Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure

Abstract

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.

Appendix: Some Lemmas Related to the VAG Discretization

This appendix gathers lemmas on some properties of the VAG discretization that are independent of the continuous problem (and thus of the scheme). In what follows, $\mathcal {D}= (\mathcal {M},\mathcal {T})$ denote a discretization of $\Omega$ as prescribed in Sect. 2.1.1, and $\pi _\mathcal {T}, \pi _\mathcal {M}, \pi _\mathcal {D}$ and $\varvec{\nabla }_\mathcal {T}$ are the corresponding reconstruction operators.

Lemma 6.1

For $\kappa \in \mathcal {M}$, let $\mathbf{A}_\kappa = \left( a^\kappa _{\mathrm{s},\mathrm{s}'} \right) _{\mathrm{s},\mathrm{s}' \in \mathcal {V}_\kappa }$ be the matrix defined by (34), then there exists C depending only on ${\varvec{\Lambda }}$, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ (but not on $\kappa$) such that $\mathrm{Cond} _2(\mathbf{A}_\kappa ) \le C$.

Proof

Following [26, Lemma 3.2], there exist $C_1,C_2 >0$ depending only on $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that, for all ${\varvec{u}}\in W_\mathcal {D}$ and all $\kappa \in \mathcal {M}$, one has

$$\begin{aligned} C_1 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( u_\mathrm{s}- u_\kappa \right) ^2 \le {\Vert \varvec{\nabla }_\mathcal {T}{\varvec{u}}\Vert } ^2_{L^2(\kappa )} \le C_2 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( u_\mathrm{s}- u_\kappa \right) ^2, \end{aligned}$$

where $h_\kappa$ denotes the diameter of the cell $\kappa \in \mathcal {M}$. As a consequence, one has

$$\begin{aligned} \lambda _\star C_1 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} |{\varvec{\delta }}_\kappa {\varvec{u}}|^2 \le {\varvec{\delta }}_\kappa {\varvec{u}}\cdot \mathbf{A}_\kappa {\varvec{\delta }}_\kappa {\varvec{u}}= \int _\kappa {\varvec{\Lambda }}\varvec{\nabla }_\mathcal {T}{\varvec{u}}\cdot \varvec{\nabla }_\mathcal {T}{\varvec{u}}\, \mathrm{d} {\varvec{x}}\le \lambda ^\star C_2 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} |{\varvec{\delta }}_\kappa {\varvec{u}}|^2. \end{aligned}$$

Since the application ${\varvec{\delta }}_\kappa : W_\mathcal {D}\rightarrow \mathbb {R}^{\ell _\kappa }$ is onto, we deduce that

$$\begin{aligned} \lambda _\star C_1 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} |{\varvec{v}}|^2 \le {\varvec{v}}\cdot \mathbf{A}_\kappa {\varvec{v}}\le \lambda ^\star C_2 \frac{\mathrm{meas}(\kappa )}{(h_\kappa )^2} |{\varvec{v}}|^2, \quad \forall {\varvec{v}}\in \mathbb {R}^{\ell _\kappa }, \end{aligned}$$

and thus that $\mathrm{Cond} _2(\mathbf{A}_\kappa ) \le \frac{\lambda ^\star C_2}{\lambda _\star C_1}.$ $\square$

Lemma 6.2

There exists $C\ge 1$ depending only on ${\varvec{\Lambda }}$, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that, for all $\kappa \in \mathcal {M}$ and all ${\varvec{v}}= {(v_\mathrm{s})} _{\mathrm{s}\in \mathcal {V}_\kappa } \in \mathbb {R}^{\ell _\kappa }$, one has

$$\begin{aligned} \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) \left( v_\mathrm{s}\right) ^2 \le C\; {\varvec{v}}\cdot \mathbf{A}_\kappa {\varvec{v}}. \end{aligned}$$

