Raviart–Thomas Mixed Finite Element Method

• The Raviart–Thomas mixed finite element method is a discretization framework for conservation PDEs that employs H(div)-conforming vector spaces and piecewise scalar functions.
• It directly computes both the primary variable and its flux within a coupled system, preserving local conservation and linking finite element with finite volume approaches.
• Its Petrov–Galerkin variants use dual basis functions to create sparse matrices and achieve optimal first-order error estimates, enhancing computational efficiency.

The Raviart–Thomas mixed finite element method (RT-MFEM) provides a framework for discretizing partial differential equations (PDEs) in mixed variational form, especially those expressing conservation of flux, such as the Poisson equation, Darcy flow, and diffusion processes in porous media. The method is characterized by the use of $H(\mathrm{div})$-conforming vector-valued finite element spaces (Raviart–Thomas spaces) for flux variables, combined with suitable approximation spaces for scalar variables (typically piecewise constants or discontinuous polynomials). This allows the direct computation of both the primary variable (e.g., pressure or potential) and its flux within a single, strongly coupled system, preserving key structural properties such as local conservation and enabling a direct link between finite element and finite volume methodologies.

1. Variational Principles and Mixed Formulation

The RT-MFEM originates from rewriting second-order elliptic PDEs in mixed form. For the Poisson equation with Dirichlet boundary conditions,

$$-\Delta u = f \quad \text{in } \Omega, \quad u = 0 \text{ on } \partial \Omega,$$

a flux variable $p = \nabla u$ is introduced, producing the first-order system: $$\begin{cases} p = \nabla u, \ \operatorname{div} p + f = 0. \end{cases}$$ The corresponding mixed variational formulation seeks $(p, u)$ such that

$$(p, q) + (u, \operatorname{div} q) = 0 \qquad \forall q \in H(\operatorname{div}, \Omega), \ -(\operatorname{div} p, v) = (f, v) \qquad \forall v \in L^2(\Omega).$$

Here, $H(\operatorname{div}, \Omega)$ is the space of square-integrable vector fields with square-integrable divergence. This framework sets the stage for discretization with the RT spaces and their finite-dimensional analogues.

2. Raviart–Thomas Finite Element Spaces and Discretization

The lowest-order Raviart–Thomas space on a triangle $K$ is given by

$$RT_0(K) = \left\{ \mathbf{q} \mid \mathbf{q}(x) = a_K + b_K x, \; a_K \in \mathbb{R}^2, \; b_K \in \mathbb{R} \right\},$$

with degrees of freedom corresponding to the normal components of $\mathbf{q}$ over each edge of the element. Globally, the discrete flux space is assembled to ensure $H(\mathrm{div})$ conformity across elements.

The standard RT mixed finite element method discretizes the problem as: $$\text{Find } (u_{\mathcal{T}}, p_{\mathcal{T}}) \in L^2(\Omega)_T \times H_T(\operatorname{div}, \Omega) \text{ such that}$$

$$\begin{aligned} (p_{\mathcal{T}}, q) + (u_{\mathcal{T}}, \operatorname{div} q) &= 0, \quad \forall q \in H_T(\operatorname{div},\Omega), \ (\operatorname{div} p_{\mathcal{T}}, v) + (f, v) &= 0, \quad \forall v \in L^2(\Omega)_T, \end{aligned}$$

where $L^2(\Omega)_T$ is the space of piecewise constants, and $H_T(\operatorname{div}, \Omega)$ is spanned by RT basis functions indexed by mesh edges.

A haLLMark of the RT discretization is exact local conservation: for every element $K$,

$$\int_K \operatorname{div} p_{\mathcal{T}} \, dx + \int_K f\, dx = 0,$$

directly reflecting the PDE’s conservation law at the discrete level.

3. Petrov-Galerkin Variants and Dual Raviart–Thomas Bases

In classical RT methods, the test and trial spaces for the flux variable coincide (i.e., a Galerkin approach). Petrov-Galerkin variants employ a different test space, producing notable computational and structural benefits. This is accomplished by constructing a dual RT basis $\{ \varphi_a^* \}$, defined by

$(\varphi_a, \varphi_b^*) = \delta_{ab},$

where $a,b$ are edge indices and $(\cdot,\cdot)$ is the $L^2$ inner product. These dual basis functions have local support (over a patch of $6$ triangles in the two-dimensional case), ensure $H(\mathrm{div})$ conformity, and crucially:

• Yield a diagonal (or sparse) mass matrix,
• Permit computation of the discrete flux across an edge purely from the values of $u_T$ in a local neighborhood,
• Bridge the algebraic gap between finite element and finite volume methods.

