**Proof.** The previous reasoning shows that the bilinear form $\mathcal{A}$ is uniformly $T_h$-coercive for the mapping

$$ T_h : V_h \times Q_h \longrightarrow V_h \times Q_h $$ 
$$ (u_h, p_h) \longrightarrow (v C_{\text{div}}^2 C_{\pi}^2 u_h + \Pi_h(v p_h), -v C_{\text{div}}^2 C_{\pi}^2 p_h) $$

as long as the two conditions (92) and (93) are fulfilled for all $p_h \in Q_h$, which is the case if (96) and (97) hold true. The stability estimate (98) then follows by using (94) and (95) in (88). $\square$

The conditions (96) and (97) correspond exactly to the assumptions of an abstract result known as Fortin's lemma [32]. Above, the T-coercivity approach allowed us to recover these two conditions in a straightforward way. Moreover, we recall that, since the form $b$ fulfills an inf-sup condition (3), those conditions (96)-(97) are equivalent to the so-called uniform discrete inf-sup condition on the form $b$

$$ \exists \underline{\beta'} > 0, \forall h, \inf_{q_h \in Q_h \setminus \{0\}} \sup_{v_h \in V_h \setminus \{0\}} \frac{\int_{\Omega} q_h \operatorname{div} v_h \, dx}{\|\nabla v_h\| \|q_h\|} \geq \underline{\beta'}, $$

see for instance [33, Lemma II.1.1].

Finally, we recall that, provided there is a basic approximability property (i.e. any element of $V \times Q$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$), the convergence of the discrete solutions to the exact one is a consequence of (98).

## 7.2. Approximation of saddle-point problems

We now derive a conforming approximation of the abstract problem (15), starting from the variational expressions (16) or (17), the latter with the form

$$ \mathcal{A}((u, p), (v, q)) = a(u, v) + b(v, p) + b(u, q). $$

So, let $(V_h)_h$, resp. $(Q_h)_h$, be two families of finite dimensional subspaces of $V$, resp. $Q$. Starting from (16), the discrete variational formulation writes

$$ \left\{ \begin{array}{l} \text{Find } (u_h, p_h) \in V_h \times Q_h \text{ such that} \\ \forall v_h \in V_h, \quad a(u_h, v_h) + b(v_h, p_h) = \langle f, v_h \rangle_{V', V} \\ \forall q_h \in Q_h, \quad b(u_h, q_h) = \langle g, q_h \rangle_{Q', Q}. \end{array} \right. $$

while, starting from (17), the all-in-one discrete variational formulation writes

$$ \left\{ \begin{array}{l} \text{Find } (u_h, p_h) \in V_h \times Q_h \text{ such that} \\ \forall (v_h, q_h) \in V_h \times Q_h, \quad \mathcal{A}((u_h, p_h), (v_h, q_h)) = \langle f, v_h \rangle_{V', V} + \langle g, q_h \rangle_{Q', Q}. \end{array} \right. $$

In abstract form, the uniform discrete inf-sup condition on the form $b$ writes

$$ \exists \underline{\beta'} > 0, \forall h, \inf_{q_h \in Q_h \setminus \{0\}} \sup_{v_h \in V_h \setminus \{0\}} \frac{b(v_h, q_h)}{\|v_h\|_V \|q_h\|_Q} \geq \underline{\beta'}. \quad (99) $$

**Remark 35.** We note that if the form $\mathcal{A}$ is uniformly $T_h$-coercive in the sense of Definition 30, then the form $b$ automatically fulfills a uniform discrete inf-sup condition. As a matter of fact, let $q_h \in Q_h \setminus \{0\}$ be given, and introduce $(w_h, r_h) = T_h(0, q_h)$. Using Definition 30 with the test function $(0, q_h)$ leads to

$$ |b(w_h, q_h)| \geq \alpha^* \|q_h\|_Q^2 \quad \text{and} \quad \|(w_h, r_h)\|_{V \times Q} \leq \beta^* \|q_h\|_Q. $$

In particular, $w_h \neq 0$, and it follows that

$$ \sup_{v_h \in V_h \setminus \{0\}} \frac{b(v_h, q_h)}{\|v_h\|_V \|q_h\|_Q} \geq \frac{b(w_h, q_h)}{\|w_h\|_V \|q_h\|_Q} \geq \alpha^* \frac{\|q_h\|_Q}{\|w_h\|_V} \geq \frac{\alpha^*}{\beta^*}. $$

The condition (99) holds with $\underline{\beta'} = \alpha^* / \beta^* > 0$.
**Theorem 34.** If there exist a family of operators $(\Pi_h)_h$ and a constant $C_\pi > 0$ such that, for all $h$,

$$ \int_{\Omega} q_h \operatorname{div}(\Pi_h \boldsymbol{v}) \, dx = \int_{\Omega} q_h \operatorname{div} \boldsymbol{v} \, dx, \quad \forall \boldsymbol{v} \in \mathbf{H}_0^1(\Omega), \forall q_h \in Q_h, \qquad (95) $$

$$ \|\nabla(\Pi_h(\boldsymbol{v}))\| \le C_\pi \|\nabla \boldsymbol{v}\|, \quad \forall \boldsymbol{v} \in \mathbf{H}_0^1(\Omega), \qquad (96) $$

then Problem (88) is well-posed for all $h$ and

$$ \| (\boldsymbol{u} - \boldsymbol{u}_h, p - p_h) \|_{\mathbf{H}_0^1(\Omega) \times L_0^2(\Omega)} \le C \inf_{(v_h, q_h) \in \mathbf{V}_h \times Q_h} \| (\boldsymbol{u} - \boldsymbol{v}_h, p - q_h) \|_{\mathbf{H}_0^1(\Omega) \times L_0^2(\Omega)}, \qquad (97) $$

with

$$ C = 1 + \frac{2 \max(v, 2) \max(\sqrt{2} v C_{\text{div}}^2 C_{\pi}^2, C_{\text{div}} C_{\pi} (2 + v^2 C_{\text{div}}^2 C_{\pi}^2)^{1/2})}{\min(v^2 C_{\text{div}}^2 C_{\pi}^2, 1)}. $$

**Proof.** The previous reasoning shows that the bilinear form $\mathcal{A}$ is uniformly $T_h$-coercive for the mapping

$$ T_h : \mathbf{V}_h \times Q_h \longrightarrow \mathbf{V}_h \times Q_h \\ (\boldsymbol{u}_h, p_h) \longmapsto (v C_{\text{div}}^2 C_{\pi}^2 \boldsymbol{u}_h + \Pi_h (\boldsymbol{v}_h p_h), -v C_{\text{div}}^2 C_{\pi}^2 p_h) $$

as long as the two conditions (91) and (92) are fulfilled for all $p_h \in Q_h$, which is the case if (95) and (96) hold true. The stability estimate (97) then follows by using (93) and (94) in (87). $\square$

The conditions (95) and (96) correspond exactly to the assumptions of an abstract result known as Fortin's lemma [32]. Above, the T-coercivity approach allowed us to recover these two conditions in a straightforward way. Moreover, we recall that, since the form $b$ fulfills an inf-sup condition (3), those conditions (95)-(96) are equivalent to the so-called *uniform discrete inf-sup condition on the form b*

$$ \exists \underline{\beta}' > 0, \quad \forall h, \quad \inf_{q_h \in Q_h \setminus \{0\}} \sup_{\boldsymbol{v}_h \in \mathbf{V}_h \setminus \{0\}} \frac{\int_{\Omega} q_h \operatorname{div} \boldsymbol{v}_h \, dx}{\|\nabla \boldsymbol{v}_h\| \|q_h\|} \ge \underline{\beta}', $$

see for instance [33, Lemma II.1.1].

Finally, we recall that, provided there is a basic approximability property (i.e. any element of $\mathbf{V} \times \mathcal{Q}$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$), the convergence of the discrete solutions to the exact one is a consequence of (97).

## 7.2. Approximation of saddle-point problems

We now derive a conforming approximation of the abstract problem (15), starting from the variational expressions (16) or (17), the latter with the form

$$ \mathcal{A}((u, p), (v, q)) = a(u, v) + b(v, p) + b(u, q). $$

So, let $(V_h)_h$, resp. $(Q_h)_h$, be two families of finite dimensional subspaces of $\mathbf{V}$, resp. $\mathcal{Q}$. Starting from (16), the discrete variational formulation writes

$$ \begin{cases} \text{Find } (u_h, p_h) \in V_h \times Q_h \quad \text{such that} \\ \forall v_h \in V_h, \quad a(u_h, v_h) + b(v_h, p_h) = \langle f, v_h \rangle_{V', V} \\ \forall q_h \in Q_h, \quad b(u_h, q_h) = \langle g, q_h \rangle_{Q', Q}. \end{cases} $$

While, starting from (17), the all-in-one discrete variational formulation writes

$$ \begin{cases} \text{Find } (u_h, p_h) \in V_h \times Q_h \quad \text{such that} \\ \forall (v_h, q_h) \in V_h \times Q_h, \quad \mathcal{A}((u_h, p_h), (v_h, q_h)) = \langle f, v_h \rangle_{V', V} + \langle g, q_h \rangle_{Q', Q}. \end{cases} $$
Let us denote by $\Pi_{Q_h}: Q \to Q_h$ the orthogonal projection operator defined by

$$ (\Pi_{Q_h} p, q_h)_Q = (p, q_h)_Q, \quad \forall p \in Q, \forall q_h \in Q_h. $$

We denote by $B_h: V_h \to Q_h$ the discrete version of the operator B given by $B_h = \Pi_{Q_h} B_{|V_h}$. In other words, we have

$$ (B_h v_h, q_h)_Q = (B v_h, q_h)_Q = b(v_h, q_h), \quad \forall (v_h, q_h) \in V_h \times Q_h. $$

With such notations, the straightforward discrete counterpart of Theorem 8 is

**Theorem 36.** The following three statements are equivalent:

(i) There exists $\beta' > 0$ such that the form b fulfills the uniform discrete inf-sup condition (99).

