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A Structural Relationship Between Crouzeix-Raviart Immersed Finite Elements for Elliptic and Stokes Interface Problems

Jiaying Ren, Guozhu Yu, Xu Zhang

arXiv:2609.12462v1math.NAcs.NA

TL;DR

The paper addresses the understudied algebraic connection between scalar elliptic and Stokes immersed finite element spaces. It relates their local matrices and shows that Stokes unisolvence follows from the scalar elliptic case; numerical results confirm optimal convergence for three-dimensional velocity and pressure.

  • Problem

    The algebraic relationship between scalar elliptic and vector Stokes IFE spaces has not been systematically studied, although the connection matters for clarifying Stokes IFE structure and higher-dimensional analysis.

    Source: §1 · p. 3

  • Method

    The paper relates elliptic and gradient-form Stokes local IFE matrices, then uses incompressibility to connect gradient and stress formulations.

    Source: §1 · p. 3, §3.2 · p. 9

  • Results

    The Stokes IFE matrix determinant factors in terms of the scalar elliptic matrix, and three-dimensional experiments confirm optimal convergence for both velocity and pressure.

    Source: §1 · p. 3, §6 · p. 20

  • Takeaways & Limitations

    Stokes IFE unisolvence can be analyzed through the scalar elliptic case, simplifying the analysis in three dimensions.

    Source: §1 · p. 3, §6 · p. 20

    The three-dimensional interface-tetrahedron configuration assumes the interface intersects each tetrahedron edge at most once.

    Source: §4.1 · p. 11

Abstract

from arXiv · show

In this paper, we study the structural relationship between immersed finite element (IFE) spaces for scalar elliptic and Stokes interface problems on unfitted meshes. Although IFE methods have been widely developed for individual partial differential equation models, the algebraic connection between scalar and vector IFE spaces has not been systematically explored. We establish a precise unisolvence relationship between the immersed Crouzeix-Raviart (CR) element for elliptic interface problems and the immersed CR-$P_0$ element for Stokes interface problems in both gradient and stress formulations. By analyzing the block structure of the local IFE matrices, we show that the determinant of the Stokes IFE matrix factorizes in terms of the determinant of the corresponding elliptic IFE matrix. This factorization reveals an intrinsic algebraic link between the two classes of immersed elements and explains the unisolvence of the Stokes IFE spaces through the scalar elliptic case. The same result extends naturally to three dimensions and significantly simplifies the corresponding analysis. Numerical experiments confirm optimal convergence rates for both velocity and pressure in three-dimensional Stokes interface problems.

1 Introduction

The paper addresses the algebraic relationship between scalar elliptic and Stokes immersed finite element spaces on meshes that need not align with the interface. It connects their local constructions and uses the scalar elliptic case to analyze Stokes unisolvence.

1 Introduction

  • Unfitted meshes avoid costly interface-conforming mesh generation, while IFE shape functions incorporate interface jump conditions locally.Discontinuous coefficients across interfaces can reduce solution regularity and produce nonsmooth behavior.Source: §1 · p. 2
  • The algebraic relationship between scalar and vector IFE spaces had not been systematically studied despite extensive development of each for individual PDE models.Source: §1 · p. 3
  • The analysis links elliptic IFE spaces to gradient-form Stokes through their shared gradient structure, then relates stress and gradient formulations using incompressibility.The same scalar reduction is applied to the three-dimensional CR-P0 Stokes element.Source: §1 · p. 3
  • The local Stokes IFE matrix determinant factors in terms of the scalar elliptic IFE matrix, so scalar elliptic unisolvence implies Stokes IFE unisolvence.Source: §1 · p. 3

2 IFE Spaces for Elliptic and Stokes Interface Problems

This section defines scalar elliptic and vector Stokes CR immersed finite element spaces on unfitted meshes. The scalar construction uses edge-average degrees of freedom and interface jump conditions; the Stokes pair couples velocity and pressure through formulation-specific interface conditions.

2.1 Preliminaries

  • An interface element is cut by the interface, approximated inside the element by a line segment; mesh-interface assumptions restrict how often edges are intersected.Source: §2.1 · p. 4

2.2 Scalar CR IFE space for (P0)

  • On interface triangles, the scalar CR IFE space uses piecewise linear functions on the two subelements, with edge averages as local degrees of freedom and interface jump conditions.Source: §2.2 · p. 5

2.3 Vector CR-P0 IFE spaces for (P1)-(P2)

  • The CR-P0 Stokes pair approximates velocity with CR functions and pressure with piecewise constants, coupling them through interface jump conditions.The stress and gradient formulations use different traction jump conditions.Source: §2.3 · pp. 5–6

3 Unisolvence Relationships in Two Dimensions

The section links scalar elliptic and Stokes CR immersed elements through local-matrix structure, reducing Stokes unisolvence to the scalar elliptic case. It establishes scalar CR unisolvence, extends the result to both Stokes formulations, and notes applicability to rotated Q1-Q0 elements.

