Mixed Non-Conforming FEM

• Mixed non-conforming FEM are finite element approaches that use non-conforming spaces for primary or auxiliary variables within a mixed formulation to improve flexibility and local conservation.
• These methods leverage saddle-point theory and bubble enrichment to maintain stability and achieve optimal convergence even in the presence of complex interfaces and discontinuous coefficients.
• They are widely applied in fields like fluid mechanics, elasticity, and multi-physics simulations where mesh flexibility and accurate enforcement of interface conditions are essential.

A mixed non-conforming finite element method (FEM) is a finite element methodology wherein either the approximation spaces for the primary variables or the auxiliary variables (such as Lagrange multipliers) are chosen to be non-conforming—meaning these spaces are not subspaces of the regularity class naturally required by the weak form—while at the same time, a mixed formulation is used to treat multiple field variables or to enforce constraints. These methods are of central importance in the numerical solution of interface problems, incompressible fluid mechanics, elasticity, and a broad array of transmission and multi-physics problems, especially where geometric complexity or the need for local conservation properties makes strictly conforming discretizations undesirable or inefficient.

1. Core Concepts and Variational Foundations

Mixed non-conforming FEMs arise when the variational problem involves multiple field variables and coupling via constraints or conservation laws, and at least one finite element space is chosen to be non-conforming, i.e., not a subset of the required energy space. The canonical mixed variational form seeks $(u, p) \in V \times Q$ such that

$$\begin{aligned} a(u, v) + b(v, p) &= f(v) &\forall v \in V \ b(u, q) &= g(q) &\forall q \in Q \ \end{aligned}$$

where $a(\cdot,\cdot)$ is coercive on $\ker b$, $b(\cdot, \cdot)$ encodes constraints (e.g., divergence, interfacial transmission), $V$ a trial/test space (not necessarily conforming to, e.g., $H^1$), and $Q$ a multiplier or secondary variable space.

Non-conformity may appear in:

• The primary variable space (e.g., velocity in incompressible Stokes problems with Crouzeix-Raviart elements, or stress in certain elasticity elements).
• The multiplier/auxiliary variable space (e.g., discontinuous Lagrange multiplier for enforcing constraints on unfitted domains or at interfaces).

The methodology is distinguished by the use of saddle point (Babuška–Brezzi) theory for well-posedness, with stability relying critically on the "inf-sup" (LBB) condition between the chosen non-conforming spaces.

2. Structural Advantages: Unfitted and Interface Problems

A principal advantage of mixed non-conforming FEMs is their flexibility for interface and transmission problems, particularly in settings with complex, possibly non-matching or unfitted meshes. For instance, in elliptic interface problems with jump coefficients (such as

$-\nabla \cdot (\beta \nabla u) = f$

in $\Omega$, with $\beta$ discontinuous across an interface $\Gamma$), mixed non-conforming approaches—via fictitious domain and distributed Lagrange multiplier (DLM) frameworks—permit the use of meshes that do not align with $\Gamma$. The saddle-point formulation

$$\begin{aligned} (\beta \nabla u, \nabla v)_\Omega + \langle \lambda, v|_{\Omega_2} \rangle &= (f, v)_\Omega, \quad \forall v \in V \ (\beta_2 - \beta) \nabla u_2, \nabla v_2)_{\Omega_2} - \langle\lambda, v_2\rangle &= (f_2 - f, v_2)_{\Omega_2}, \forall v_2 \in V_2 \ \langle \mu, u|_{\Omega_2} - u_2 \rangle &= 0, \forall \mu \in \Lambda \end{aligned}$$

enables robust enforcement of interfacial continuity and flux jump conditions even on nonmatching, possibly independent mesh partitions of $\Omega_1$ and $\Omega_2$ (Alshehri et al., 2022).

Employing a discontinuous (piecewise constant) multiplier, and enriching spaces with bubble functions as needed, these schemes support greater flexibility in mesh design, crucial for simulations involving evolving domains or moving interfaces, as in fluid-structure interaction.

3. Typical Finite Element Spaces and Stability Techniques

Mixed non-conforming FEM systems are instantiated using spaces such as:

• Non-conforming piecewise polynomial vector fields for primary variables (e.g., Crouzeix–Raviart $P_1$, Park–Sheen on quads, or enriched $P_2$ elements with bubbles in 3D (Zhang, 2 Aug 2024)).
• Discontinuous Lagrange multiplier or flux spaces (e.g., $P_0$ piecewise constants).
• Bubble-enriched subspaces for stability (e.g., $Q_1+(B)$—bilinear plus interior bubbles—required for discrete inf-sup stability in DLM formulations).