(ii) For all $h$, $B_h: (\text{Ker } B_h)^\perp \to Q_h$ is an isomorphism, and

$$ \|B_h v_h\|_Q \ge \beta' \|v_h\|_V, \quad \forall v_h \in (\text{Ker } B_h)^\perp. \qquad (100) $$

(iii) For all $h$, there exists an isomorphic operator $L_{B,h}: Q_h \to (\text{Ker } B_h)^\perp$ such that

$$ B_h(L_{B,h} q_h) = q_h \quad \text{and} \quad \|q_h\|_Q \ge \beta' \|L_{B,h} q_h\|_V, \quad \forall q_h \in Q_h. \qquad (101) $$

**Remark 37.** Obviously, this result also holds if the value of the constant in the discrete inf-sup condition on the form b depends on $h$, i.e. for each $h$ it holds for some $\beta'(h) > 0$, with $\lim_{h \to 0} \beta'(h) = 0$. In this case however, getting error estimates can be more intricate.

As mentioned above for the Stokes system, one has the Fortin lemma (cf. [33, Lemma II.1.1]).

**Theorem 38.** Assume that the form b fulfills an inf-sup condition (18). The uniform discrete inf-sup condition on the form b (99) holds if, and only if, there exist a family of operators $(\Pi_h)_h$, with $\Pi_h: V \to V_h$, and a constant $C_\pi > 0$ such that, for all $h$,

$$ \begin{gathered} b(\Pi_h v, q_h) = b(v, q_h), \quad \forall v \in V, \forall q_h \in Q_h, \\ \sup_h \|\Pi_h\| \le C_\pi. \end{gathered} \qquad (102) $$

Operators $(\Pi_h)_h$ that fulfill conditions (102) are called Fortin operators.

Let us now proceed with the derivation of conditions to ensure that the form $\mathcal{A}$ is uniformly $T_h$-coercive. As a general rule, the proofs of the results follow very closely the proofs that were given in the exact case. The straightforwardness of the procedure when going from the continuous to the discrete level is one of the main features of the $T$-coercivity approach. We give next the discrete counterparts of Theorems 9 and 11.

**Theorem 39.** Assume that the form a is symmetric and positive, that there exists a constant $\alpha' > 0$ such that

$$ a(u_h, u_h) \ge \alpha' \|u_h\|_V^2, \quad \forall u_h \in V_h, \qquad (103) $$

and that the uniform discrete inf-sup condition (99) on the form b holds true. Then the form $\mathcal{A}$ is uniformly $T_h$-coercive.

The property (103) is sometimes called the uniform discrete coercivity property.

**Proof.** Let $h$ be given. We introduce the mapping

$$ \begin{gathered} T_h: V_h \times Q_h \to V_h \times Q_h \\ (u_h, p_h) \mapsto (\lambda u_h + L_{B,h} p_h, -\lambda p_h). \end{gathered} $$

We then compute

$$ \begin{aligned} \mathcal{A}((u_h, p_h), T_h(u_h, p_h)) &= a(u_h, \lambda u_h) + a(u_h, L_{B,h} p_h) + b(\lambda u_h, p_h) + b(L_{B,h} p_h, p_h) - b(u_h, \lambda p_h) \\ &= \lambda a(u_h, u_h) + a(u_h, L_{B,h} p_h) + \|p_h\|_Q^2, \quad \text{according to (101)-left.} \end{aligned} $$
In abstract form, the uniform discrete inf-sup condition on the form b writes

$$ \exists \underline{\beta}' > 0, \forall h, \quad \inf_{q_h \in Q_h \setminus \{0\}} \sup_{v_h \in V_h \setminus \{0\}} \frac{b(v_h, q_h)}{\|v_h\|_V \|q_h\|_Q} \ge \underline{\beta}'. \quad (98) $$

**Remark 35.** We note that if the form $\mathcal{A}$ is uniformly $T_h$-coercive in the sense of Definition 30, then the form $b$ automatically fulfills a uniform discrete inf-sup condition. As a matter of fact, let $q_h \in Q_h \setminus \{0\}$ be given, and introduce $(w_h, r_h) = T_h(0, q_h)$. Using Definition 30 with the test function $(0, q_h)$ leads to

$$ |b(w_h, q_h)| \ge \alpha^* \|q_h\|_Q^2 \quad \text{and} \quad \|(w_h, r_h)\|_{V \times Q} \le \beta^* \|q_h\|_Q. $$

In particular, $w_h \ne 0$, and it follows that

$$ \sup_{v_h \in V_h \setminus \{0\}} \frac{b(v_h, q_h)}{\|v_h\|_V \|q_h\|_Q} \ge \frac{b(w_h, q_h)}{\|w_h\|_V \|q_h\|_Q} \ge \alpha^* \frac{\|q_h\|_Q}{\|w_h\|_V} \ge \frac{\alpha^*}{\beta^*}. $$

The condition (98) holds with $\underline{\beta}' = \alpha^* / \beta^* > 0$.

Let us denote by $\Pi_{Q_h} : Q \to Q_h$ the orthogonal projection operator defined by

$$ (\Pi_{Q_h} p, q_h)_Q = (p, q_h)_Q, \quad \forall p \in Q, \forall q_h \in Q_h. $$

We denote by $\mathbb{B}_h : V_h \to Q_h$ the discrete version of the operator $\mathbb{B}$ given by $\mathbb{B}_h = \Pi_{Q_h} \mathbb{B}_{|V_h}|$. In other words, we have

$$ (\mathbb{B}_h v_h, q_h)_Q = (\mathbb{B} v_h, q_h)_Q = b(v_h, q_h), \quad \forall (v_h, q_h) \in V_h \times Q_h. $$

With such notations, the straightforward discrete counterpart of Theorem 8 is

**Theorem 36.** The following three statements are equivalent:

(1) There exists $\underline{\beta}' > 0$ such that the form $b$ fulfills the uniform discrete inf-sup condition (98).

(2) For all $h$, $\mathbb{B}_h : (\text{Ker} \mathbb{B}_h)^\perp \to Q_h$ is an isomorphism, and

$$ \|\mathbb{B}_h v_h\|_Q \ge \underline{\beta}' \|v_h\|_V, \quad \forall v_h \in (\text{Ker} \mathbb{B}_h)^\perp. \quad (99) $$

(3) For all $h$, there exists an isomorphic operator $\text{L}_{\mathbb{B},h} : Q_h \to (\text{Ker} \mathbb{B}_h)^\perp$ such that

$$ \mathbb{B}_h (\text{L}_{\mathbb{B},h} q_h) = q_h \quad \text{and} \quad \|q_h\|_Q \ge \underline{\beta}' \|\text{L}_{\mathbb{B},h} q_h\|_V, \quad \forall q_h \in Q_h. \quad (100) $$

**Remark 37.** Obviously, this result also holds if the value of the constant in the discrete inf-sup condition on the form $b$ depends on $h$, i.e. for each $h$ it holds for some $\underline{\beta}'(h) > 0$, with $\lim_{h \to 0} \underline{\beta}'(h) = 0$. In this case however, getting error estimates can be more intricate.

As mentioned above for the Stokes system, one has the Fortin lemma (cf. [33, Lemma II.1.1]).

**Theorem 38.** Assume that the form $b$ fulfills an inf-sup condition (18). The uniform discrete inf-sup condition on the form $b$ (98) holds if, and only if, there exist a family of operators $(\Pi_h)_h$, with $\Pi_h : V \to V_h$, and a constant $C_\pi > 0$ such that, for all $h$,

$$ \begin{gathered} b(\Pi_h v, q_h) = b(v, q_h), \quad \forall v \in V, \forall q_h \in Q_h, \\ \sup_h \|\Pi_h\| \le C_\pi. \end{gathered} \quad (101) $$

Operators $(\Pi_h)_h$ that fulfill conditions (101) are called Fortin operators.

Let us now proceed with the derivation of conditions to ensure that the form $\mathcal{A}$ is uniformly $T_h$-coercive. As a general rule, the proofs of the results follow very closely the proofs that were given in the exact case. The straightforwardness of the procedure when going from the continuous to the discrete level is one of the main features of the $T$-coercivity approach. We give next the discrete counterparts of Theorems 9 and 11.
Because the form $a$ is symmetric and positive, we can apply Young's inequality: for any $\eta > 0$,

$$a(u_h, L_{B,h} p_h) \geq -\frac{\eta}{2} a(u_h, u_h) - \frac{1}{2\eta} a(L_{B,h} p_h, L_{B,h} p_h).$$

According now to (101)-right, we find

$$a(L_{B,h} p_h, L_{B,h} p_h) \leq C_a \|L_{B,h} p_h\|_V^2 \leq C_a (\underline{\beta}')^{-2} \|p_h\|_Q^2.$$

Using assumption (103), if $\lambda - \frac{\eta}{2} > 0$, it follows that

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \alpha' \left(\lambda - \frac{\eta}{2}\right) \|u_h\|_V^2 + \left(1 - \frac{C_a(\underline{\beta}')^{-2}}{2\eta}\right) \|p_h\|_Q^2.$$

Setting $\eta = \lambda = C_a(\underline{\beta}')^{-2}$ as in the exact case, we infer that

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{2} \min(\alpha' C_a(\underline{\beta}')^{-2}, 1) \|(u_h, p_h)\|_{V \times Q}^2$$

which proves that $\mathcal{A}$ is $T_h$-coercive, with a $T$-coercivity constant $\frac{1}{2} \min(\alpha' C_a(\underline{\beta}')^{-2}, 1) > 0$ that is independent of $h$.

Since $T_h(u_h, p_h) = (C_a(\underline{\beta}')^{-2}u_h + L_{B,h}p_h, -C_a(\underline{\beta}')^{-2}p_h)$, one finds that

$$\|T_h(u_h, p_h)\|_{V \times Q}^2 \leq 2(C_a(\underline{\beta}')^{-2})^2 \|u_h\|_V^2 + 2\|L_{B,h}p_h\|_V^2 + (C_a(\underline{\beta}')^{-2})^2 \|p_h\|_Q^2 \\ \leq 2(C_a(\underline{\beta}')^{-2})^2 \|u_h\|_V^2 + (2(\underline{\beta}')^{-2} + (C_a(\underline{\beta}')^{-2})^2) \|p_h\|_Q^2,$$

where the last inequality follows from (101)-right. The bound is valid for all $h$, which yields

$$\sup_h \|T_h\| \leq \max\left(\sqrt{2C_a(\underline{\beta}')^{-2}}, \beta(2 + C_a^2(\underline{\beta}')^{-2})^{1/2}\right),$$

so the form $\mathcal{A}$ is uniformly $T_h$-coercive.

