--- title: Immersed Raviart–Thomas Space Overview url: https://www.emergentmind.com/topics/immersed-raviart-thomas-space type: topic --- # Immersed Raviart–Thomas Space Overview Searching arXiv for the cited papers to ground the article in current records. Immersed Raviart–Thomas space denotes a family of constructions in which the classical Raviart–Thomas (RT) finite element structure is modified or embedded so that flux unknowns better reflect geometric or interface information that is not represented by a standard fitted simplicial mesh. In recent work, the term is used in at least three technically distinct senses: a Petrov–Galerkin trial flux space whose boundary degrees of freedom are shifted from a polyhedral approximation to the true curved boundary of a smooth domain; a locally modified RT space on unfitted interface elements; and an RT-bubble enrichment embedded into Scott–Vogelius velocity spaces to restore inf–sup stability on general simplicial grids [2602.21197] [2105.03227] [2206.01242] [2604.02153]. ## 1. Terminology and range of meanings The terminology is not uniform. In the curved-domain mixed formulation of Raviart–Thomas elements on straight-edged tetrahedra, the immersed space is the trial flux space \(P_h^k\), whose RT face-normal degrees of freedom on Neumann faces are imposed at points on the true boundary \(\Gamma\) rather than on the polyhedral boundary \(\Gamma_h\) [2602.21197]. In unfitted interface methods, the immersed RT space is a local element space on cut cells, typically piecewise RT on the subcells \(T^\pm\) or \(T_i\), with additional interface constraints that encode continuity of normal flux and a tangential constitutive condition [2105.03227] [2604.02153]. By contrast, in the Scott–Vogelius stabilization literature, “immersed RT” explicitly refers to embedding a small RT bubble subspace into an \(H^1\)-conforming velocity space; it does **not** refer to unfitted or immersed boundary/interface discretization [2206.01242]. | Usage | Defining mechanism | Representative paper | |---|---|---| | Curved-boundary trial space | Boundary RT DOFs shifted from \(\Gamma_h\) to \(\Gamma\) | [2602.21197] | | Unfitted interface space | Piecewise RT on cut cells with interface constraints | [2105.03227], [2604.02153] | | Scott–Vogelius enrichment | RT bubbles embedded into \(V_k\) | [2206.01242] | This suggests that “immersed” functions less as the name of a single canonical finite element family than as a design principle: standard RT structure is retained locally, while geometry-, interface-, or stability-critical information is injected through modified constraints, modified degrees of freedom, or embedded bubble components. ## 2. Classical Raviart–Thomas structure retained under immersion All recent variants preserve the algebraic core of the classical RT family. On a simplex \(K\), the local space is \[ RT_k(K)= [\mathcal P_k(K)]^d + x\,\mathcal P_k(K), \] with \[ \nabla\cdot RT_k(K)=\mathcal P_k(K), \] and the usual unisolvent degrees of freedom are face-normal flux moments together with suitable interior moments for \(k\ge 1\) [2602.21197] [2206.01242] [2604.02153]. In the lowest-order triangular case, \[ RT(T)=\{v(x)=a+c\,x: a\in\mathbb R^2,\ c\in\mathbb R\}, \] so the divergence is constant on each element and the edge-normal flux moments determine the field [2105.03227]. The contravariant Piola transform remains central. For an affine map \(F:\hat K\to K\) with Jacobian \(J=\nabla F\), the RT pullback is \[ v(x)=\frac{1}{\det J}\,J\,\hat v(F^{-1}(x)), \] which preserves normal fluxes and transforms divergence by \[ \nabla\cdot v = \frac{1}{\det J}\,\hat\nabla\cdot \hat v. \] This is exploited both in the curved-domain tetrahedral setting, where the geometry is kept straight-edged and “immersion” is realized only through