Mixed FEM for the Poisson equation (Raviart-Thomas)

Intuition Beginner

The usual Poisson equation can be solved for a scalar potential first, then its flux is computed afterward. Mixed finite elements reverse the emphasis: they solve for the flux and the scalar unknown together.

This matters when flux is the quantity engineers care about. In heat flow, porous media, and conservation laws, the amount crossing each face of a mesh cell must be reliable.

Raviart-Thomas elements are designed for this. Their normal component matches across element faces, so flux does not leak or duplicate across the mesh.

The method is "mixed" because it uses two different spaces: one for the flux and one for the scalar variable.

Visual Beginner

The arrows crossing a shared face agree in the normal direction, so neighboring elements share one flux.

Worked example Beginner

Imagine heat flowing through a triangular mesh. A conforming scalar method gives temperatures at nodes and then estimates heat flux from the temperature gradient.

A mixed method instead treats the heat flux as an unknown from the start. Each element records a flux field, and neighboring elements agree on how much heat crosses their shared side.

What this tells us: mixed methods trade a larger saddle-point system for better direct control of conservative quantities.

Check your understanding Beginner

Formal definition Intermediate+

Consider the Poisson problem on a bounded domain $\Omega$: $$ -\operatorname{div}(K\nabla u)=f $$ with suitable boundary conditions. Introduce the flux $$ \sigma=-K\nabla u. $$ Then the first-order system is $$ K^{-1}\sigma+\nabla u=0,\qquad \operatorname{div}\sigma=f. $$

The mixed weak form seeks $$ (\sigma,u)\in H(\operatorname{div};\Omega)\times L^2(\Omega) $$ such that $$ (K^{-1}\sigma,\tau)-(u,\operatorname{div}\tau)=\ell(\tau) \quad\text{for all }\tau\in H(\operatorname{div};\Omega), $$ and $$ (\operatorname{div}\sigma,v)=(f,v) \quad\text{for all }v\in L^2(\Omega). $$ The boundary data are encoded in $\ell$ or in the normal trace space, depending on the boundary condition.

A Raviart-Thomas discrete method chooses $$ \Sigma_h\subset H(\operatorname{div};\Omega),\qquad V_h\subset L^2(\Omega), $$ typically with ${\Sigma }_h=R{T}_r$ and $V_h$ piecewise polynomials of matching degree. The discrete problem is the same pair of equations tested only against ${\tau }_h\in {\Sigma }_h$ and $v_h\in V_h$.

Counterexamples to common slips

• Mixed Poisson is not just conforming scalar FEM rewritten. It changes the unknowns and the stability condition.
• Flux conformity means normal continuity, not full vector continuity.
• The scalar variable in the basic mixed method lives naturally in $L^2$, not necessarily in $H^1$.

Key theorem with proof Intermediate+

Theorem (mixed Poisson stability pattern). Suppose the mixed spaces satisfy continuity, coercivity on the divergence-free kernel, and the Babuška-Brezzi inf-sup condition for the divergence pairing. Then the mixed Poisson weak problem is well-posed, and a discrete Raviart-Thomas pair with uniform discrete inf-sup stability is stable.

Proof. Put the problem in the saddle-point form of 24.01.04 with $$ a(\sigma,\tau)=(K^{-1}\sigma,\tau), \qquad b(\tau,u)=-(u,\operatorname{div}\tau). $$ The form $a$ is continuous. If $K$ is uniformly positive definite, then $a$ is coercive on all of $H(\mathrm{div})$, hence in particular on the kernel of the divergence constraint.

The inf-sup condition says that every scalar test direction in $L^2$ is detected by the divergence of some flux test direction. With this condition, the Babuška-Brezzi theorem gives existence, uniqueness up to the boundary-condition convention, and continuous dependence on the data.

For the discrete problem, the same argument applies on ${\Sigma }_h\times V_h$ if the discrete inf-sup constant is bounded below independently of mesh size. Raviart-Thomas spaces are built so that the divergence maps the flux space onto the chosen scalar polynomial space and so that a commuting interpolation/Fortin argument transfers stability from the continuous level to the discrete level. $\square$

Bridge. This unit uses the weak formulation framework of 24.01.03, the inf-sup theory of 24.01.04, and the $H(\mathrm{div})$ interpretation from 24.01.02. It prepares 24.03.02 and 24.03.03, where Raviart-Thomas and Brezzi-Douglas-Marini spaces are reinterpreted as polynomial differential-form families in FEEC.

Exercises Intermediate+

Advanced results Master

On a simplex $K\subset {\mathbb{R}}^n$, the Raviart-Thomas family has the local form $$ RT_r(K)=\mathcal P_r(K;\mathbb R^n)+x,\mathcal P_r(K), $$ with equivalent shifted-index conventions in the literature. Its degrees of freedom include normal moments on faces and interior moments. These degrees of freedom enforce $H(\mathrm{div})$ conformity when elements are assembled over a mesh.

The divergence of a Raviart-Thomas field lies in a polynomial scalar space, and the pair is chosen so that $$ \operatorname{div}\Sigma_h=V_h $$ locally and globally under the usual mesh assumptions. This surjectivity is the algebraic source of the discrete inf-sup condition.

The commuting projection is the key stability device. If ${\Pi }_h$ maps sufficiently smooth fluxes into ${\Sigma }_h$ and satisfies $$ \operatorname{div}\Pi_h\sigma=P_h(\operatorname{div}\sigma), $$ where $P_h$ is the $L^2$ projection onto $V_h$, then the divergence pairing is preserved on discrete scalar tests. This is the mixed-method precursor of the FEEC cochain projection.