<figure><img src="image_7.png" /></figure>

**Remark 40.** As for the Stokes problem, the discrete right-inverse $L_{B,h}$ is connected to the Fortin operator $\Pi_h$. As a matter of fact, if there exists a family of discrete projectors $(\Pi_h)_{h \in \mathbb{N}}$ verifying (102), the operator defined by $L_{B,h} = \Pi_h(L_B)$ satisfies (101) with $\underline{\beta}' = (C_\pi \beta)^{-1}$ since for all $q_h \in Q_h$

$$\|\Pi_h(L_B q_h)\|_V \leq C_\pi \|L_B q_h\|_V \leq C_\pi \beta \|q_h\|_Q,$$

according to (22). As a consequence, to perform stability estimates at the discrete level using $T_h$-coercivity, one has only to replace $\beta$ by $C_\pi \beta$ in the computations done at the continuous level.

**Theorem 41.** Assume that the form $a$ is symmetric and positive.

In this case, if the uniform discrete inf-sup condition (99) on the form $b$ holds true and if there exists a constant $\alpha'_0 > 0$ such that

$$a(u_{0,h}, u_{0,h}) \geq \alpha'_{0} \|u_{0,h}\|_{V}^{2}, \quad \forall u_{0,h} \in \text{Ker } B_{h}, \qquad (104)$$

then the form $\mathcal{A}$ is uniformly $T_h$-coercive.

The property (104) is sometimes called the uniform discrete coercivity property on the kernels.

**Proof.** Let $h$ be given. We consider the mapping

$$T_h : V_h \times Q_h \to V_h \times Q_h \\ (u_h, p_h) \mapsto (\lambda u_h + L_{B,h} p_h, -\lambda p_h + \lambda \mu B_h u_h).$$

As in the proof of Theorem 11, we can compute

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) = \lambda a(u_h, u_h) + a(u_h, L_{B,h} p_h) + \|p_h\|_Q^2 + \lambda \mu \|B_h u_h\|_Q^2$$
**Theorem 39.** Assume that the form $a$ is symmetric and positive, that there exists a constant $\alpha' > 0$ such that

$$a(u_h, u_h) \geq \alpha' \|u_h\|_V^2, \quad \forall u_h \in V_h, \qquad (102)$$

and that the uniform discrete inf-sup condition (98) on the form $b$ holds true. Then the form $\mathcal{A}$ is uniformly $T_h$-coercive.

The property (102) is sometimes called the uniform discrete coercivity property.

**Proof.** Let $h$ be given. We introduce the mapping

$$T_h : V_h \times Q_h \to V_h \times Q_h \\ (u_h, p_h) \mapsto (\lambda u_h + L_{B,h} p_h, -\lambda p_h).$$

We then compute

$$\begin{aligned} \mathcal{A}((u_h, p_h), T_h(u_h, p_h)) &= a(u_h, \lambda u_h) + a(u_h, L_{B,h} p_h) + b(\lambda u_h, p_h) + b(L_{B,h} p_h, p_h) - b(u_h, \lambda p_h) \\ &= \lambda a(u_h, u_h) + a(u_h, L_{B,h} p_h) + \|p_h\|_Q^2, \quad \text{according to (100)-left.} \end{aligned}$$

Because the form $a$ is symmetric and positive, we can apply Young's inequality: for any $\eta > 0$,

$$a(u_h, L_{B,h} p_h) \geq -\frac{\eta}{2} a(u_h, u_h) - \frac{1}{2\eta} a(L_{B,h} p_h, L_{B,h} p_h).$$

According now to (100)-right, we find

$$a(L_{B,h} p_h, L_{B,h} p_h) \leq C_a \|L_{B,h} p_h\|_V^2 \leq C_a (\underline{\beta}')^{-2} \|p_h\|_Q^2.$$

Using assumption (102), if $\lambda - \frac{\eta}{2} > 0$, it follows that

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \alpha' \left(\lambda - \frac{\eta}{2}\right) \|u_h\|_V^2 + \left(1 - \frac{C_a(\underline{\beta}')^{-2}}{2\eta}\right) \|p_h\|_Q^2.$$

Setting $\eta = \lambda = C_a(\underline{\beta}')^{-2}$ as in the exact case, we infer that

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{2} \min \left( \alpha' C_a (\underline{\beta}')^{-2}, 1 \right) \|(u_h, p_h)\|_{V \times Q}^2$$

which proves that $\mathcal{A}$ is $T_h$-coercive, with a $T$-coercivity constant $\frac{1}{2} \min(\alpha' C_a(\underline{\beta}')^{-2}, 1) > 0$ that is independent of $h$.

Since $T_h(u_h, p_h) = (C_a(\underline{\beta}')^{-2}u_h + L_{B,h}p_h, -C_a(\underline{\beta}')^{-2}p_h)$, one finds that

$$\begin{aligned} \|T_h(u_h, p_h)\|_{V \times Q}^2 &\leq 2 \left(C_a(\underline{\beta}')^{-2}\right)^2 \|u_h\|_V^2 + 2 \|L_{B,h} p_h\|_V^2 + \left(C_a(\underline{\beta}')^{-2}\right)^2 \|p_h\|_Q^2 \\ &\leq 2 \left(C_a(\underline{\beta}')^{-2}\right)^2 \|u_h\|_V^2 + \left(2(\underline{\beta}')^{-2} + \left(C_a(\underline{\beta}')^{-2}\right)^2\right) \|p_h\|_Q^2, \end{aligned}$$

where the last inequality follows from (100)-right. The bound is valid for all $h$, which yields

$$\sup_h \|T_h\| \leq \max\left(\sqrt{2C_a(\underline{\beta}')^{-2}}, \beta(2 + C_a^2(\underline{\beta}')^{-2})^{1/2}\right),$$

so the form $\mathcal{A}$ is uniformly $T_h$-coercive. $\square$

<figure><img src="image_8.png" /></figure>

**Remark 40.** As for the Stokes problem, the discrete right-inverse $L_{B,h}$ is connected to the Fortin operator $\Pi_h$. As a matter of fact, if there exists a family of discrete projectors $(\Pi_h)_h$ verifying (101), the operator defined by $L_{B,h} = \Pi_h(L_B)$ satisfies (100) with $\underline{\beta}' = (C_\pi \beta)^{-1}$ since for all $q_h \in Q_h$

$$\|\Pi_h(L_B q_h)\|_V \leq C_\pi \|L_B q_h\|_V \leq C_\pi \beta \|q_h\|_Q,$$

according to (22). As a consequence, to perform stability estimates at the discrete level using $T_h$-coercivity, one has only to replace $\beta$ by $C_\pi \beta$ in the computations done at the continuous level.
because $b(L_{B,h}p_h, p_h) = \|p_h\|_Q^2$. Since the form $a$ is symmetric and positive, one may use Young's inequality. By proceeding as in the proof of Theorem 39 and after setting $\lambda = C_a(\underline{\beta}')^{-2}$, we find that

$$ \lambda a(u_h, u_h) + a(u_h, L_{B,h}, p_h) + \|p_h\|_Q^2 \geq \frac{1}{2} C_a(\underline{\beta}')^{-2} a(u_h, u_h) + \frac{1}{2} \|p_h\|_Q^2, $$

and

$$ \mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{2} C_a(\underline{\beta}')^{-2} (a(u_h, u_h) + 2\mu \|B_h u_h\|_Q^2) + \frac{1}{2} \|p_h\|_Q^2. $$

Then, we use the decomposition $u_h = u_{0,h} + \bar{u}_h$ with $u_{0,h} \in \text{Ker} B_h$ and $\bar{u}_h \in (\text{Ker} B_h)^\perp$. As before, Young's inequality yields

$$ a(u_h, u_h) \geq (1 - \theta)a(u_{0,h}, u_{0,h}) + \left(C_a - \frac{C_a}{\theta}\right) \|\bar{u}_h\|_V^2 $$

for all $0 < \theta < 1$. Moreover, $\|B_h u_h\|_Q^2 = \|B_h \bar{u}_h\|_Q^2 \geq (\underline{\beta}')^2 \|\bar{u}_h\|_V^2$ according to (100), so that

$$ a(u_h, u_h) + 2\mu \|B_h u_h\|_Q^2 \geq (1 - \theta)a(u_{0,h}, u_{0,h}) + \left(C_a - \frac{C_a}{\theta} + 2\mu(\underline{\beta}')^2\right) \|\bar{u}_h\|_V^2. $$

Choosing $\theta = \frac{1}{2}$ and $\mu = \frac{3}{4} C_a(\underline{\beta}')^{-2}$, it holds that

$$ \begin{aligned} a(u_h, u_h) + 2\mu \|B_h u_h\|_Q^2 &\geq \frac{1}{2} a(u_{0,h}, u_{0,h}) + \frac{C_a}{2} \|\bar{u}_h\|_V^2 \\ &\geq \frac{\alpha_0'}{2} \|u_{0,h}\|_V^2 + \frac{\alpha_0'}{2} \|\bar{u}_h\|_V^2 = \frac{\alpha_0'}{2} \|u_h\|_V^2, \end{aligned} $$

where we used assumption (104) and $C_a \geq \alpha_0'$ on the second line.

Finally, we conclude that

$$ \mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{4} \alpha_0' C_a(\underline{\beta}')^{-2} \|u_h\|_V^2 + \frac{1}{2} \|p_h\|_Q^2, $$

which yields that $\mathcal{A}$ is $T_h$-coercive, with a $T$-coercivity constant $\min(\frac{1}{4}\alpha_0' C_a(\underline{\beta}')^{-2}, \frac{1}{2}) > 0$ that is independent of $h$.

From the above, we have $T_h(u_h, p_h) = (C_a(\underline{\beta}')^{-2}u_h + L_{B,h}p_h, -C_a(\underline{\beta}')^{-2}p_h + \frac{3}{4}(C_a(\underline{\beta}')^{-2})^2 B_h u_h)$, and, noting that $\|B_h u_h\|_Q \leq C_b \|u_h\|_V$, one concludes that

$$ \sup_h \|T_h\| \leq \infty, $$

so the form $\mathcal{A}$ is uniformly $T_h$-coercive. $\square$

**Remark 42.** The reciprocal of Theorem 41 is also true. To see this, one simply needs to mimick the proof of Theorem 11 - item 2. at the discrete level.

Again, provided a basic approximability property holds, that is, any element of $V \times Q$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$, convergence will follow under the assumptions of Theorem 39 or Theorem 41.