shifted boundary evaluations, and in RT-bubble enrichment, where elementwise reference-space constructions transfer by simple scaling [2602.21197] [2206.01242]. What changes from one immersed variant to another is therefore not the local RT polynomial ansatz itself, but the manner in which continuity, boundary data, interface laws, or divergence-surjectivity are enforced. ## 3. Curved domains with straight-edged tetrahedra For smooth curved three-dimensional domains, the mixed model considered in recent work is the first-order system \[ p-\nabla u=0 \quad \text{in }\Omega,\qquad -\nabla\cdot p + v u = f \quad \text{in }\Omega, \] with \(u=0\) on \(\Gamma_0\) and \(p\cdot n = 0\) on \(\Gamma_1\), where \(v\ge 0\) and \(v=0\) is allowed. The mixed weak formulation uses \[ Q:=\{q\in H(\mathrm{div};\Omega): q\cdot n=0 \text{ on }\Gamma_1\}, \] and \(V:=L^2(\Omega)\), with the standard mean-zero condition when \(v=0\) [2602.21197]. On a straight-edged tetrahedral mesh \(\mathcal T_h\) approximating \(\Omega\) by a polyhedron \(\Omega_h\), three discrete spaces are introduced. The scalar space \(V_h^k\) consists of piecewise polynomials of total degree at most \(k\). The test flux space \(Q_h^k\) is the standard \(H(\mathrm{div};\Omega_h)\)-conforming RT space with \(q\cdot n_T=0\) on polyhedral Neumann faces \(F_T\subset \Gamma_{1,h}\). The immersed trial flux space \(P_h^k\) is also elementwise \(RT_k(T)\), but on Neumann faces its boundary DOFs are enforced at points \(P\) on the true boundary \(\Gamma_1\), obtained by shifting designated face quadrature points \(M\in F_T\) along half-lines \(OM\) to their nearest intersections with \(\Gamma_1\). The number of shifted points is \[ m_k=\frac{(k+2)(k+1)}{2}; \] for \(k=0\) there is one point, the face centroid, and for \(k=1\) there are three symmetric barycentric points \((2/3,1/6,1/6)\), \((1/6,2/3,1/6)\), \((1/6,1/6,2/3)\) [2602.21197]. The resulting Petrov–Galerkin formulation keeps testing on the polyhedral boundary but imposes trial-side normal flux conditions against the true normal \(n(P)\) on the true boundary. In operational terms, the local RT shape functions are built on straight tetrahedra and their face traces are evaluated at the shifted points \(P_i\), so no nonaffine curved-element mapping is required. For general Neumann data \(g\), the curved-boundary enforcement is represented by \[ \int_{F,\mathrm{true}} (p_h\cdot n)\,q\,ds \approx \sum_{i=1}^{m_k} w_i\, (p_h(P_i)\cdot n(P_i))\, q(P_i) = \sum_{i=1}^{m_k} w_i\, g(P_i)\, q(P_i). \] The principal motivation is the observed accuracy downgrade that occurs when true flux DOFs live on \(\Gamma\) but are imposed on \(\Gamma_h\). The immersed trial space is designed to avoid that boundary shift. Numerical evidence for the two lowest orders supports the design. For \(RT_0\), the immersed Petrov–Galerkin method and the “do-nothing” Galerkin method both achieve first order in \(H(\mathrm{div})\) for \(p\) and its divergence on the unit ball with \(v=1\), but the immersed method is clearly more accurate in \(u_h\), with \(L^2\) experimental order of convergence \(\approx 1.48\) versus \(\approx 1.58\) for Galerkin and consistently smaller errors. For \(v=0\), first-order convergence of \(p\) persists and the immersed method is more accurate across all reported metrics. On a hollow ellipsoid with outer Neumann and inner Dirichlet data, optimal first-order rates are recovered for \(p\) and its divergence, while \(u_h\) shows near-second-order behavior for both formulations, with the immersed method slightly more accurate [2602.21197]. The same paper also studies the Hermite analog \(HRT0\), where \(u_h\) is approximated in an incomplete quadratic polynomial space \(U_H\) and \(p_h\approx V_h u_h\) via the broken gradient. The bilinear form \[ a_h(u_h,v_h)= (V_hu_h,V_hv_h)_{\Omega_h} - (\Delta_h u_h,v_h)_{\Omega_h} + (u_h,\Delta_h v_h)_{\Omega_h} + v(u_h,v_h)_{\Omega_h} \] delivers second-order \(L^2\) convergence for \(u\) and first-order convergence for \(p\) in \(H(\mathrm{div})\). In the reported tests, immersed \(HRT0\) consistently outperforms its Galerkin counterpart for \(u\), sometimes with superconvergent behavior for \(p\) [2602.21197]. For \(RT_1\) with \(v>0\), a symmetric enriched Petrov–Galerkin variant is introduced: \[ -(\nabla\cdot p_h,v_h)_{\Omega_h}+v(u_h,v_h)_{\Omega_h}=(f,v_h)_{\Omega_h}, \] \[ (p_h,q_h)_{\Omega_h}-(u_h,\nabla\cdot q_h)_{\Omega_h} +2v^{-1}(\nabla\cdot p_h,\nabla\cdot q_h)_{\Omega_h} =-2v^{-1}(f,\nabla\cdot q_h)_{\Omega_h}. \] This formulation yields SPD linear systems. The reported implementation uses conjugate gradients and static condensation of seven element-internal DOFs, leaving only face-normal flux DOFs globally. In the unit ball test, immersed \(RT_1\) achieves EOCs \(\approx 1.85\)–\(1.88\) for \(\|p-p_h\|\) and \(\|\nabla\cdot(p-p_h)\|\), versus \(\approx 1.96\)–\(1.97\) for Galerkin, but with significantly smaller errors, especially for \(u_h\). The ellipsoidal and hollow-ball tests show the same pattern: near-optimal rates with improved accuracy relative to the do-nothing strategy, and no significant order loss from the inner polyhedral approximation. Stability with respect to small \(v\) is also reported down to \(v=10^{-8}\) on fixed meshes [2602.21197]. ## 4. Unfitted interface formulations on cut elements In unfitted interface problems, the immersed RT construction is local to elements cut by the interface. For a polygonal domain \(\Omega\subset \mathbb R^2\) split by a smooth closed interface \(\Gamma\) into \(\Omega^-\) and \(\Omega^+\), one writes the mixed system for \(p=\beta\nabla u\) as \[ B p-\nabla u=0,\qquad -\nabla\cdot p = f \quad \text{in }\Omega\setminus \Gamma, \] with interface conditions \[ [u]_\Gamma=0,\qquad [p\cdot n]_\Gamma=0,\qquad [Bp\cdot t]_\Gamma=0, \] where \(B=\beta^{-1}\) and \(t\) is the unit tangent [2105.03227]. On an interface triangle \(T\), the local immersed space \(IRT(T)\) is defined by piecewise RT fields \[ \phi(x)= \begin{cases} \phi^+(x), & x\in T^+,\\ \phi^-(x), & x\in T^-, \end{cases} \qquad \phi^\pm\in RT(T), \] subject to three interface constraints: continuity of \(\phi\cdot n_h\) along the interface chord \(\Gamma_h\cap T\), continuity of \(B_T\phi\cdot t_h\) at one point \(x_T\in \Gamma_h\cap T\), and equality of the divergence on both sides. The degrees of freedom remain the three standard edge-normal fluxes \[ N_{i,T}(\phi)=\frac{1}{|e_i|}\int_{e_i} \phi\cdot n_{i,T}\,ds,\qquad i=1,2,3. \] Under the maximum-angle condition \(\alpha_{\max}\le \pi/2\), these DOFs are unisolvent. The construction reduces to standard RT when the coefficient jump disappears [2105.03227]. A central property is the commuting relation \[ \nabla\cdot(\Pi_T^{IFE} q)=P_0(\nabla\cdot q), \] which holds elementwise for the immersed interpolant. Together with piecewise \(H^1\) regularity assumptions, it yields the approximation estimates \[ \|q-\Pi_h^{IFE}q\|_{0,\Omega}\le C h \sum_{s=\pm} \|q\|_{H^1(\Omega^s)}, \] and \[ \|\nabla\cdot(q-\Pi_h^{IFE}q)\|_{0,T} \le C h_T \sum_{s=\pm} \|\nabla\cdot q\|_{H^1(T\cap\Omega^s)}. \] Because the global space \(IRT(\mathcal T_h)\) is generally nonconforming in \(H(\mathrm{div};\Omega)\), a symmetric interior penalty on interface edges is added: \[ S_h(p_h,q_h)=\tau\sum_{e\in E_h^\Gamma}\int_e [p_h\cdot n_e][q_h\cdot n_e]\,ds. \] The resulting lowest-order immersed mixed method is stable, and for \(\tau=O(1)\) it satisfies an optimal first-order a priori estimate for \(\|p-p_h\|_{0,\Omega}+\|u-u_h\|_{0,\Omega}\). The same source emphasizes that the penalty is not needed for stability but is needed for optimal convergence; \(\tau=0\) yields suboptimal rates [2105.03227]. A related but distinct use appears in conservative flux reconstruction for CutFEM interface discretizations. There the local immersed space on a cut triangle \(T\) is \[ IRT^0(T)=\{\psi\in H(\mathrm{div};T): \psi|_{T_i}\in RT_0(T_i),\ [\![\psi\cdot n_\Gamma]\!]