINI, Taylor–Hood, and higher-order conforming pairs.The paper also gives a general construction for inf-sup stable pairs, although the resulting operator is not local.
Novelty: The authors present trace-preserving constructions for finite element pairs with continuous pressures, recasting existing correction methods in a more constructive form.They describe the trace-preserving framework as new for continuous pressure spaces and relate the corrections to local macro-element techniques.
Applications: The framework is applied to non-Newtonian flow, inhomogeneous Dirichlet conditions, and uniform inf-sup stability for p∈(1,∞) under domain approximation and moving domains.The domain result uses locality of the Fortin operator together with uniform continuous inf-sup stability on sequences of John domains.

2. Preliminaries on mixed finite element methods

The preliminaries define the function spaces, triangulations, finite element neighborhoods, bubbles, and quasi-interpolation operators used to construct Fortin maps. Scott–Zhang-type operators provide the baseline trace preservation, local stability, and approximation properties.

Function spaces: The paper sets up bounded Lipschitz domains, Sobolev spaces, zero and normal-trace subspaces, and conforming velocity-pressure finite element spaces.The velocity and pressure spaces are denoted Xh⊂W 1,∞(Ω)d and Qh⊂L∞(Ω), respectively.
Triangulations: A shape-regular conforming triangulation is described through simplices, maximal mesh size, facets, boundary facets, and element inradii.The notation includes hT, the facet sets F(Th) and F∂(Th), and the shape-regularity parameter χ.
Locality: Element, facet, edge, and vertex neighborhoods define the local domains on which correction and bubble constructions are supported.The notation ωh(Sh), Ωh(σ), and Ωz records neighboring simplices and their covered domains.
Finite element spaces: The finite element setup introduces polynomial, Lagrange, bubble, and facet-bubble spaces together with their global and boundary-restricted counterparts.These spaces are used to specify local velocity enrichments and support the later Fortin constructions.
Quasi-interpolation: Scott–Zhang-type quasi-interpolation operators provide zero-trace preservation, local W 1,1 stability, and local approximation estimates.The operators can also preserve discrete traces and traces on resolved boundary subsets, and the construction extends to enriched spaces containing polynomial subspaces.
Fortin operators: The paper constructs a single Fortin operator with the desired combined properties because the abstract Fortin lemma generally does not guarantee locality, projection structure, broad W 1,p stability, or trace preservation.This motivates the subsequent constructive framework.

3. Properties of Fortin operators

The paper specifies Fortin-operator properties covering projection, divergence and trace preservation, global and local stability, approximation, and locality. It also records which finite element pairs satisfy the full property set.

The Fortin operator is defined as a linear projection onto the discrete velocity space Xh.
It preserves the discrete divergence when tested against every discrete pressure function.
Trace preservation includes zero traces and zero normal traces, with additional mean-normal-trace preservation among the stated properties.
Global stability and approximation hold for p ∈ [1, ∞], while local versions hold elementwise with constants controlled by mesh shape regularity and finite-element dimensions.
The framework can preserve zero traces on a resolved boundary subset without imposing zero traces on the remainder, supporting mixed boundary conditions.
For the Pd −P0, lowest-order Bernardi–Raugel, conforming Crouzeix–Raviart, and Guzmán–Neilan elements, the constructions satisfy all properties (P0)–(P4).

4. Applications

The constructed Fortin operators support quasi-optimal Stokes estimates, non-Newtonian-flow analysis, and uniform discrete inf-sup stability under domain approximation. These applications rely especially on trace preservation, locality, and uniform continuous stability.

Trace preservation yields quasi-optimal Stokes error estimates with inhomogeneous Dirichlet data and without extra data-approximation terms.The estimate is uniform in h when the Fortin operator preserves traces of discrete velocity functions.
Locality is important for handling nonlinear terms in quasi-norm-based a priori estimates for non-Newtonian fluid flows.
The framework verifies the inhomogeneous divergence-preservation assumptions for the Bernardi–Raugel, Pd −P0, conforming Crouzeix–Raviart, enriched MINI, and Taylor–Hood elements.
The domain sequences may consist of John domains with polyhedral boundaries and need not be Lipschitz.
Uniform discrete inf-sup stability follows for p ∈ (1, ∞) on suitable domain sequences when local Fortin operators and uniform continuous inf-sup stability are available.
The uniform inf-sup constant depends on p, the continuous stability bound, mesh shape regularity, dimension, and local finite-element dimensions, but not on the domain-sequence index.

5. Local Fortin operators

The paper presents a unified construction for local Fortin operators across discontinuous- and continuous-pressure finite element pairs. Existing constructions are retained for several discontinuous-pressure elements, while continuous-pressure cases require modifications or enrichment.

The framework organizes constructions for Bernardi–Raugel, Pd −P0, conforming Crouzeix–Raviart, MINI, and Taylor–Hood finite element pairs.
For Bernardi–Raugel, Pd −P0, and conforming Crouzeix–Raviart elements, homogeneous-case Fortin operators require no modification to satisfy properties (P0)–(P4).
The MINI construction modifies the velocity space to obtain a local trace-preserving Fortin operator.
For generalized Taylor–Hood elements with k ≥ d, the paper gives a local construction satisfying properties (P0)–(P4).

5.1. General construction principle.

The general construction starts from a local Scott–Zhang-type quasi-interpolation operator and applies trace and divergence corrections. Locality, preservation, and approximation then follow from corresponding properties of the ingredients and correction operators.