### 7.3. Approximation of augmented saddle-point problems

We now approximate the abstract problem (36), starting from the variational expression (37), with the form

$$ \mathcal{A}_c((u, p), (v, q)) = a(u, v) + b(v, p) + b(u, q) - c(p, q), $$

where $c(\cdot, \cdot)$ is a form defined on $Q \times Q$ that fulfills (35); in particular, $c(\cdot, \cdot)$ is *positive*. So, let again $(V_h)_h$, resp. $(Q_h)_h$, be two families of finite dimensional subspaces of $V$, resp. $Q$. The discrete variational formulation writes

$$ \left\{ \begin{array}{ll} \text{Find } (u_h, p_h) \in V_h \times Q_h & \text{such that} \\ \forall (v_h, q_h) \in V_h \times Q_h, & \mathcal{A}_c((u_h, p_h), (v_h, q_h)) = \langle f, v_h \rangle_{V', V} + \langle g, q_h \rangle_{Q', Q}, \end{array} \right. $$
**Theorem 41.** Assume that the form a is symmetric and positive.

In this case, if the uniform discrete inf-sup condition (98) on the form b holds true and if there exists a constant $\alpha'_0 > 0$ such that

$$a(u_{0,h}, u_{0,h}) \geq \alpha'_0 \|u_{0,h}\|_V^2, \quad \forall u_{0,h} \in \text{Ker} \mathbb{B}_h, \quad (103)$$

then the form $\mathcal{A}$ is uniformly $T_h$-coercive.

The property (103) is sometimes called the uniform discrete coercivity property on the kernels.

**Proof.** Let h be given. We consider the mapping

$$T_h : V_h \times Q_h \to V_h \times Q_h \\ (u_h, p_h) \mapsto (\lambda u_h + L_{B,h} p_h, -\lambda p_h + \lambda \mu \mathbb{B}_h u_h).$$

As in the proof of Theorem 11, we can compute

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) = \lambda a(u_h, u_h) + a(u_h, L_{B,h} p_h) + \|p_h\|_Q^2 + \lambda \mu \|\mathbb{B}_h u_h\|_Q^2$$

because $b(L_{B,h} p_h, p_h) = \|p_h\|_Q^2$. Since the form a is symmetric and positive, one may use Young's inequality. By proceeding as in the proof of Theorem 39 and after setting $\lambda = C_a(\underline{\beta}')^{-2}$, we find that

$$\lambda a(u_h, u_h) + a(u_h, L_{B,h} p_h) + \|p_h\|_Q^2 \geq \frac{1}{2} C_a (\underline{\beta}')^{-2} a(u_h, u_h) + \frac{1}{2} \|p_h\|_Q^2,$$

and

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{2} C_a (\underline{\beta}')^{-2} (a(u_h, u_h) + 2\mu \|\mathbb{B}_h u_h\|_Q^2) + \frac{1}{2} \|p_h\|_Q^2.$$

Then, we use the decomposition $u_h = u_{0,h} + \bar{u}_h$ with $u_{0,h} \in \text{Ker} \mathbb{B}_h$ and $\bar{u}_h \in (\text{Ker} \mathbb{B}_h)^\perp$. As before, Young's inequality yields

$$a(u_h, u_h) \geq (1 - \theta)a(u_{0,h}, u_{0,h}) + \left(C_a - \frac{C_a}{\theta}\right) \|\bar{u}_h\|_V^2$$

for all $0 < \theta < 1$. Moreover, $\|\mathbb{B}_h u_h\|_Q^2 = \|\mathbb{B}_h \bar{u}_h\|_Q^2 \geq (\underline{\beta}')^2 \|\bar{u}_h\|_V^2$ according to (99), so that

$$a(u_h, u_h) + 2\mu \|\mathbb{B}_h u_h\|_Q^2 \geq (1 - \theta) a(u_{0,h}, u_{0,h}) + \left(C_a - \frac{C_a}{\theta} + 2\mu (\underline{\beta}')^2\right) \|\bar{u}_h\|_V^2.$$

Choosing $\theta = \frac{1}{2}$ and $\mu = \frac{3}{4} C_a (\underline{\beta}')^{-2}$, it holds that

$$\begin{aligned} a(u_h, u_h) + 2\mu \|\mathbb{B}_h u_h\|_Q^2 &\geq \frac{1}{2} a(u_{0,h}, u_{0,h}) + \frac{C_a}{2} \|\bar{u}_h\|_V^2 \\ &\geq \frac{\alpha'_0}{2} \|u_{0,h}\|_V^2 + \frac{\alpha'_0}{2} \|\bar{u}_h\|_V^2 = \frac{\alpha'_0}{2} \|u_h\|_V^2, \end{aligned}$$

where we used assumption (103) and $C_a \geq \alpha'_0$ on the second line.

Finally, we conclude that

$$\mathcal{A}((u_h, p_h), T_h(u_h, p_h)) \geq \frac{1}{4} \alpha'_0 C_a (\underline{\beta}')^{-2} \|u_h\|_V^2 + \frac{1}{2} \|p_h\|_Q^2,$$

which yields that $\mathcal{A}$ is $T_h$-coercive, with a $T$-coercivity constant $\min(\frac{1}{4} \alpha'_0 C_a (\underline{\beta}')^{-2}, \frac{1}{2}) > 0$ that is independent of $h$.

From the above, we have $T_h(u_h, p_h) = (C_a(\underline{\beta}')^{-2}u_h + L_{B,h}p_h, -C_a(\underline{\beta}')^{-2}p_h + \frac{3}{4}(C_a(\underline{\beta}')^{-2})^2 \mathbb{B}_h u_h)$, and, noting that $\|\mathbb{B}_h u_h\|_Q \leq C_b \|u_h\|_V$, one concludes that

$$\sup_h \|T_h\| \leq \infty,$$

so the form $\mathcal{A}$ is uniformly $T_h$-coercive.

□

**Remark 42.** The reciprocal of Theorem 41 is also true. To see this, one simply needs to mimick the proof of Theorem 11 - item 2. at the discrete level.
To ensure that the form $\mathcal{A}_c$ is uniformly $T_h$-coercive, the proofs once more follow very closely those that were given in the exact case. We give next the discrete counterparts of Theorems 13, 15 and 21.

**Theorem 43.** Assume that the form $a$ is symmetric, positive, fulfills the uniform discrete coercivity property (103), and that the uniform discrete inf-sup condition (99) on the form $b$ holds true. Then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

**Theorem 44.** Assume that the form $a$ is symmetric and positive, fulfills the uniform discrete coercivity property on the kernels (104), and that the uniform discrete inf-sup condition (99) on the form $b$ holds true. If moreover the form $c$ is like in (40), where $\epsilon$ is small enough, namely

$$ \epsilon \le \frac{1}{2C_a(C_\pi\beta)^4C_b^2}\left(2 - \frac{\alpha_0}{C_a}\right), $$

then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

**Theorem 45.** Assume that (52) holds true and that the bilinear forms $a$ and $c$ are both symmetric and positive. If there exists a constant $\alpha'_B > 0$ such that

$$ a(u_h, u_h) + \frac{\gamma}{2C_c^2} \|B_h u_h\|_Q^2 \ge \alpha'_B \|u_h\|_V^2, \quad \forall u_h \in V_h, \quad (105) $$

then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

As before, provided a basic approximability property holds, that is, any element of $V \times Q$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$, convergence will follow under the assumptions of Theorem 43, Theorem 44 or Theorem 45.

## 7.4. Applications

Let us briefly see how the T-coercivity approach can be used to discretize the mixed problems, that is for Stokes, electromagnetism, nearly-incompressible elasticity and finally neutron diffusion. For each problem, we propose one or several possibilities. Note that, since there is a vast literature on this topic, there is no need to devise new approximation techniques. On the contrary, the simple framework of the T-coercivity approach provides elementary guidelines to help us choose among existing techniques. In most cases, we emphasize that this leads to explicit discrete operators $T_h$. And, as we shall see next, the degree of explicitness depends on the problem that is studied.

In each case, the first step is to choose a conforming finite element discretization adapted to the space $V$ under consideration. We assume for simplicity that $\Omega$ is a polyhedron for $d = 3$, or a polygon for $d = 2$, so one can use meshes made of simplices for the discretization by finite elements. For $k \ge 1$, $\mathcal{P}_k$ stands for the Lagrange finite elements of order $k$. For Stokes and elasticity, we note that the space $H_0^1(\Omega)$ may be approximated using $(\mathcal{P}_k)^d$ finite elements with $k \ge 2$. For electromagnetism, we have to deal with the space $H_0(\text{curl}; \Omega)$, which can be discretized using the (first-kind) Nédélec finite elements of order $k \ge 1$, denoted by $\mathcal{N}_k$. Lastly, for neutron diffusion, we have to deal with the space $H(\text{div}; \Omega)$, discretized with the help of the Raviart-Thomas elements of order $k \ge 0$, denoted by $\mathcal{RT}_k$. We refer to [5] for details.

The next step is to choose the conforming finite element discretization in the space $Q$ in such a way that convergence of the discrete solutions to the exact one is guaranteed. This occurs as soon as one achieves uniform $T_h$-coercivity for the all-in-one form $\mathcal{A}$. To that aim, one simply has to build discrete operators $T_h$ similarly as in the continuous case but using (when applicable)
Again, provided a basic approximability property holds, that is, any element of $V \times Q$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$, convergence will follow under the assumptions of Theorem 39 or Theorem 41.

### 7.3. Approximation of augmented saddle-point problems

We now approximate the abstract problem (36), starting from the variational expression (37), with the form

$$ \mathcal{A}_c((u, p), (v, q)) = a(u, v) + b(v, p) + b(u, q) - c(p, q), $$

where $c(\cdot, \cdot)$ is a form defined on $Q \times Q$ that fulfills (35); in particular, $c(\cdot, \cdot)$ is positive. So, let again $(V_h)_h$, resp. $(Q_h)_h$, be two families of finite dimensional subspaces of $V$, resp. $Q$. The discrete variational formulation writes

$$ \begin{cases} \text{Find } (u_h, p_h) \in V_h \times Q_h & \text{such that} \\ \forall (v_h, q_h) \in V_h \times Q_h, \quad \mathcal{A}_c((u_h, p_h), (v_h, q_h)) = \langle f, v_h \rangle_{V', V} + \langle g, q_h \rangle_{Q', Q} \end{cases} $$

To ensure that the form $\mathcal{A}_c$ is uniformly $T_h$-coercive, the proofs once more follow very closely those that were given in the exact case. We give next the discrete counterparts of Theorems 13, 15 and 21.