Proof

Denoting by $\Vert \cdot \Vert _{q}$ the usual matrix q-norm, one has

$$\begin{aligned} \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) \left( v_\mathrm{s}\right) ^2 \le {\Vert \mathbf{A}_\kappa \Vert } _1 | {\varvec{v}}|^2. \end{aligned}$$

Since the dimension of the space $\mathbb {R}^{\ell _\kappa }$ is bounded by $\ell _\mathcal {D}$, there exists $C_1$ depending only on $\ell _\mathcal {D}$ such that $\Vert \mathbf{A}_\kappa \Vert _1 \le C_1 \Vert \mathbf{A}_\kappa \Vert _2$, so that

$$\begin{aligned} \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) \left( v_\mathrm{s}\right) ^2 \le C_1 {\Vert \mathbf{A}_\kappa \Vert } _2 | {\varvec{v}}|^2. \end{aligned}$$

(120)

On the other hand, since $\mathbf{A}_\kappa$ is symmetric definite and positive, one has

$$\begin{aligned} {\varvec{v}}\cdot \mathbf{A}_\kappa {\varvec{v}}\ge \frac{{\Vert \mathbf{A}_\kappa \Vert } _2}{\mathrm{Cond} _2(\mathbf{A}_\kappa )} |{\varvec{v}}|^2. \end{aligned}$$

Using Lemma 6.1, we obtain that there exists $C_2>0$ depending only on ${\varvec{\Lambda }}$, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that

$$\begin{aligned} {\varvec{v}}\cdot \mathbf{A}_\kappa {\varvec{v}}\ge C_2 {{\Vert \mathbf{A}_\kappa \Vert } _2} | {\varvec{v}}|^2. \end{aligned}$$

(121)

Putting (120) and (121) together, we conclude the proof of Lemma 6.2 by choosing $C = \frac{C_1}{C_2}$. $\square$

Lemma 6.3

Let $\kappa \in \mathcal {M}$ and $\mathbf{A}_\kappa = (a^\kappa _{\mathrm{s},\mathrm{s}'})_{\mathrm{s},\mathrm{s}'\in \mathcal {V}_\kappa } \in \mathbb {R}^{\ell _\kappa \times \ell _\kappa }$ be the matrix defined by (34). Let ${\varvec{\mu }}_\kappa = \left( \mu _{\kappa ,\mathrm{s}} \right) _{\mathrm{s}\in \mathcal {V}_\kappa } \in \mathbb {R}^{\ell _\kappa }$ and ${\varvec{v}}\in W_\mathcal {D}$, then

$$\begin{aligned}&\sum _{\mathrm{s}\in \mathcal {V}_\kappa } \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } (v_\mathrm{s}- v_\kappa ) \mu _{\kappa ,\mathrm{s}} a^\kappa _{\mathrm{s},\mathrm{s}'} \mu _{\kappa ,\mathrm{s}'} (v_{\mathrm{s}'} - v_\kappa ) \\&\quad \le \max _{\mathrm{s}\in \mathcal {V}_\kappa } (\mu _{\kappa ,\mathrm{s}})^2 \sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) \left( v_\mathrm{s}- v_\kappa \right) ^2. \end{aligned}$$

Proof

Using $ab \le \frac{a^2}{2} + \frac{b^2}{2}$, we obtain that

$$\begin{aligned}&\sum _{\mathrm{s}\in \mathcal {V}_\kappa } \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } (v_\mathrm{s}- v_\kappa ) \mu _{\kappa ,\mathrm{s}} a^\kappa _{\mathrm{s},\mathrm{s}'} \mu _{\kappa ,\mathrm{s}'} (v_{\mathrm{s}'} - v_\kappa ) \\&\quad \le \max _{\mathrm{s}\in \mathcal {V}_\kappa } (\mu _{\kappa ,\mathrm{s}})^2\sum _{\mathrm{s}\in \mathcal {V}_\kappa } \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |v_\mathrm{s}- v_\kappa | |a^\kappa _{\mathrm{s},\mathrm{s}'} | |v_{\mathrm{s}'} - v_\kappa | \\&\quad \le \frac{\max _{\mathrm{s}\in \mathcal {V}_\kappa } (\mu _{\kappa ,\mathrm{s}})^2}{2}\sum _{\mathrm{s}\in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}' \in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) (v_\mathrm{s}- v_\kappa )^2 \\&\qquad + \frac{\max _{\mathrm{s}\in \mathcal {V}_\kappa } (\mu _{\kappa ,\mathrm{s}})^2}{2}\sum _{\mathrm{s}' \in \mathcal {V}_\kappa } \left( \sum _{\mathrm{s}\in \mathcal {V}_\kappa } |a^\kappa _{\mathrm{s},\mathrm{s}'} |\right) (v_\mathrm{s}' - v_\kappa )^2. \end{aligned}$$