If both the interpolation and correction operators are local, the resulting Fortin operator is local.
The Fortin operator is built by correcting a quasi-interpolation operator with operators that enforce trace and divergence conditions.
The correction operators are required to be bounded, locally stable, divergence-preserving, and compatible with facet normal-trace constraints.
The construction preserves divergence because the correction satisfies the discrete divergence identity.
Trace preservation follows from zero-trace properties of the interpolation and correction operators, including mean normal traces on boundary facets.
Local approximation and global stability follow from correction stability, interpolation approximation, inverse estimates, scaling, and mesh shape regularity.
If a correction operator is nonlocal, the construction still provides properties (P0)–(P3) but not locality.

5.2. Elements with discontinuous pressure space.

For discontinuous pressure spaces, the framework constructs local Fortin operators through divergence corrections, with trace-preserving and stability properties verified across several finite element families. The Bernardi–Raugel, P_d-P_0, conforming Crouzeix–Raviart, and related constructions achieve the stated Fortin properties under their respective degree and dimension conditions.

General framework: For discontinuous pressures, divergence preservation is local, simplifying the construction of the Fortin operator.The framework verifies properties (P0)–(P4) without modifying the construction used for the homogeneous case.
Bernardi–Raugel element: The lowest-order Bernardi–Raugel operator satisfies (P0)–(P4) with k = 1 and s = 1, and can also preserve traces on a selected boundary part.Its quasi-interpolation operator itself provides the Fortin construction, with normal-trace preservation ensured by facet degrees of freedom.
Bernardi–Raugel element: Higher-order Bernardi–Raugel constructions also satisfy all stated properties, including trace preservation, for the described second-order three-dimensional case.The corresponding quasi-interpolation operator satisfies (P0)–(P4) with k = 2 and s = 1.
P_d-P_0 element: The same Bernardi–Raugel construction is a Fortin operator for P_d-P_0, while a Scott–Zhang-based variant recovers approximation order k = d.The lowest-order Bernardi–Raugel interpretation gives reduced approximation order, whereas the corrected Scott–Zhang operator achieves (P0)–(P4) with k = d.
Conforming Crouzeix–Raviart element: For conforming Crouzeix–Raviart elements, the correction approach yields Fortin operators satisfying (P0)–(P4) for k ≥ d in dimensions d ∈ {2, 3}.The construction uses a projection onto the velocity space and can also satisfy the selected-boundary trace property (P2)_Γ.

5.3. Elements with continuous pressure space.

For continuous pressure spaces, trace preservation requires corrections that exploit integration by parts and, in low-order cases, enrich the velocity space. The framework establishes the desired Fortin properties for modified MINI and generalised Taylor–Hood elements, while identifying specific enrichment and boundary-trace trade-offs.

Modified MINI element: The modified MINI Fortin operator satisfies (P0)–(P4) with k = s = 1 and can also satisfy (P2)_Γ.The enrichment ensures both correction operators map into the modified velocity space.
Modified MINI element: The modified MINI construction uses element and facet-bubble corrections to handle continuous pressures and preserve traces for nonzero boundary data.Continuity of the pressure permits integration by parts, while boundary facet bubbles control the resulting trace terms.
Section synthesis: The modified MINI and generalised Taylor–Hood constructions extend the framework to continuous pressure spaces while retaining local Fortin-operator properties.The resulting operators are designed around the pressure-space continuity and the associated integration-by-parts structure.
Modified MINI element: Boundary-facet enrichment uses bubbles one degree higher than the alternative modification, reducing the number of added facets but increasing implementation difficulty.Only boundary facet bubbles are needed in this approach, although the higher degree raises its implementation challenge.
Generalised Taylor–Hood element: For generalised Taylor–Hood elements, the framework constructs a local Fortin operator for k ≥ d using vertex-patch corrections and proves the required local stability and divergence preservation.The construction relies on finitely many reference vertex-patch types and positive-definite local matrices.
Generalised Taylor–Hood element: The Taylor–Hood construction adds an extra correction operator to obtain all trace-preservation properties, but the lowest-order case requires velocity-space enrichment.This contrasts with earlier constructions that used boundary tangential edge bubbles without preserving zero traces.

6. Abstract global Fortin operators

The section constructs trace-preserving Fortin operators by combining interpolation with separate trace and divergence corrections. The resulting framework applies to general inf-sup stable pairs, while local constructions provide stronger locality and stability properties.

Construction framework: The construction combines an interpolation operator with trace and divergence corrections to obtain a Fortin operator preserving both traces and discrete divergence.The trace correction handles boundary and, where needed, interior facet traces; the divergence correction completes the Fortin construction.
Scope and trade-offs: The abstract construction applies to all inf-sup stable finite element pairs, but unlike the local constructions it does not ensure locality.Its broader applicability comes with fewer properties than the element-specific local operators.
Trace correction: The boundary trace correction is linear, locally W 1,p-stable for p ∈[1, ∞], supported in a boundary layer, and preserves the relevant traces.Its locality follows from support properties near boundary facets, and its stability is obtained through scaling, inverse estimates, and Hölder’s inequality.
Global correction: The global divergence correction is built from a homogeneous Fortin operator and preserves discrete divergence while providing global stability.The correction is generally based on a global operator, so its stability scaling is not optimal.