The stability of the mixed non-conforming discretization is analyzed via the Babuška–Brezzi conditions:

• Discrete kernel coercivity (Elker condition): Stability on the kernel of $b$ is ensured by mesh-independent coercivity constants.
• Discrete inf-sup (LBB) condition: For any $\mu_h$ in the multiplier space, there must exist $(v_h, v_{2h})$ in the trial spaces such that the constraint is enforced robustly: $$\sup_{(v_h, v_{2h})} \frac{(\mu_h, v_h|_{\Omega_2} - v_{2h})_{\Omega_2}} {(\|v_h\|_V^2 + \|v_{2h}\|_{V_2}^2)^{1/2}} \ge \gamma \|\mu_h\|_\Lambda$$ Bubble function enrichment is often necessary; omitting these can cause loss of stability and convergence (e.g., $Q_1-Q_1-P_0$ without bubbles fails the inf-sup test).

4. Error Bounds, Convergence, and Mesh Regularity

Convergence analysis yields optimal a priori error estimates under standard regularity, modulated by interface geometry and coefficient jumps. For elliptic interface problems,

$$\| u - u_h \|_V + \| u_2 - u_{2h} \|_{V_2} + \| \lambda - \lambda_h \|_\Lambda \leq C \Big( h^{r-1} \|u\|_{H^r(\Omega)} + \max\left( h_2^{s-1}, h_2^{1-t} \right) \|u_2\|_{H^s(\Omega_2)} + \ldots \Big)$$

with $r$ and $s$ reflecting solution regularity near the interface.

For non-conforming elements on anisotropic or mixed meshes (e.g., triangles plus quadrilaterals near interfaces), optimal order approximation and consistency errors are established—often without classical shape regularity or quasi-regularity assumptions, relying only on conditions like maximum angle or regular decomposition property (RDP) for macro-element subdivision (Wang et al., 18 Jun 2025).

5. Applications: Fluid-Structure, Heterogeneous Media, Interface Conservation

These methods have found application in diverse scientific computing domains:

• Elliptic interface and transmission problems: Fictitious domain mixed non-conforming FEMs provide robust and flexible treatment of jump conditions, enabling practical simulation in composite and multi-physics domains (Alshehri et al., 2022, Sacco et al., 2018).
• Incompressible and Stokes flows: Non-conforming elements (e.g., Crouzeix–Raviart, bubble-enriched $P_2$/$P_1$ (Zhang, 2 Aug 2024), non-conforming MsFEM (Muljadi et al., 2014)) yield stable saddle-point formulations with favorable mass conservation and robustness to mesh artifacts.
• Elasticity, especially in 3D: Non-conforming mixed elements for stress/displacement (e.g., Arnold–Awanou–Winther (Arnold et al., 2012), Awanou (Awanou, 2010, Hu et al., 2016)) reduce complexity compared to conforming alternatives, simplify implementation, and offer strong symmetry with minimal DOFs.
• Thermodynamically coupled or active fluid problems: Mixed non-conforming methods, often with auxiliary variables and weak constraint enforcement, enable efficient discretization of high-order PDEs and enforce divergence-free or interface conditions (Zheng et al., 23 Sep 2025).

6. Numerical Evidence and Robustness

Comprehensive numerical studies validate these methods across variable interface geometries, high coefficient jumps, and anisotropic mesh configurations. Key observations include:

• Stability and convergence rates are optimal if bubble enrichment and inf-sup conditions are satisfied.
• The performance is robust to mesh mismatch and interface complexity; no re-meshing is needed for moving geometries.
• Local conservation properties (e.g., flux or mass) improve when using discontinuous (piecewise constant) multipliers, paralleling classical mixed FEM advantages in flow and transport.
• Failing to include necessary enrichment or having insufficiently robust non-conforming space selection leads to loss of stability and deteriorated convergence.

Scheme	Main Variable Spaces	Multiplier Space	Stability	Convergence
$Q_1-(Q_1+B)-P_0$	Bilinear, bubble-enriched	Piecewise constant	Stable	Optimal
$Q_2-Q_2-P_0$	Biquadratic	Piecewise constant	Stable	Slightly superior
$Q_1-Q_1-P_0$	Bilinear (no bubbles)	Piecewise constant	Unstable	Fails

7. Broader Context and Significance

Mixed non-conforming FEMs generalize and connect diverse finite element families—spanning methods for incompressible flow (Crouzeix–Raviart, MINI), elasticity (Arnold–Winther, Awanou), domain decomposition, mortar, hybrid and fictitious domain techniques, to contemporary unfitted and immersed boundary approaches.

Their flexibility, local conservation, and mesh-agnostic enforcement of interface conditions—along with rigorous theory for stability and error—make them essential tools in computational mathematics and engineering. Their continuous development now includes error estimation, adaptive refinement, and extension to strongly nonlinear, multiphysics settings. The methods provide both a robust practical discretization strategy and a framework for understanding the interplay between discretization, conservation, and interface handling within the mathematical theory of finite element approximation.