One concludes the proof of Lemma 6.3 by noticing that, since $\mathbf{A}_\kappa$ is symmetric, the two terms in the right-hand side of the above inequality are equal. $\square$

Lemma 6.4

There exists C depending only on $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that

$$\begin{aligned} \mathrm{meas}(T) \le \mathrm{meas}(\kappa ) \le C \mathrm{meas}(T), \quad \forall \kappa \in \mathcal {M}, \; \forall T \in \mathcal {T}\text {with } T \subset \kappa . \end{aligned}$$

Proof

Let $\kappa \in \mathcal {M}$, then there exist $T_1, \ldots , T_{r}$ simplexes, with $r=\ell _\kappa$ if $d = 2$ and $r = 2 \#\mathcal {E}_\kappa$ if $d=3$, such that

$$\begin{aligned} \bigcup _{i=1} ^{r_\kappa } {\overline{T}}_i = {\overline{\kappa }}, \quad T_i \cap T_j = \emptyset \; \text {if } \; i \ne j. \end{aligned}$$

The Euler-Descartes theorem ensures that $r \le 4 (\ell _\mathcal {D}- 1)$ if $d=3$.

If $T_i$ and $T_j$ share a common edge, one gets that

$$\begin{aligned} \mathrm{meas}(T_i) \le \theta ^d \; \mathrm{meas}(T_j). \end{aligned}$$

Let $i_0, i_1 \in \{1,\ldots , r_\kappa \}$ be arbitrary but different, we deduce from the previous inequality the following non-optimal estimate:

$$\begin{aligned} \mathrm{meas}(T_{i_0}) \le \theta ^{4(\ell _\mathcal {D}- 1) d} \; \mathrm{meas}(T_{i_1}). \end{aligned}$$

Let $i_\mathrm{max}$ be such that $\mathrm{meas}(T_{i_\mathrm{max} }) = \max _{1 \le i \le r} \mathrm{meas}(T_i)$, then

$$\begin{aligned} \mathrm{meas}(\kappa ) \le r \; \mathrm{meas}(T_{i_\mathrm{max} }) \le 4 (\ell _\mathcal {D}- 1) \theta ^{4(\ell _\mathcal {D}- 1) d} \mathrm{meas}(T_i), \quad \forall i \in \{1,\ldots , r\}. \end{aligned}$$

$\square$

We state now a slight generalization of [26, Lemma 3.4], where the same result is proven in the particular case $q=2$. The straightforward adaptation of the proof given in [26] to the case $q \ne 2$ is left to the reader.

Lemma 6.5

There exists C depending only on $\ell _\mathcal {D}$ and $\theta _\mathcal {T}$ defined in (24) and (21), respectively, such that, for all ${\varvec{v}}\in W_{\mathcal {D}}$ and all $q \in [1,\infty ]$, one has

$$\begin{aligned} \left\| \pi _{\mathcal {D}} {\varvec{v}}- \pi _{\mathcal {T}} {\varvec{v}}\right\| _{L^q(\Omega )} + \left\| \pi _{\mathcal {D}} {\varvec{v}}- \pi _{\mathcal {M}} {\varvec{v}}\right\| _{L^q(\Omega )} \le C h_\mathcal {T}\left\| \varvec{\nabla }_{\mathcal {T}} {\varvec{v}}\right\| _{L^q(\Omega )}. \end{aligned}$$