**Theorem 43.** Assume that the form $a$ is symmetric, positive, fulfills the uniform discrete coercivity property (102), and that the uniform discrete inf-sup condition (98) on the form $b$ holds true. Then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

**Theorem 44.** Assume that the form $a$ is symmetric and positive, fulfills the uniform discrete coercivity property on the kernels (103), and that the uniform discrete inf-sup condition (98) on the form $b$ holds true. If moreover the form $c$ is like in (40), where $\epsilon$ is small enough, namely

$$ \epsilon \le \frac{1}{2C_a(C_\pi\beta)^4 C_b^2} \left(2 - \frac{\alpha'_0}{C_a}\right), $$

then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

**Theorem 45.** Assume that (51) holds true and that the bilinear forms $a$ and $c$ are both symmetric and positive. If there exists a constant $\alpha'_B > 0$ such that

$$ a(u_h, u_h) + \frac{\gamma}{2C_c^2} \|B_h u_h\|_Q^2 \ge \alpha'_B \|u_h\|_V^2, \quad \forall u_h \in V_h, \quad (104) $$

then the form $\mathcal{A}_c$ is uniformly $T_h$-coercive.

As before, provided a basic approximability property holds, that is, any element of $V \times Q$ can be approximated by a sequence of elements of $(V_h \times Q_h)_h$, convergence will follow under the assumptions of Theorem 43, Theorem 44 or Theorem 45.

### 7.4. Applications

Let us briefly see how the $T$-coercivity approach can be used to discretize the mixed problems, that is for Stokes, electromagnetism, nearly-incompressible elasticity and finally neutron diffusion. For each problem, we propose one or several possibilities. Note that, since there is a vast literature on this topic, there is no need to devise new approximation techniques. On the contrary, the simple framework of the $T$-coercivity approach provides elementary guidelines to help us choose among existing techniques. In most cases, we emphasize that this leads to explicit discrete operators $T_h$. And, as we shall see next, the degree of explicitness depends on the problem that is studied.
the Fortin operators defined in Theorem 38 to project the lifting $L_B$ on the discrete space. Note that according to classical theory [5, 33], the existence of such operators is equivalent to the uniform discrete inf-sup condition (99) on the form $b$. Doing so, the discrete stability estimates then follow from the continuous case by changing the constant $\beta$ to take into account the influence of the Fortin operators, see Remark 40.

First, for Stokes and elasticity, and for $k = 2$, setting $Q_h = \mathcal{P}_1$ leads to Fortin operators $\Pi_h^\mathcal{P}: H_0^1(\Omega) \to (\mathcal{P}_k)^d$ satisfying (96)-(97), or the abstract counterpart (102): the pair $((\mathcal{P}_2)^d, \mathcal{P}_1)$ is called the Taylor-Hood finite element. In this case, building Fortin operators is a very technical issue. We do not go into the details, and refer instead to Section 8.8 in [5]: we note that the resulting expression of the operator $\Pi_h^\mathcal{P}$ is quite involved. This leads to a uniform discrete inf-sup condition on the form $b$. Regarding uniform $T_h$-coercivity, one uses (91) to define the discrete operators $T_h$ for Stokes. While, for nearly-incompressible elasticity, the bilinear form defined in (80) is uniformly $T_h$-coercive for the mapping

$$ T_h : (\mathcal{P}_2)^d \times \mathcal{P}_1 \to (\mathcal{P}_2)^d \times \mathcal{P}_1 \\ (\mathbf{u}_h, p_h) \to (2\mu(C_{\pi,\mathcal{P}}C_{\text{div}})^2\mathbf{u}_h + \Pi_h^\mathcal{P}(\mathbf{v}_{-p_h}), -2\mu(C_{\pi,\mathcal{P}}C_{\text{div}})^2 p_h), $$

according to Theorem 43 and (81).

For electromagnetism in an anisotropic medium (with smooth tensor fields $\varepsilon$ and $\mu$), for $k \ge 1$, setting $Q_h = \mathcal{P}_k$ leads to Fortin operators $\Pi_h^\mathcal{N}: H_0(\text{curl}; \Omega) \to \mathcal{N}_k$ satisfying (102). Let us explain below how to proceed. In (102), the operators must fulfill the compatibility conditions

$$ (\varepsilon \Pi_h^\mathcal{N} \mathbf{v}, \nabla q_h)_{L^2(\Omega)} = (\varepsilon \mathbf{v}, \nabla q_h)_{L^2(\Omega)}, \quad \forall \mathbf{v} \in H_0(\text{curl}; \Omega), \forall q_h \in \mathcal{P}_k. \quad (106) $$

Under the assumptions (57) on $\varepsilon$ (and $\mu$), the Helmholtz decomposition of $\mathbf{v}$, orthogonal with respect to the inner product $(\cdot, \cdot)_{\varepsilon, \mu^{-1}\text{curl}}$, writes $\mathbf{v} = \mathbf{k}_{\mathbf{v}} + \nabla\phi_{\mathbf{v}}$, where $\mathbf{k}_{\mathbf{v}} \in \mathcal{K}_N(\Omega; \varepsilon)$ and $\phi_{\mathbf{v}} \in H_0^1(\Omega)$ solves (75). Since $q_h \in H_0^1(\Omega)$, we note that

$$ (\varepsilon \mathbf{v}, \nabla q_h)_{L^2(\Omega)} = (\varepsilon \nabla \phi_{\mathbf{v}}, \nabla q_h)_{L^2(\Omega)}, \quad \forall \mathbf{v} \in H_0(\text{curl}; \Omega), \forall q_h \in \mathcal{P}_k. $$

This leads to the "natural" choice

$$ \Pi_h^\mathcal{N} \mathbf{v} = \nabla(P_h^k \phi_{\mathbf{v}}), $$

where $P_h^k: H_0^1(\Omega) \to \mathcal{P}_k$ is the orthogonal projection on $\mathcal{P}_k$ with respect to the inner product $(\cdot, \cdot)_{1, \varepsilon}$, namely

$$ (\varepsilon \nabla(P_h^k \phi), \nabla q_h)_{L^2(\Omega)} = (\varepsilon \nabla \phi, \nabla q_h)_{L^2(\Omega)}, \quad \forall \phi \in H_0^1(\Omega), \forall q_h \in \mathcal{P}_k. \quad (107) $$

Indeed, the above problem is well-posed thanks to the assumptions (57) on $\varepsilon$, and $\Pi_h^\mathcal{N} \mathbf{v}$ automatically belongs to $\mathcal{N}_k$ because, by design, the finite element space $\mathcal{N}_k$ contains $\nabla[\mathcal{P}_k]$. With this definition of the operator $\Pi_h^\mathcal{N}$, the compatibility conditions (106) immediately follow. Furthermore, the uniform bound on the norm of the operators in (102) is obtained via

$$ \begin{align*} \|\Pi_h^\mathcal{N} \mathbf{v}\|_{\varepsilon, \mu^{-1}\text{curl}}^2 &= \|\nabla(P_h^k \phi_{\mathbf{v}})\|_{\varepsilon, \mu^{-1}\text{curl}}^2 \\ &= (\varepsilon \nabla(P_h^k \phi_{\mathbf{v}}), \nabla(P_h^k \phi_{\mathbf{v}}))_{L^2(\Omega)} \\ \text{(cf. (107))} \quad &= (\varepsilon \nabla \phi_{\mathbf{v}}, \nabla(P_h^k \phi_{\mathbf{v}}))_{L^2(\Omega)} \\ &\le \|\nabla \phi_{\mathbf{v}}\|_{\varepsilon, \mu^{-1}\text{curl}} \|\nabla(P_h^k \phi_{\mathbf{v}})\|_{\varepsilon, \mu^{-1}\text{curl}} \\ &= \|\nabla \phi_{\mathbf{v}}\|_{\varepsilon, \mu^{-1}\text{curl}} \|\Pi_h^\mathcal{N} \mathbf{v}\|_{\varepsilon, \mu^{-1}\text{curl}}, \end{align*} $$

so that $\|\Pi_h^\mathcal{N} \mathbf{v}\|_{\varepsilon, \mu^{-1}\text{curl}} \le \|\mathbf{v}\|_{\varepsilon, \mu^{-1}\text{curl}}$ by orthogonality of the Helmholtz decomposition.

On the other hand, the uniform coercivity condition on the discrete kernels in Theorem 41 is
In each case, the first step is to choose a conforming finite element discretization adapted to the space V under consideration. We assume for simplicity that Ω is a polyhedron for d = 3, or a polygon for d = 2, so one can use meshes made of simplices for the discretization by finite elements. For $k \ge 1$, $\mathcal{P}_k$ stands for the Lagrange finite elements of order k. For Stokes and elasticity, we note that the space $H_0^1(\Omega)$ may be approximated using $(\mathcal{P}_k)^d$ finite elements with $k \ge 2$. For electromagnetism, we have to deal with the space $H_0(\text{curl}; \Omega)$, which can be discretized using the (first-kind) Nédélec finite elements of order $k \ge 1$, denoted by $\mathcal{N}_k$. Lastly, for neutron diffusion, we have to deal with the space $H(\text{div}; \Omega)$, discretized with the help of the Raviart-Thomas elements of order $k \ge 0$, denoted by $\mathcal{RT}_k$. We refer to [5] for details.

The next step is to choose the conforming finite element discretization in the space Q in such a way that convergence of the discrete solutions to the exact one is guaranteed. This occurs as soon as one achieves uniform $T_h$-coercivity for the all-in-one form $\mathcal{A}$. To that aim, one simply has to build discrete operators $T_h$ similarly as in the continuous case but using (when applicable) the Fortin operators defined in Theorem 38 to project the lifting $L_B$ on the discrete space. Note that according to classical theory [5, 33], the existence of such operators is equivalent to the uniform discrete inf-sup condition (98) on the form b. Doing so, the discrete stability estimates then follow from the continuous case by changing the constant $\beta$ to take into account the influence of the Fortin operators, see Remark 40.