3 Unisolvence Relationships in Two Dimensions

  • The two-dimensional analysis uses block factorization to connect scalar elliptic CR and Stokes CR-P0 local IFE matrices.Source: §3 · p. 6

3.1 Unisolvence of Scalar CR IFE Functions for Elliptic Interface Problem (P0)

  • Scalar CR-IFE shape functions are uniquely determined by prescribed edge values and jump conditions, regardless of interface location or coefficient jump.This unisolvence result supplies the scalar building block for the Stokes analysis.Source: §3.1 · p. 7

3.2 Unisolvence of CR-P0 IFE functions for Stokes Interface Problems

  • The gradient-form Stokes CR-P0 element is unisolvent whenever the scalar elliptic CR-IFE element is unisolvent.The local Stokes system is algebraically reducible to the scalar elliptic case.Source: §3.2 · pp. 8–9
  • The stress-form CR-P0 element is also unisolvent, and its extra traction term adds no independent constraint beyond the existing CR degrees of freedom and divergence constraint.The determinant remains unchanged by row reduction to the gradient-form matrix.Source: §3.2 · p. 10
  • The determinant and unisolvence results also extend to rotated Q1-Q0 IFE spaces on rectangular meshes.Source: §3.2 · p. 10

4 Three-Dimensional IFE Spaces for Stokes Problems

The paper develops three-dimensional CR-P0 immersed finite element spaces for Stokes interface problems in gradient and stress formulations. Their local-matrix determinant factorization links them to the scalar elliptic IFE element, yielding unisolvence.

4.1 CR-P0 IFE Spaces

  • The construction assumes at most one interface intersection per edge and at most two per face, yielding Type I and Type II interface tetrahedra.For Type II, three selected intersection points define an approximation plane; a level-set-based alternative is also described, and the algebraic results apply to it.Source: §4.1 · p. 11
  • Velocity is approximated by vector-valued linear polynomials and pressure by piecewise constants, with extra care needed for the third velocity component and interface-tetrahedron geometry.The local conditions uniquely determine the immersed shape-function coefficients.Source: §4.1 · pp. 11–12
  • Each tetrahedral element has 13 local degrees of freedom: face-average values for each velocity component and an element-average pressure value.The coefficients are determined by these degrees of freedom and the applicable interface jump conditions.Source: §4.1 · pp. 11–12

4.2 Unisolvence

  • The three-dimensional gradient-form Stokes IFE matrix factorizes through the scalar elliptic IFE matrix, and the corresponding CR-P0 space is unisolvent.The factorization is dimension-independent and follows from the block structure and interface constraints; unisolvence follows from the scalar elliptic result.Source: §4.2 · pp. 12, 14
  • The stress-form local matrix has the same determinant as the gradient-form matrix, establishing the structural link for both formulations.The paper states the equivalence for admissible interface configurations and positive µ±.Source: §4.2 · p. 15

5 Immersed Finite Element Methods and Numerical Examples

Three-dimensional numerical tests of the CR-P0 IFE methods examine interpolation and solution errors on unfitted meshes for gradient and stress formulations. The reported convergence rates are optimal across the tested cases.

5 Immersed Finite Element Methods and Numerical Examples

  • The experiments focus on three dimensions, measuring velocity in L2 and broken H1 norms and pressure in L2 on unfitted Cartesian meshes subdivided into tetrahedra.The numerical tests use both gradient and stress formulations.Source: §5 · pp. 15–16

5.1 Weak formulation for Problem (P1)

  • Second-order velocity L2 interpolation convergence and first-order broken H1 velocity and L2 pressure convergence are observed for the planar-interface example; its solution errors also converge optimally.These results are consistent with those reported for the two-dimensional case.Source: §5.1 · p. 17
  • For this problem, interpolation errors (Table 3) and IFE solution errors (Table 4), including reversed coefficient contrast (Table 5) and a large coefficient jump (Table 6), show optimal convergence rates.Source: §5.1 · p. 18

5.2 Weak formulation for Problem (P2)

  • The spherical-interface stress formulation also shows optimal convergence rates in all reported solution-error norms.Interpolation errors are omitted because they are very similar to those in the gradient-form example.Source: §5.2 · p. 19

6 Conclusion

The conclusion links scalar elliptic and Stokes Crouzeix–Raviart immersed finite element spaces through a determinant factorization. This relation explains Stokes-space unisolvence and extends to both formulations and three dimensions.

6 Conclusion

  • The local CR-P0 Stokes IFE matrix has a determinant factorization in terms of the scalar elliptic CR IFE matrix, providing a unified explanation of Stokes-space unisolvence.The factorization applies to both gradient and stress formulations.Source: §6 · pp. 19–20
  • The factorization extends naturally to three dimensions and reduces the Stokes unisolvence analysis to the scalar elliptic case.Source: §6 · p. 20
  • Numerical results confirm optimal convergence for both velocity and pressure.Source: §6 · p. 20

Declarations

The paper reports funding support, no conflicts of interest, and that its findings are supported by information contained in the article.

Funding

  • Guozhu Yu and Xu Zhang report partial support from their respective funding agencies.The funding statement lists grant numbers 12561071 and DMS-2110833.Source: Funding · p. 20

Conflict of Interest

Data Availability Statement

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