Lemma 6.6

Let $\mathcal {D}$ be a discretization of $\Omega$ as introduced in Sect. 2.1.1 such that $\zeta _\mathcal {D}>0$, then there exist $C_1 >0$ depending only on q, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ and $C_2$ depending moreover on $\zeta _\mathcal {D}$ such that

$$\begin{aligned} C_1 \Vert \pi _{\mathcal {D}} {\varvec{v}}\Vert _{L^q(\Omega )} \le \Vert \pi _{\mathcal {T}} {\varvec{v}}\Vert _{L^q(\Omega )} \le C_2 \Vert \pi _{\mathcal {D}} {\varvec{v}}\Vert _{L^q(\Omega )}, \quad \forall {\varvec{v}}\in W_{\mathcal {D}}. \end{aligned}$$

(122)

Proof

Let ${\widehat{T}}$ be a reference tetrahedron, and let ${\widehat{v}}: {\widehat{T}} \rightarrow \mathbb {R}$ be an affine function with nodal values $v_i$, $i \in \{1,\ldots , 4\}$, then for all $q>0$, there exists C depending on q such that

$$\begin{aligned} \frac{1}{C} \sum _{i = 1} ^4 |v_i|^q \le \Vert {\widehat{v}} \Vert _{L^q({\widehat{T}})} ^q \le C \sum _{i = 1} ^4 |v_i|^q. \end{aligned}$$

Therefore, using classical properties of the affine change of variable between simplexes, one gets the existence of C depending only on q, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that, for all ${\varvec{v}}\in W_{\mathcal {D}}$,

$$\begin{aligned}&\frac{1}{C} \sum _{\kappa \in \mathcal {M}} \mathrm{meas}(\kappa ) \left( |v_\kappa |^q + \sum _{\mathrm{s}\in \mathcal {V}_\kappa } | v_\mathrm{s}|^q \right) \nonumber \\&\quad \le \Vert \pi _{\mathcal {T}} {\varvec{v}}\Vert _{L^q(\Omega )} ^q \le C \sum _{\kappa \in \mathcal {M}} \mathrm{meas}(\kappa ) \left( |v_\kappa |^q + \sum _{\mathrm{s}\in \mathcal {V}_\kappa } | v_\mathrm{s}|^q \right) . \end{aligned}$$

(123)

On the other hand, one has

$$\begin{aligned} \Vert \pi _{\mathcal {D}} {\varvec{v}}\Vert _{L^q(\Omega )} ^q = \sum _{\kappa \in \mathcal {M}} m_\kappa |v_\kappa |^q + \sum _{\mathrm{s}\in \mathcal {V}} m_\mathrm{s}|v_\mathrm{s}|^q. \end{aligned}$$

A classical geometrical property and (29) yield

$$\begin{aligned} m_\kappa \le \mathrm{meas}(\kappa ) = d \int _{\Omega } \pi _\mathcal {T}\mathbf{e}_{\kappa } ({\varvec{x}}) \mathrm{d} {\varvec{x}}\le \frac{d}{\zeta _\mathcal {D}} m_\kappa \quad \forall \kappa \in \mathcal {M}, \end{aligned}$$

(124)

and similarly

$$\begin{aligned} m_\mathrm{s}\le d \int _{\Omega } \pi _\mathcal {T}\mathbf{e}_{\mathrm{s}} ({\varvec{x}}) \mathrm{d} {\varvec{x}}\le \frac{d}{\zeta _\mathcal {D}} m_\mathrm{s}, \quad \; \forall \mathrm{s}\in \mathcal {V}. \end{aligned}$$

Notice now that the following geometrical identity holds:

$$\begin{aligned} d \int _{\Omega } \pi _\mathcal {T}\mathbf{e}_{\mathrm{s}} ({\varvec{x}}) \mathrm{d} {\varvec{x}}= \sum _{\begin{array}{c} T \in \mathcal {T}\\ {\varvec{x}}_\mathrm{s}\in \partial T \end{array} } \mathrm{meas}(T), \quad \forall \mathrm{s}\in \mathcal {V}. \end{aligned}$$