First, for Stokes and elasticity, and for $k = 2$, setting $Q_h = \mathcal{P}_1$ leads to Fortin operators $\Pi_h^\mathcal{P}: H_0^1(\Omega) \to (\mathcal{P}_k)^d$ satisfying (95)-(96), or the abstract counterpart (101): the pair $((\mathcal{P}_2)^d, \mathcal{P}_1)$ is called the Taylor-Hood finite element. In this case, building Fortin operators is a very technical issue. We do not go into the details, and refer instead to [5, Section 8.8]: we note that the resulting expression of the operator $\Pi_h^\mathcal{P}$ is quite involved. This leads to a uniform discrete inf-sup condition on the form b. Regarding uniform $T_h$-coercivity, one uses (90) to define the discrete operators $T_h$ for Stokes. While, for nearly-incompressible elasticity, the bilinear form defined in (79) is uniformly $T_h$-coercive for the mapping

$$ T_h : (\mathcal{P}_2)^d \times \mathcal{P}_1 \to (\mathcal{P}_2)^d \times \mathcal{P}_1 \\ (\mathbf{u}_h, p_h) \to \left( 2\mu(C_{\pi,\mathcal{P}}C_{\text{div}})^2 \mathbf{u}_h + \Pi_h^\mathcal{P} (\mathbf{v}_{-p_h}), -2\mu(C_{\pi,\mathcal{P}}C_{\text{div}})^2 p_h \right), $$

according to Theorem 43 and (80).

For electromagnetism in an anisotropic medium (with smooth tensor fields $\varepsilon$ and $\mu$), for $k \ge 1$, setting $Q_h = \mathcal{P}_k$ leads to Fortin operators $\Pi_h^\mathcal{N}: H_0(\text{curl}; \Omega) \to \mathcal{N}_k$ satisfying (101). Let us explain below how to proceed. In (101), the operators must fulfill the compatibility conditions

$$ (\underline{\varepsilon} \Pi_h^\mathcal{N} \mathbf{v}, \nabla q_h)_{L^2(\Omega)} = (\underline{\varepsilon} \mathbf{v}, \nabla q_h)_{L^2(\Omega)}, \quad \forall \mathbf{v} \in H_0(\text{curl}; \Omega), \forall q_h \in \mathcal{P}_k. \quad (105) $$

Under the assumptions (56) on $\underline{\varepsilon}$ (and $\underline{\mu}$), the Helmholtz decomposition of $\mathbf{v}$, orthogonal with respect to the inner product $(\cdot, \cdot)_{\underline{\varepsilon}, \underline{\mu}-1|\text{curl}}$, writes $\mathbf{v} = \mathbf{k}_\mathbf{v} + \nabla\phi_\mathbf{v}$, where $\mathbf{k}_\mathbf{v} \in \mathbf{K}_N(\Omega; \underline{\varepsilon})$ and $\phi_\mathbf{v} \in H_0^1(\Omega)$ solves (74). Since $q_h \in H_0^1(\Omega)$, we note that

$$ (\underline{\varepsilon} \mathbf{v}, \nabla q_h)_{L^2(\Omega)} = (\underline{\varepsilon} \nabla \phi_\mathbf{v}, \nabla q_h)_{L^2(\Omega)}, \quad \forall \mathbf{v} \in H_0(\text{curl}; \Omega), \forall q_h \in \mathcal{P}_k. $$

This leads to the “natural” choice

$$ \Pi_h^\mathcal{N} \mathbf{v} = \nabla (P_h^k \phi_\mathbf{v}), $$

where $P_h^k: H_0^1(\Omega) \to \mathcal{P}_k$ is the orthogonal projection on $\mathcal{P}_k$ with respect to the inner product $(\cdot, \cdot)_{1,\underline{\varepsilon}}$, namely

$$ (\underline{\varepsilon} \nabla (P_h^k \phi), \nabla q_h)_{L^2(\Omega)} = (\underline{\varepsilon} \nabla \phi, \nabla q_h)_{L^2(\Omega)}, \quad \forall \phi \in H_0^1(\Omega), \forall q_h \in \mathcal{P}_k. \quad (106) $$

Indeed, the above problem is well-posed thanks to the assumptions (56) on $\underline{\varepsilon}$, and $\Pi_h^\mathcal{N} \mathbf{v}$ automatically belongs to $\mathcal{N}_k$ because, by design, the finite element space $\mathcal{N}_k$ contains $\nabla[\mathcal{P}_k]$.
obtained for instance in [25, Theorem 3]. It hinges on the crucial property stating that, given a field $\mathbf{v} = \nabla q$ with $q \in H_0^1(\Omega)$, the result of the interpolation of $\mathbf{v}$ with the Nédélec interpolation operator may be expressed as $\nabla q_h$ for some $q_h \in \mathcal{P}_k$.

Hence, for electromagnetism in an anisotropic medium we infer from Theorem 41 that the bilinear form $\mathcal{A}_{\varepsilon, \mu}$ defined in (70) is uniformly $T_h$-coercive for the mapping

$$ T_h : \mathcal{N}_k \times \mathcal{P}_k \to \mathcal{N}_k \times \mathcal{P}_k \\ (\mathbf{E}_h, \tilde{\mathbf{p}}_h) \to \left( (C_{\pi, \mathcal{N}})^2 \mathbf{E}_h + \Pi_h^\mathcal{N} (\nabla \tilde{\mathbf{p}}_h), -(C_{\pi, \mathcal{N}})^2 \tilde{\mathbf{p}}_h + \frac{3}{4} (C_{\pi, \mathcal{N}})^4 \phi_{E_h} \right), $$

where $\phi_{E_h} \in Q_h$ satisfies the discrete counterpart of (75), namely

$$ (\varepsilon \nabla \phi_{E_h}, \nabla q_h)_{L^2(\Omega)} = (\varepsilon \mathbf{E}_h, \nabla q_h)_{L^2(\Omega)}, \quad \forall q_h \in \mathcal{P}_k. $$

Alternatively, one can follow Section 4.2. In this case, adding indices and superscripts $h$ in the definition of $T_{opt}$, one considers

$$ (T_{opt})_h : \mathcal{N}_k \times \mathcal{P}_k \to \mathcal{N}_k \times \mathcal{P}_k \\ (\mathbf{E}_h, \tilde{\mathbf{p}}_h) \to (\mathbf{k}_{E_h}^h + \nabla \tilde{\mathbf{p}}_h, \phi_{E_h}^h), $$

with the decomposition $\mathbf{E}_h = \mathbf{k}_{E_h}^h + \nabla \phi_{E_h}^h$. The crucial idea is now to use a discrete Helmholtz decomposition to define $\mathbf{k}_{E_h}^h$ and $\phi_{E_h}^h$. Namely, for $\mathbf{v}_h \in \mathcal{N}_k$, $\phi_{\mathbf{v}_h}^h \in \mathcal{P}_k$ is defined by

$$ (\varepsilon \nabla \phi_{\mathbf{v}_h}^h, \nabla q_h)_{L^2(\Omega)} = (\varepsilon \mathbf{v}_h, \nabla q_h)_{L^2(\Omega)}, \quad \forall q_h \in \mathcal{P}_k, $$

and $\mathbf{k}_{\mathbf{v}_h}^h = \mathbf{v}_h - \nabla \phi_{\mathbf{v}_h}^h$. Doing so, one obtains an (orthogonal) discrete Helmholtz decomposition that is uniformly stable, and it can be checked that the form $\mathcal{A}_{\varepsilon, \mu}$ is uniformly $T_h$-coercive. The results also hold for piecewise smooth tensor fields $\varepsilon$ and $\mu$. Details can be found in [25, Proposition 13].

Last, for neutron diffusion and for $k \ge 1$, one only needs to select $Q_h$ in such a way that (22) is fulfilled, namely $\mathrm{div}(\mathcal{R}\mathcal{T}_k) \subset Q'_h$. To do so, we set $Q_h = \mathcal{P}_k^{pw}$, for $k \ge 0$, where the superscript $^{pw}$ stands for piecewise Lagrange finite elements of order $k$. Assuming for simplicity that $\sigma$ restricted to any simplex is constant, we introduce the discrete mapping [28, 41]

$$ T_h : \mathcal{R}\mathcal{T}_k \times \mathcal{P}_k^{pw} \to \mathcal{R}\mathcal{T}_k \times \mathcal{P}_k^{pw} \\ (\mathbf{p}_h, u_h) \to \left( \mathbf{p}_h, \frac{1}{2}(-u_h + \sigma^{-1} \mathrm{div} \, \mathbf{p}_h) \right), $$

and the property $\mathrm{div}(\mathcal{R}\mathcal{T}_k) \subset \mathcal{P}_k^{pw}$ guarantees the uniform $T_h$-coercivity of the bilinear form (86) in virtue of Theorem 45.

All basic approximability properties are established in [5], which guarantees convergence in all of the above cases. We again refer to [5] for details and possible extensions, such as the generalized Taylor-Hood elements ($k \ge 3$) or the MINI element for Stokes or elasticity.

## 8. Conclusion and perspectives

We have demonstrated the flexibility of the $T$-coercivity approach, here applied to classical linear mixed problems, both for the theoretical study of the problems and for their numerical approximation by finite elements. Let us mention some possible extensions, such as nonconforming discretization methods for Stokes [40], multigroup diffusion [34] or DDM for diffusion [28]. It is our belief that numerous applications can be studied with the $T$-coercivity approach, both theoretically and numerically. Recent works include application in poromechanics [3, 4], time-harmonic Maxwell's equations with impedance surfaces [43], and the applications listed in [39].
With this definition of the operator $\Pi_h^{\mathcal{N}}$, the compatibility conditions (105) immediately follow. Furthermore, the uniform bound on the norm of the operators in (101) is obtained via

$$
\begin{align*}
\| \Pi_h^{\mathcal{N}} \mathbf{v} \|_{\mathcal{E}, \mu^{-1} \text{curl}}^2 &= \| \nabla (P_h^k \phi_v) \|_{\mathcal{E}, \mu^{-1} \text{curl}}^2 \\
&= (\varepsilon \nabla (P_h^k \phi_v), \nabla (P_h^k \phi_v))_{L^2(\Omega)} \\
(\text{cf. (106)}) &= (\varepsilon \nabla \phi_v, \nabla (P_h^k \phi_v))_{L^2(\Omega)} \\
&\le \| \nabla \phi_v \|_{\mathcal{E}, \mu^{-1} \text{curl}} \| \nabla (P_h^k \phi_v) \|_{\mathcal{E}, \mu^{-1} \text{curl}} \\
&= \| \nabla \phi_v \|_{\mathcal{E}, \mu^{-1} \text{curl}} \| \Pi_h^{\mathcal{N}} \mathbf{v} \|_{\mathcal{E}, \mu^{-1} \text{curl}},
\end{align*}
$$

so that $\|\Pi_h^{\mathcal{N}} \mathbf{v}\|_{\mathcal{E}, \mu^{-1} \text{curl}} \le \|\mathbf{v}\|_{\mathcal{E}, \mu^{-1} \text{curl}}$ by orthogonality of the Helmholtz decomposition.