The Brezzi-Douglas-Marini family is another major $H(\mathrm{div})$-conforming family. In FEEC language, Raviart-Thomas and BDM appear as parts of the two polynomial differential-form families, making their relationship systematic rather than ad hoc.

Optimal mixed estimates typically control the flux in an $H(\mathrm{div})$ or $L^2$ norm and the scalar variable in $L^2$. The precise order depends on polynomial degree, regularity of the exact solution, mesh regularity, and the chosen family.

Synthesis. Mixed Poisson is the canonical example where the unknown dictated by physics is not only the scalar potential but also the conservative flux. Raviart-Thomas spaces solve the geometric conformity problem by matching normal fluxes across faces. Babuška-Brezzi stability solves the analytic saddle-point problem. FEEC later explains both facts as consequences of placing the method inside a discrete de Rham complex.

Full proof set Master

Proposition 1 (local conservation). If ${\sigma }_h$ satisfies the mixed discrete divergence equation, then on each element $K$, $$ \int_{\partial K}\sigma_h\cdot n,ds=\int_K f_h,dx $$ for the projected right-hand side $f_h$.

Proof. Test the scalar equation with the element indicator or the corresponding piecewise constant test function on $K$ when it belongs to the scalar space. The equation gives $$ \int_K \operatorname{div}\sigma_h,dx=\int_K f_h,dx. $$ The divergence theorem gives $$ \int_K \operatorname{div}\sigma_h,dx=\int_{\partial K}\sigma_h\cdot n,ds. $$ Combining the equalities proves local conservation. $\square$

Proposition 2 (commuting projection gives divergence compatibility). If $\mathrm{div}{\Pi }_h=P_h\mathrm{div}$, then the projected flux has the projected divergence.

Proof. Apply the commuting identity directly to $\sigma$: $$ \operatorname{div}(\Pi_h\sigma)=P_h(\operatorname{div}\sigma). $$ Thus projecting the flux and then taking divergence gives the same discrete scalar result as taking divergence first and projecting into $V_h$. $\square$

Proposition 3 (discrete inf-sup from commuting projection pattern). A bounded commuting projection with $\mathrm{div}{\Sigma }_h=V_h$ transfers continuous divergence inf-sup stability to the Raviart-Thomas pair.

Proof. For a discrete scalar $v_h\in V_h$, continuous inf-sup gives a flux test $\tau \in H(\mathrm{div})$ whose divergence detects $v_h$. Apply the bounded commuting projection ${\Pi }_h$. The pairing with $v_h$ is preserved because $$ (v_h,\operatorname{div}\Pi_h\tau)=(v_h,P_h\operatorname{div}\tau)=(v_h,\operatorname{div}\tau). $$ Boundedness of ${\Pi }_h$ controls the denominator in the $H(\mathrm{div})$ norm. This gives a mesh-independent lower bound for the discrete supremum. $\square$

Connections Master

• Classical conforming FEM 24.02.01. Mixed FEM uses Galerkin testing but changes the unknowns and stability mechanism.

• Babuška-Brezzi condition 24.01.04. Mixed Poisson is a model saddle-point application of inf-sup stability.

• Sobolev spaces of differential forms 24.01.02. The flux space is the vector proxy for $H{\Lambda }^{n-1}$.

• Nédélec and trimmed form elements 24.03.02. Raviart-Thomas elements are part of the FEEC polynomial-form families.

• Polynomial differential form spaces 24.03.03. FEEC identifies RT and BDM spaces through ${\mathcal{P}}_r^{-}{\Lambda }^k$ and ${\mathcal{P}}_r{\Lambda }^k$.

Historical & philosophical context Master

Raviart and Thomas introduced their mixed finite element method for second-order elliptic problems in the 1970s ^{[Raviart-Thomas]}. Its lasting insight was to build finite element spaces that approximate fluxes while preserving normal continuity and local conservation.

Brezzi, Douglas, and Marini later introduced another major family of mixed elements for second-order elliptic problems ^{[Brezzi-Douglas-Marini]}. Together with the Babuška-Brezzi theory, these spaces became the classical mixed-method toolkit.

FEEC did not discard this classical theory. It explained it. In the FEEC dictionary, Raviart-Thomas and BDM elements are not isolated inventions; they are instances of polynomial differential-form spaces in a discrete de Rham complex ^{[Arnold-Falk-Winther]}.

Bibliography Master

@incollection{RaviartThomas1977,
  author = {Raviart, Pierre-Arnaud and Thomas, Jean-Marie},
  title = {A mixed finite element method for 2nd order elliptic problems},
  booktitle = {Mathematical Aspects of Finite Element Methods},
  series = {Lecture Notes in Mathematics},
  volume = {606},
  publisher = {Springer},
  pages = {292--315},
  year = {1977}
}

@article{BrezziDouglasMarini1985,
  author = {Brezzi, Franco and Douglas, Jim and Marini, L. Donatella},
  title = {Two families of mixed finite elements for second order elliptic problems},
  journal = {Numerische Mathematik},
  volume = {47},
  pages = {217--235},
  year = {1985}
}

@book{BrezziFortin1991MixedPoisson,
  author = {Brezzi, Franco and Fortin, Michel},
  title = {Mixed and Hybrid Finite Element Methods},
  series = {Springer Series in Computational Mathematics},
  volume = {15},
  publisher = {Springer},
  year = {1991}
}

@article{ArnoldFalkWinther2006MixedPoisson,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Finite element exterior calculus, homological techniques, and applications},
  journal = {Acta Numerica},
  volume = {15},
  pages = {1--155},
  year = {2006}
}