Lemma 6.4 yields the existence of $C>0$ depending on $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that

$$\begin{aligned} \frac{1}{C} \sum _{\kappa \in \mathcal {M}_\mathrm{s}} \mathrm{meas}(\kappa ) \le d \int _{\Omega } \pi _\mathcal {T}\mathbf{e}_{\mathrm{s}} ({\varvec{x}}) \mathrm{d} {\varvec{x}}\le \sum _{\kappa \in \mathcal {M}_\mathrm{s}} \mathrm{meas}(\kappa ), \quad \forall \mathrm{s}\in \mathcal {V}, \end{aligned}$$

and the result of Lemma 6.6 follows. $\square$

Lemma 6.7

Let $\mathcal {D}$ be a discretization of $\Omega$ as introduced in Sect. 2.1.1 such that $\zeta _\mathcal {D}>0$, then, for all $q \in [1,\infty ]$, one has

$$\begin{aligned} \left\| \pi _{\mathcal {M}} {\varvec{v}}\right\| _{L^q(\Omega )} \le \left( \frac{d}{\zeta _\mathcal {D}} \right) ^{1/q} \left\| \pi _{\mathcal {D}} {\varvec{v}}\right\| _{L^q(\Omega )}, \quad \forall {\varvec{v}}\in W_{\mathcal {D}}. \end{aligned}$$

Proof

Let ${\varvec{v}}= \left( v_\kappa , v_\mathrm{s}\right) _{\kappa \in \mathcal {M}, \mathrm{s}\in \mathcal {V}} \in W_{\mathcal {D}},$ then it follows from (124) that

$$\begin{aligned} \left\| \pi _{\mathcal {M}} {\varvec{v}}\right\| _{L^q(\Omega )} ^q&= \sum _{\kappa \in \mathcal {M}} \mathrm{meas}(\kappa ) \left| v_\kappa \right| ^q \\&\le \left( \frac{d}{\zeta _\mathcal {D}} \right) \sum _{\kappa \in \mathcal {M}} m_\kappa \left| v_\kappa \right| ^q \le \left( \frac{d}{\zeta _\mathcal {D}} \right) \left\| \pi _{\mathcal {D}} {\varvec{v}}\right\| _{L^q(\Omega )} ^q. \end{aligned}$$

Lemma 6.8

Let ${\varvec{v}}= {(v_\kappa , v_\mathrm{s})} _{\kappa ,\mathrm{s}} \in W_\mathcal {D}$ be such that $v_\beta \ge 0$ for all $\beta \in \mathcal {M}\cup \mathcal {V}$, and define ${\overline{{\varvec{v}}}}= {({\overline{v}}_\kappa , {\overline{v}}_\mathrm{s})} _{\kappa ,\mathrm{s}} \in W_\mathcal {D}$ by

$$\begin{aligned} {\overline{v}}_\mathrm{s}= 0, \quad {\overline{v}}_\kappa = \max \left( v_\kappa , \max _{\mathrm{s}' \in \mathcal {V}_\kappa } v_{\mathrm{s}'} \right) , \quad \forall \mathrm{s}\in \mathcal {V}, \forall \kappa \in \mathcal {M}. \end{aligned}$$

Then there exists C depending only on $\theta _\mathcal {T}$, $\ell _\mathcal {D}$ and $\zeta _\mathcal {D}$ such that

$$\begin{aligned} \left\| \pi _\mathcal {M}{\overline{{\varvec{v}}}}\right\| _{L^1(\Omega )} \le C \left\| \pi _\mathcal {D}{\varvec{v}}\right\| _{L^1(\Omega )}. \end{aligned}$$