On the other hand, the uniform coercivity condition on the discrete kernels in Theorem 41 is obtained for instance in [25, Theorem 3]. It hinges on the crucial property stating that, given a field $\mathbf{v} = \nabla q$ with $q \in H_0^1(\Omega)$, the result of the interpolation of $\mathbf{v}$ with the Nédélec interpolation operator may be expressed as $\nabla q_h$ for some $q_h \in \mathcal{P}_k$.

Hence, for electromagnetism in an anisotropic medium we infer from Theorem 41 that the bilinear form $\mathcal{A}_{\mathcal{E},\mu}$ defined in (69) is uniformly $T_h$-coercive for the mapping

$$
T_h : \mathcal{N}_k \times \mathcal{P}_k \to \mathcal{N}_k \times \mathcal{P}_k \\
(E_h, \bar{p}_h) \to \left( (C_{\pi,\mathcal{N}})^2 E_h + \Pi_h^{\mathcal{N}} (\nabla \bar{p}_h), -(C_{\pi,\mathcal{N}})^2 \bar{p}_h + \frac{3}{4} (C_{\pi,\mathcal{N}})^4 \phi_{E_h} \right),
$$

where $\phi_{E_h} \in Q_h$ satisfies the discrete counterpart of (74), namely

$$
(\varepsilon \nabla \phi_{E_h}, \nabla q_h)_{L^2(\Omega)} = (\varepsilon E_h, \nabla q_h)_{L^2(\Omega)}, \quad \forall q_h \in \mathcal{P}_k.
$$

Alternatively, one can follow Section 4.2. In this case, adding indices and superscripts $h$ in the definition of $T_{opt}$, one considers

$$
(T_{opt})_h : \mathcal{N}_k \times \mathcal{P}_k \to \mathcal{N}_k \times \mathcal{P}_k \\
(E_h, \bar{p}_h) \to (\mathbf{k}_{E_h}^h + \nabla \bar{p}_h, \phi_{E_h}^h),
$$

with the decomposition $E_h = \mathbf{k}_{E_h}^h + \nabla \phi_{E_h}^h$. The crucial idea is now to use a discrete Helmholtz decomposition to define $\mathbf{k}_{E_h}^h$ and $\phi_{E_h}^h$. Namely, for $\mathbf{v}_h \in \mathcal{N}_k$, $\phi_{\mathbf{v}_h}^h \in \mathcal{P}_k$ is defined by

$$
(\varepsilon \nabla \phi_{\mathbf{v}_h}^h, \nabla q_h)_{L^2(\Omega)} = (\varepsilon \mathbf{v}_h, \nabla q_h)_{L^2(\Omega)}, \quad \forall q_h \in \mathcal{P}_k,
$$

and $\mathbf{k}_{\mathbf{v}_h}^h = \mathbf{v}_h - \nabla \phi_{\mathbf{v}_h}^h$. Doing so, one obtains an (orthogonal) discrete Helmholtz decomposition that is uniformly stable, and it can be checked that the form $\mathcal{A}_{\mathcal{E},\mu}$ is uniformly $T_h$-coercive. The results also hold for piecewise smooth tensor fields $\varepsilon$ and $\mu$. Details can be found in [25, Proposition 13].

Last, for neutron diffusion, we use $Q_h = \mathcal{P}_k^{pw}$, for $k \ge 0$, where the superscript $^{pw}$ stands for piecewise Lagrange finite elements of order $k$. Assuming for simplicity that $\sigma$ restricted to any simplex is constant, we introduce the discrete mapping [28, 41]

$$
T_h : \mathcal{RT}_k \times \mathcal{P}_k^{pw} \to \mathcal{RT}_k \times \mathcal{P}_k^{pw} \\
(\mathbf{p}_h, u_h) \to \left( \mathbf{p}_h, \frac{1}{2} (-u_h + \sigma^{-1} \operatorname{div} \mathbf{p}_h) \right),
$$

and the property $\operatorname{div}(\mathcal{RT}_k) \subset \mathcal{P}_k^{pw}$ guarantees the uniform $T_h$-coercivity of the bilinear form (85) in virtue of Theorem 45.
Dedication

This article is dedicated to Professor Franco Brezzi by the second author, with admiration.

References

[1] F. Assous, P. Ciarlet Jr, S. Labrunie, *Mathematical foundations of computational electromagnetism*, Springer, 2018.

[2] I. Babuška, "The finite element method with Lagrangian multipliers", *Numerische Mathematik* **20** (1973), no. 3, p. 179-192.

[3] M. Barré, C. Grandmont, P. Moireau, "Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials", *Evolution Equations and Control Theory* **12** (2023), no. 3, p. 846-906.

[4] ——, "Numerical analysis of an incompressible soft material poromechanics model using T-coercivity", *Comptes Rendus. Mécanique* **351** (2023), no. S1, p. 1-36.

[5] D. Boffi, F. Brezzi, M. Fortin *et al.*, *Mixed finite element methods and applications*, vol. 44, Springer, 2013.

[6] A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Jr, "Mesh requirements for the finite element approximation of problems with sign-changing coefficients", *Numer. Math.* **138** (2018), p. 801–838.

[7] A.-S. Bonnet-Ben Dhia, L. Chesnel, P. Ciarlet, Jr, "*T*-coercivity for scalar interface problems between dielectrics and metamaterials", *Math. Mod. Num. Anal.* **46** (2012), p. 1363-1387.

[8] ——, "*T*-coercivity for the Maxwell problem with sign-changing coefficients", *Communications in Partial Differential Equations* **39** (2014), p. 1007-1031.

[9] ——, "Two-dimensional Maxwell's equations with sign-changing coefficients", *Appl. Numer. Math.* **79** (2014), p. 29-41.

[10] A.-S. Bonnet-Ben Dhia, L. Chesnel, X. Claeys, "Radiation condition for a non-smooth interface between a dielectric and a metamaterial", *Mathematical Models and Methods in Applied Sciences* **23** (2013), no. 09, p. 1629-1662.

[11] A.-S. Bonnet-Ben Dhia, P. Ciarlet Jr, C. M. Zwölf, "Time harmonic wave diffraction problems in materials with sign-shifting coefficients", *Journal of Computational and Applied Mathematics* **234** (2010), no. 6, p. 1912-1919.

[12] F. Brezzi, "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers", *Publications mathématiques et informatique de Rennes* (1974), no. S4, p. 1-26.

[13] A. Buffa, S. Christiansen, "A dual finite element complex on the barycentric refinement", *C. R. Math. Acad. Sci. Paris* **340** (2005), no. 6, p. 461-464.

[14] A. Buffa, "Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations", *SIAM J. Numer. Anal.* **43** (2005), p. 1-18.

[15] A. Buffa, S. H. Christiansen, "The electric field integral equation on Lipschitz screens: definitions and numerical approximation", *Numerische Mathematik* **94** (2003), no. 2, p. 229-267.

[16] A. Buffa, M. Costabel, C. Schwab, "Boundary element methods for Maxwell's equations on non-smooth domains", *Numerische Mathematik* **92** (2002), no. 4, p. 679-710.

[17] R. Bunoiu, L. Chesnel, K. Ramdani, M. Rihani, "Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients", in *Annales de la Faculté des Sciences de Toulouse. Mathématiques.*, 2020.

[18] R. Bunoiu, K. Ramdani, "Homogenization of materials with sign changing coefficients", *Communications in Mathematical Sciences* **14** (2016), no. 4, p. 1137-1154.

[19] R. Bunoiu, K. Ramdani, C. Timofte, "T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains", *Electronic Journal of Differential Equations* (2021), p. 1-22.

[20] ——, "Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump", *Communications in Mathematical Sciences* **21** (2023), no. 7, p. 2029-2049.

[21] C. Carvalho, L. Chesnel, P. Ciarlet, Jr, "Eigenvalue problems with sign-changing coefficients", *C. R. Math. Acad. Sci. Paris* **355** (2017), no. 6, p. 671-675.

[22] L. Chesnel, "Bilaplacian problems with a sign-changing coefficient", *Mathematical Methods in the Applied Sciences* **39** (2016), no. 17, p. 4964-4979.

[23] L. Chesnel, P. Ciarlet Jr, "T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients", *Numerische Mathematik* **124** (2013), no. 1, p. 1-29.

[24] P. Ciarlet Jr, "T-coercivity: Application to the discretization of Helmholtz-like problems", *Computers & Mathematics with Applications* **64** (2012), no. 1, p. 22-34.

[25] ——, "Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient", Technical Report HAL, 2020, https://hal.inria.fr/hal-02567484v1.

[26] ——, "Lecture notes on Maxwell's equations and their approximation (in French)", Master's degree Analysis, Modelling and Simulation from Paris-Saclay University and Institut Polytechnique de Paris, 2021, https://hal.inria.fr/hal-03153780.

[27] ——, "On the approximation of electromagnetic fields by edge finite elements – Part 4: analysis of the model with one sign-changing coefficient", *Numer. Math.* **152** (2022), p. 223-257.
All basic approximability properties are established in [5], which guarantees convergence in all of the above cases. We again refer to [5] for details and possible extensions, such as the generalized Taylor-Hood elements ($k \ge 3$) or the MINI element for Stokes or elasticity.

## 8. Conclusion and perspectives

We have demonstrated the flexibility of the T-coercivity approach, here applied to classical linear mixed problems, both for the theoretical study of the problems and for their numerical approximation by finite elements. Let us mention some possible extensions, such as nonconforming discretization methods for Stokes [40], multigroup diffusion [34] or DDM for diffusion [28]. It is our belief that numerous applications can be studied with the T-coercivity approach, both theoretically and numerically. Recent works include application in poromechanics [3, 4], time-harmonic Maxwell's equations with impedance surfaces [43], and the applications listed in [39].

## Declaration of interests

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

## References

[1] F. Assous, P. Ciarlet, Jr., S. Labrunie, *Mathematical foundations of computational electromagnetism*, Springer, 2018.

[2] I. Babuška, "The finite element method with Lagrangian multipliers", *Numerische Mathematik* **20** (1973), no. 3, p. 179-192.

[3] M. Barré, C. Grandmont, P. Moireau, "Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials", *Evolution Equations and Control Theory* **12** (2023), no. 3, p. 846-906.

[4] ——, "Numerical analysis of an incompressible soft material poromechanics model using T-coercivity", *Comptes Rendus. Mécanique* **351** (2023), no. S1, p. 1-36.

[5] D. Boffi, F. Brezzi, M. Fortin *et al.*, *Mixed finite element methods and applications*, vol. 44, Springer, 2013.