=0,\ [\![\kappa^{-1}\psi\cdot t_\Gamma]\!]=0 \text{ on }\Gamma\cap T\}, \] while on uncut cells one uses standard \(RT_0(T)\) [2604.02153]. The three outer-edge flux moments again determine the local field uniquely. The defining feature here is that the transmission condition \([\![\psi\cdot n_\Gamma]\!]=0\) is built into the space, so local conservation on cut cells is automatic. For the reconstructed flux \(\sigma_h\), one obtains the elementwise relation \[ -(\nabla\cdot \sigma_h)|_T = \Pi_T^0 f. \] The CutFEM paper contrasts this with a naive reconstruction using independent RT fields on \(K\cap \Omega_1\) and \(K\cap \Omega_2\), which generally fails to enforce \[ [\![\tilde \sigma_h\cdot n_\Gamma]\!] = 0 \] on the interface. The immersed reconstruction is then used to define a posteriori indicators \[ \eta_T = \|\kappa^{-1/2}(\sigma_h-\kappa\nabla u_h)\|_{0,T}, \] together with additional cut-face and interface-jump contributions, and the reliability estimate \[ |u-u_h|_{1,\kappa} \le \eta + C(\eta_\Gamma + \epsilon(\Omega)), \] with \(C\) independent of \(h\), the cut configuration, and the coefficient contrast. The sharpness of the bound relies on the immersed RT space enforcing strong continuity of the normal flux across \(\Gamma\) and exact local conservation on cut cells [2604.02153]. ## 5. Raviart–Thomas enrichment of Scott–Vogelius spaces In the Stokes discretization literature, the RT component is not a standalone flux space for an elliptic mixed problem but an enrichment of the Scott–Vogelius velocity space on arbitrary simplicial meshes. The basic pair is \[ V_k := P_k(\mathcal T)^d\cap H_0^1(\Omega)^d,\qquad Q_{k-1}:=P_{k-1}^{disc}(\mathcal T)\cap L_0^2(\Omega), \] with \(\mathrm{div}(V_k)\subset Q_{k-1}\), so that a stable scheme would yield exactly divergence-free discrete velocities. On general meshes, however, inf–sup stability may fail [2206.01242]. The proposed remedy enriches \(V_k\) with a small RT bubble space: \[ V_k^{enr} := V_k \oplus V_h^R. \] The enrichment is designed so that \[ \mathrm{div}(V_k^{enr}) = Q_{k-1}, \] with \(V_h^R\) chosen from interior RT bubbles or, in low order, from \((RT_0(\mathcal T)\cap H_0(\mathrm{div};\Omega))\) plus interior RT bubbles. For \(k\ge d\), one takes \[ \hat Q_h = P_{k-d}^{disc}(\mathcal T)\cap L_0^2,\qquad V_h^R = \{v_h\in \widetilde{RT}_{k-1}^{int}(\mathcal T): \mathrm{div}\,v_h\in \hat Q_h^\perp\}, \] and the divergence map from \(\widetilde{RT}_{k-1}^{int}\) to \(\widetilde P_{k-1}^{disc}\) is bijective, making the method parameter-free [2206.01242]. The explicit constructions are local. On a simplex \(T\) with barycentric coordinates \(\phi_j\), the lowest-order RT face functions are \[ \psi_j^{RT0}(x)=\frac{1}{d|T|}(x-P_j), \] and interior bubbles are formed as \[ \psi_j^{RT1}=\phi_j\,\psi_j^{RT0}. \] Higher-order explicit bubble systems are also given. In 2D, for example, \[ \psi_j^{RT2}=(5\phi_j-2)\psi_j^{RT1},\qquad \psi_j^{RT3}=\frac{1}{7}(7\phi_j^2-6\phi_j+1)\psi_j^{RT1}, \] supplemented by an additional mixed bubble for \(k=3\). In 3D, pairwise combinations \[ \psi_{(j,k)}^{RT2}=(6\phi_j-1)\psi_k^{RT1} + (6\phi_k-1)\psi_j^{RT1} \] provide the required divergence moments [2206.01242]. The discrete bilinear form couples the \(H^1\)- and \(H(\mathrm{div})\)-parts: \[ a_h(u_h,v_h) = (\nabla u_h^{ct},\nabla v_h^{ct}) - (\Delta_{pw}u_h^{ct},v_h^R) + (\Delta_{pw}v_h^{ct},u_h^R) + a_h^D(u_h^{RT0},v_h^{RT0}), \] with \(a_h^D\) present only when \(k