Proof

Let ${\varvec{v}}\in W_\mathcal {D}$ be a vector with positives coordinates, and let ${\overline{{\varvec{v}}}}$ be constructed as above. It follows from the construction of ${\overline{{\varvec{v}}}}$ that

$$\begin{aligned} {\overline{v}}_\kappa \le v_\kappa + \sum _{\mathrm{s}\in \mathcal {V}_\kappa } v_\mathrm{s}, \quad \forall \kappa \in \mathcal {M}, \end{aligned}$$

whence, applying (123) with $q=1$, one gets

$$\begin{aligned} \Vert \pi _\mathcal {M}{\overline{{\varvec{v}}}}\Vert _{L^1(\Omega )} \le C \Vert \pi _\mathcal {T}{\varvec{v}}\Vert _{L^1(\Omega )}. \end{aligned}$$

The result now directly follows from Lemma 6.6. $\square$

Lemma 6.9

Let ${\varvec{u}}= (u_\kappa , u_\mathrm{s})_{\kappa \in \mathcal {M}, \mathrm{s}\in \mathcal {V}} \in W_\mathcal {D}$, then for all $\kappa \in \mathcal {M}$, we define ${\overline{{\varvec{\delta }}}}{\varvec{u}}= \left( {\overline{\delta }}_\kappa {\varvec{u}}, {\overline{\delta }}_\mathrm{s}{\varvec{u}}\right) _{\kappa \in \mathcal {M}, \mathrm{s}\in \mathcal {V}} \in W_\mathcal {D}$ by

$$\begin{aligned} {\overline{\delta }}_\mathrm{s}{\varvec{u}}= 0 \quad \text {and } \quad {\overline{\delta }}_\kappa {\varvec{u}}= \max _{\mathrm{s}' \in \mathcal {V}_\kappa } | u_\kappa - u_\mathrm{s}|, \quad \forall \kappa \in \mathcal {M}, \; \forall \mathrm{s}\in \mathcal {V}, \end{aligned}$$

then, for all $q \in [1,\infty ]$, there exists C depending only on q, $\theta _\mathcal {T}$, and $\ell _\mathcal {D}$ such that

$$\begin{aligned} \left\| \pi _\mathcal {M}{\overline{{\varvec{\delta }}}}{\varvec{u}}\right\| _{L^q(\Omega )} \le C h_\mathcal {T}\Vert \varvec{\nabla }_\mathcal {T}{\varvec{u}}\Vert _{L^q(\Omega )}. \end{aligned}$$

(125)

Proof

Let $\kappa \in \mathcal {M}$ and $\mathrm{s}\in \mathcal {V}_\kappa$, then there exists a simplicial subelement $T \in \mathcal {T}$ of $\kappa \in \mathcal {M}$ such that ${\varvec{x}}_\kappa$ and ${\varvec{x}}_\mathrm{s}$ are vertices of T. Then it follows from classical finite element arguments (see, e.g. [40, 46]) that

$$\begin{aligned} \mathrm{meas}(T)^{1/q} |u_\kappa - u_\mathrm{s}| \le c \frac{\left( h_T\right) ^2}{\rho _T} \Vert \varvec{\nabla }_\mathcal {T}{\varvec{u}}\Vert _{L^q(T)} \le C h_\mathcal {T}\Vert \varvec{\nabla }_\mathcal {T}{\varvec{u}}\Vert _{L^q(\kappa )}, \end{aligned}$$

where c depends only on the dimension d and on q, while C depends additionally on $\theta _\mathcal {T}$. Thanks to Lemma 6.4, we get the existence of C depending on d, q, $\theta _\mathcal {T}$ and $\ell _\mathcal {D}$ such that,

$$\begin{aligned} \mathrm{meas}(\kappa )^{1/q} |u_\kappa - u_\mathrm{s}| \le C h_\mathcal {T}\left\| \varvec{\nabla }_\mathcal {T}{\varvec{u}}\right\| _{L^q(\kappa )}, \quad \forall \kappa \in \mathcal {M}, \; \forall \mathrm{s}\in \mathcal {V}_\kappa . \end{aligned}$$

Summing over $\kappa \in \mathcal {M}$ provides that (125) holds. $\square$