[6] A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Jr., "Mesh requirements for the finite element approximation of problems with sign-changing coefficients", *Numer. Math.* **138** (2018), p. 801-838.

[7] A.-S. Bonnet-Ben Dhia, L. Chesnel, P. Ciarlet, Jr., "T-coercivity for scalar interface problems between dielectrics and metamaterials", *Math. Mod. Num. Anal.* **46** (2012), p. 1363-1387.

[8] ——, "T-coercivity for the Maxwell problem with sign-changing coefficients", *Communications in Partial Differential Equations* **39** (2014), p. 1007-1031.

[9] ——, "Two-dimensional Maxwell's equations with sign-changing coefficients", *Appl. Numer. Math.* **79** (2014), p. 29-41.

[10] A.-S. Bonnet-Ben Dhia, L. Chesnel, X. Claeys, "Radiation condition for a non-smooth interface between a dielectric and a metamaterial", *Mathematical Models and Methods in Applied Sciences* **23** (2013), no. 09, p. 1629-1662.

[11] A.-S. Bonnet-Ben Dhia, P. Ciarlet, Jr., C. M. Zwölf, "Time harmonic wave diffraction problems in materials with sign-shifting coefficients", *Journal of Computational and Applied Mathematics* **234** (2010), no. 6, p. 1912-1919.

[12] F. Brezzi, "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers", *Publications mathématiques et informatique de Rennes* (1974), no. S4, p. 1-26.

[13] A. Buffa, S. Christiansen, "A dual finite element complex on the barycentric refinement", *C. R. Math. Acad. Sci. Paris* **340** (2005), no. 6, p. 461-464.

[14] A. Buffa, "Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations", *SIAM J. Numer. Anal.* **43** (2005), p. 1-18.

[15] A. Buffa, S. H. Christiansen, "The electric field integral equation on Lipschitz screens: definitions and numerical approximation", *Numerische Mathematik* **94** (2003), no. 2, p. 229-267.

[16] A. Buffa, M. Costabel, C. Schwab, "Boundary element methods for Maxwell's equations on non-smooth domains", *Numerische Mathematik* **92** (2002), no. 4, p. 679-710.

[17] R. Bunoiu, L. Chesnel, K. Ramdani, M. Rihani, "Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients", in *Annales de la Faculté des Sciences de Toulouse. Mathématiques.*, 2020.
[28] P. Ciarlet Jr, E. Jamelot, F. D. Kpadonou, "Domain Decomposition Methods for the diffusion equation with low-regularity solution", *Computers Math. Applic.* **74** (2017), p. 2369-2384.

[29] G. Duvaut, J. L. Lions, *Les inéquations en mécanique et en physique*, Dunod, 1972.

[30] A. Ern, J.-L. Guermond, *Finite Elements II: Galerkin approximation, elliptic and mixed PDEs*, vol. 73, Springer Nature, 2021.

[31] —, *Finite Elements III: first-order and time-dependent PDEs*, vol. 74, Springer Nature, 2021.

[32] M. Fortin, "An analysis of the convergence of mixed finite element methods", *RAIRO. Analyse numérique* **11** (1977), no. 4, p. 341-354.

[33] V. Girault, P.-A. Raviart, *Finite element methods for Navier-Stokes equations: theory and algorithms*, vol. 5, Springer Science & Business Media, 1986.

[34] L. Giret, "Numerical analysis of a non-conforming Domain Decomposition for the multigroup SPN equations", PhD Thesis, Paris-Saclay University, 2018, https://pastel.archives-ouvertes.fr/tel-01936967.

[35] M. Halla, "On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions", *IMA Journal of Numerical Analysis* drab100 (2021).

[36] M. Halla, "Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility", *Numerische Mathematik* **148** (2021), no. 2, p. 387-407.

[37] R. Hiptmair, "Finite elements in computational electromagnetics", *Acta Numerica* (2002), p. 237-339.

[38] T. Hohage, L. Nannen, "Convergence of infinite element methods for scalar waveguide problems", *BIT Numer. Math.* **55** (2015), p. 215-254.

[39] Q. Hong, J. Kraus, M. Lymbery, F. Philo, "A new practical framework for the stability analysis of perturbed saddle-point problems and applications", *Math. Comp.* **92** (2023), p. 607-634.

[40] E. Jamelot, "Improved stability estimates for solving Stokes problem with Fortin-Soulie finite elements", Technical Report HAL, 2023.

[41] E. Jamelot, P. Ciarlet Jr, "Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation", *J. Comput. Phys.* **241** (2013), p. 445-463.

[42] O. A. Ladyzhenskaya, *The mathematical theory of viscous incompressible flow*, vol. 2, Gordon and Breach New York, 1969.

[43] D. P. Levadoux, "Analyse numérique de la formulation intégrodifférentielle d'un problème de Maxwell harmonique impliquant un diélectrique traversé de surfaces exfoliées métalliques et impédantes", Technical Report HAL, 2022, https://hal.archives-ouvertes.fr/hal-03644547.

[44] S. Nicaise, J. Venel, "A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients", *J. Comput. Appl. Math.* **235** (2011), p. 4272-4282.

[45] F.-J. Sayas, T. S. Brown, M. E. Hassell, *Variational techniques for elliptic partial differential equations*, CRC Press, 2019.

[46] G. Unger, "Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems", *SN Partial Differential Equations and Applications* **39** (2021).

[47] C. Weber, "A local compactness theorem for Maxwell's equations", *Mathematical Methods in the Applied Sciences* **2** (1980), no. 1, p. 12-25.
[18] R. Bunoiu, K. Ramdani, "Homogenization of materials with sign changing coefficients", *Communications in Mathematical Sciences* **14** (2016), no. 4, p. 1137-1154.

[19] R. Bunoiu, K. Ramdani, C. Timofte, "T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains", *Electronic Journal of Differential Equations* (2021), p. 1-22.

[20] ——, "Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump", *Communications in Mathematical Sciences* **21** (2023), no. 7, p. 2029-2049.

[21] C. Carvalho, L. Chesnel, P. Ciarlet, Jr., "Eigenvalue problems with sign-changing coefficients", *C. R. Math. Acad. Sci. Paris* **355** (2017), no. 6, p. 671-675.

[22] L. Chesnel, "Bilaplacian problems with a sign-changing coefficient", *Mathematical Methods in the Applied Sciences* **39** (2016), no. 17, p. 4964-4979.

[23] L. Chesnel, P. Ciarlet, Jr., "T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients", *Numerische Mathematik* **124** (2013), no. 1, p. 1-29.

[24] P. Ciarlet, Jr., "T-coercivity: Application to the discretization of Helmholtz-like problems", *Computers & Mathematics with Applications* **64** (2012), no. 1, p. 22-34.

[25] ——, "Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient", Technical Report HAL, 2020, https://hal.inria.fr/hal-02567484v1.

[26] ——, "Lecture notes on Maxwell's equations and their approximation (in French)", Master's degree Analysis, Modelling and Simulation from Paris-Saclay University and Institut Polytechnique de Paris, 2021, https://hal.inria.fr/hal-03153780.

[27] ——, "On the approximation of electromagnetic fields by edge finite elements – Part 4: analysis of the model with one sign-changing coefficient", *Numer. Math.* **152** (2022), p. 223-257.

[28] P. Ciarlet, Jr., E. Jamelot, F. D. Kpadonou, "Domain Decomposition Methods for the diffusion equation with low-regularity solution", *Computers Math. Appl.* **74** (2017), p. 2369-2384.

[29] G. Duvaut, J. L. Lions, *Les inéquations en mécanique et en physique*, Dunod, 1972.

[30] A. Ern, J.-L. Guermond, *Finite Elements II: Galerkin approximation, elliptic and mixed PDEs*, vol. 73, Springer Nature, 2021.

[31] ——, *Finite Elements III: first-order and time-dependent PDEs*, vol. 74, Springer Nature, 2021.

[32] M. Fortin, "An analysis of the convergence of mixed finite element methods", *RAIRO. Analyse numérique* **11** (1977), no. 4, p. 341-354.

[33] V. Girault, P.-A. Raviart, *Finite element methods for Navier-Stokes equations: theory and algorithms*, vol. 5, Springer Science & Business Media, 1986.

[34] L. Giret, "Numerical analysis of a non-conforming Domain Decomposition for the multigroup SPN equations", PhD Thesis, Paris-Saclay University, 2018, https://pastel.archives-ouvertes.fr/tel-01936967.

[35] M. Halla, "On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions", *IMA Journal of Numerical Analysis* drab100 (2021).

[36] M. Halla, "Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility", *Numerische Mathematik* **148** (2021), no. 2, p. 387-407.

[37] R. Hiptmair, "Finite elements in computational electromagnetics", *Acta Numerica* (2002), p. 237-339.

[38] T. Hohage, L. Nannen, "Convergence of infinite element methods for scalar waveguide problems", *BIT Numer. Math.* **55** (2015), p. 215-254.

[39] Q. Hong, J. Kraus, M. Lymbery, F. Philo, "A new practical framework for the stability analysis of perturbed saddle-point problems and applications", *Math. Comp.* **92** (2023), p. 607-634.

[40] E. Jamelot, "Improved stability estimates for solving Stokes problem with Fortin-Soulie finite elements", Technical Report HAL, 2023.

[41] E. Jamelot, P. Ciarlet, Jr., "Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation", *J. Comput. Phys.* **241** (2013), p. 445-463.

[42] O. A. Ladyzhenskaya, *The mathematical theory of viscous incompressible flow*, vol. 2, Gordon and Breach New York, 1969.

[43] D. P. Levadoux, "Analyse numérique de la formulation intégrodifférentielle d'un problème de Maxwell harmonique impliquant un diélectrique traversé de surfaces exfoliées métalliques et impédantes", Technical Report HAL, 2022, https://hal.archives-ouvertes.fr/hal-03644547.

[44] S. Nicaise, J. Venel, "A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients", *J. Comput. Appl. Math.* **235** (2011), p. 4272-4282.

[45] F.-J. Sayas, T. S. Brown, M. E. Hassell, *Variational techniques for elliptic partial differential equations*, CRC Press, 2019.

[46] G. Unger, "Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems", *SN Partial Differential Equations and Applications* **39** (2021).

[47] C. Weber, "A local compactness theorem for Maxwell's equations", *Mathematical Methods in the Applied Sciences* **2** (1980), no. 1, p. 12-25.