The Petrov-Galerkin mixed FE scheme with dual RT test space reads: $$(p_T, q^*) + (u_T, \operatorname{div} q^*) = 0 \quad \forall q^* \in H_T^*(\operatorname{div}, \Omega), \ (\operatorname{div} p_T, v) + (f, v) = 0 \quad \forall v \in L^2(\Omega)_T,$$ where $H_T^*(\operatorname{div}, \Omega)$ is spanned by the dual basis.

A local flux formula in this context takes the form

$$\int_{SN} \nabla u_T \cdot n_a\, dy = n_a \cdot (u_{L} - u_{K}) + \alpha (u_M - u_L) + \beta (u_P - u_K) + \gamma (u_Q - u_K) + \delta (u_R - u_L),$$

with indices $K, L, M, P, Q, R$ specifying cells in the local edge patch and coefficients chosen for polynomial consistency.

4. Stability, Convergence, and Error Analysis

The analysis of the RT-MFEM leverages the abstract Babuska–Brezzi (inf-sup) theory for mixed finite elements. Essential ingredients include:

• Coercivity and boundedness of bilinear forms,
• An inf-sup condition guaranteeing unique solvability and stability, with mesh-regularity-dependent constants,
• Sufficient conditions on the dual test functions (Petrov-Galerkin case), namely bounds of the form

$$A \| q \|_0^2 \leq (q, \Pi q) \leq B \| q \|_0^2, \qquad (\operatorname{div} q, \operatorname{div} \Pi q) \geq E \| \operatorname{div} q \|_0^2,$$

where $\Pi$ is the mapping between the RT and dual RT spaces.

For well-constructed RT spaces (and especially in the Petrov-Galerkin variant with adequate dual bases), the methods yield optimal first-order error estimates in $H(\mathrm{div})$ for the flux and $L^2$ for the potential variable: $$\|u - u_T\|_{L^2} + \|p - p_T\|_{H(\mathrm{div})} \leq C h \|f\|_{L^2}.$$ These estimates are robust under mesh refinement and regularity assumptions.

5. Finite Volume Interpretation and Algebraic Equivalence

A foundational result is the algebraic equivalence of the RT-MFEM with specific finite volume schemes under certain constructions. When the dual RT basis is carefully built, the Petrov-Galerkin mixed formulation is shown to be equivalent to the "VF4" finite volume scheme of Faille, Gallouët, and Herbin, and to mass-lumping approaches (as in Baranger, Maître, Oudin). In these, the discrete flux across an edge becomes: $p_a = \frac{u_L - u_K}{c_a},$ where $c_a$ depends on the cotangents of mesh angles adjacent to the edge. This structural parallel ensures that discrete gradients and numerical fluxes are computed locally, using only neighboring cell values—a key property of finite volume methods and essential for simulating conservation-dominated phenomena.

6. Computational and Practical Implications

The RT-MFEM supports implementation and deployment in a range of applied settings:

• Sparse, local linear systems resulting from Petrov-Galerkin variants foster computational efficiency,
• Applicability to general triangulations with rigorous error control,
• Integration into finite volume codebases for engineering fields (petroleum, environmental, nuclear storage modeling),
• Flexibility for hybrid and multiphysics discretizations, linking finite element and finite volume communities.

Construction of dual RT bases involves local algebraic systems and can be technically complex for general meshes or higher dimensions. The proven stability and error bounds, however, are retained as long as key properties (local support, $L^2$-orthogonality, polynomial consistency) are preserved.

7. Advantages, Applications, and Future Directions

The RT-MFEM (and its Petrov-Galerkin variants) offers the following advantages:

• Local conservation and physical interpretability through matching of fluxes and mean-values,
• Rigorous mathematical foundation ensuring convergence (via Babuska–Brezzi theory),
• Algebraic simplicity and computational locality when dual test functions are correctly constructed,
• Seamless bridging to established finite volume methods.

Common applications include diffusion-dominated PDEs, porous media flow, and any context where conservation and flux accuracy are paramount.

Ongoing and future work aims at:

• Generalizing explicit dual RT construction to three dimensions and more complex geometries,
• Furthering the connection to volume-consistent finite difference and mass-lumped mixed methods,
• Extending the framework to multi-physics and multiphase problems, where hybrid discretizations are required.

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Summary Table: Core Formulas and Interpretations

Formula	Interpretation
$(\varphi_a, \varphi_b^*) = \delta_{ab}$	Duality of primal and dual RT basis
$p_a = \frac{u_L - u_K}{c_a}$	Local flux formula (finite volume equivalent)
Error: $$\|u-u_T\|_{L^2} + \|p-p_T\|_{H(\mathrm{div})} \leq Ch \|f\|_{L^2}$$	Optimal first-order convergence

The Raviart–Thomas mixed finite element method constitutes a central computational approach for the simulation of elliptic and conservation-law-driven problems, integrating the mathematical rigor of finite elements with the physical and computational advantages of finite volume strategies, particularly via Petrov-Galerkin and dual-basis variants.