Cotela 2016 - Revision history

Scipediacontent at 14:40, 5 January 2023

2023-01-05T14:40:37Z

Scipediacontent at 14:37, 5 January 2023

2023-01-05T14:37:29Z

Scipediacontent at 13:01, 5 January 2023

2023-01-05T13:01:57Z

Scipediacontent at 12:56, 5 January 2023

2023-01-05T12:56:46Z

Scipediacontent at 12:54, 5 January 2023

2023-01-05T12:54:45Z

Scipediacontent at 12:52, 5 January 2023

2023-01-05T12:52:43Z

Scipediacontent at 12:48, 5 January 2023

2023-01-05T12:48:49Z

Scipediacontent at 11:24, 5 January 2023

2023-01-05T11:24:02Z

Scipediacontent at 11:19, 5 January 2023

2023-01-05T11:19:43Z

Scipediacontent at 11:17, 5 January 2023

2023-01-05T11:17:03Z

@@ Line 3,002: / Line 3,002: @@
 where <math display="inline">t_i</math> represents the <math display="inline">i</math>-th component of the tractions imposed on the Neumann boundary <math display="inline">\Gamma _N</math>.
-Although Eq.&nbsp;[[#eq-3.196|3.196]] was developed using a FIC based approach, the final expression is analogous to a SUPG stabilized formulation, with <math display="inline">h_u / 2 \left\Vert u \right\Vert_{\Omega }</math> (which has dimensions of time) playing the role of the SUPG stabilization parameter <math display="inline">\tau </math>.
+Although Eq.&nbsp;[[#eq-3.196|3.196]] was developed using a FIC based approach, the final expression is analogous to a SUPG stabilized formulation, with <math display="inline">h_u / 2 \left\Vert u \right\Vert_{ }</math> (which has dimensions of time) playing the role of the SUPG stabilization parameter <math display="inline">\tau </math>.
 ===3.3.2 Gradient diffusion formulation===

@@ Line 587: / Line 587: @@
 | style="text-align: center;" | <math>   + \int _{\Sigma \, \Omega ^e}\boldsymbol{w}_s\nabla \cdot \left(\left(p_h+ p_s\right)\boldsymbol{I} - 2 \mu \left(\nabla ^s \left(u_h+ u_s\right)- \frac{1}{3} \nabla \cdot \left(u_h+ u_s\right)\boldsymbol{I} \right)\right)\, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>   + \sum _e \int _{\Gamma ^e} \boldsymbol{w}_s\left(\left(p_h+ p_s\right)\cdot \boldsymbol{n}   + 2  \mu \left(\nabla ^s \left(u_h+ u_s\right)- \frac{1}{3} \nabla \cdot \left(u_h+ u_s\right)\right)\cdot \boldsymbol{n} \right)\, \hbox{d}\Gamma </math>
+| style="text-align: center;" | <math>   + \sum _e \int _{\Gamma ^e} \boldsymbol{w}_s\left(\left(p_h+ p_s\right)\cdot \boldsymbol{n}   + 2  \mu \left(\nabla ^s \left(u_h+ u_s\right)- \frac{1}{3} \nabla \cdot \left(u_h+ u_s\right)\right)\cdot \boldsymbol{n} \right)\, \hbox{d}\Gamma _e</math>
 |-
 | style="text-align: center;" | <math>  + \int _{\Gamma _N} \boldsymbol{w}_s\cdot \rho \frac{1}{2} \left(\boldsymbol{a} \cdot \boldsymbol{n} \right)\left(u_h+ u_s\right)\, \hbox{d}\Omega   - \sum _e \int _{\Gamma ^e} \boldsymbol{w}_s\cdot \rho \frac{1}{2} \left(\boldsymbol{a} \cdot \boldsymbol{n} \right)\left(u_h+ u_s\right)\, \hbox{d}\Gamma = </math>

@@ Line 3,453: / Line 3,453: @@
 | style="text-align: center;" | <math> + \int _\Omega 2\mu \, \frac{\partial w_i}{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3}\, \delta _{ij} \right)\, \hbox{d}\Omega - \int _\Omega \frac{\partial w_i}{\partial x_i} \, p \, \, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - f_i \right)\, \hbox{d}\Omega   + \int _\Omega \frac{\partial w_i}{\partial x_j} \left(1-\beta \right)D^g_{ik} \frac{\partial u_k}{\partial x_j} \, \hbox{d}\Omega  = </math>
+| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - f_i \right)\, \hbox{d}\Omega   + \int _\Omega \frac{\partial w_i}{\partial x_j} \left(1-\beta \right)D^g_{ik} \frac{\partial u_k}{\partial x_j} \, \hbox{d}\Omega  = </math>
 |-
 | style="text-align: center;" | <math>  \int _\Omega w_i \, f_i \, \, \hbox{d}\Omega + \int _{\Gamma _N} w_i \, t_i \, \, \hbox{d}\Gamma  - \int _{\Gamma _N} \dfrac{1}{2} \rho w_i \left(u_k n_k \right)u_i \, \hbox{d}\Gamma   </math>
@@ Line 3,484: / Line 3,484: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>D^g_{ik} = \frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \delta _{ik} \qquad \tau _c = \left(\frac{3\rho \left\Vert u \right\Vert_{\Omega }}{h} + \frac{8\mu }{h^2} \right)^{-1}  \qquad \beta \in \left[0 ,1 \right] </math>
+| style="text-align: center;" | <math>D^g_{ik} = \frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{ }} \delta _{ik} \qquad \tau _c = \left(\frac{3\rho \left\Vert u \right\Vert_{ }}{h} + \frac{8\mu }{h^2} \right)^{-1}  \qquad \beta \in \left[0 ,1 \right] </math>
 |}
 |}
@@ Line 3,597: / Line 3,597: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{M}^e_{K\,ab} = \int _{\Omega _e} \rho \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \boldsymbol{N}_b \, \hbox{d}\Omega </math>
+| style="text-align: center;" | <math>\boldsymbol{M}^e_{K\,ab} = \int _{\Omega _e} \rho \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \boldsymbol{N}_b \, \hbox{d}\Omega </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.242)
 |-
@@ Line 3,603: / Line 3,603: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.243)
 |-
-| style="text-align: center;" | <math>  \boldsymbol{S}^e_{K\,ab} = \int _{\Omega _e} \rho \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \left(u_l \frac{\partial \boldsymbol{N}_b}{\partial x_l} \right)\, \hbox{d}\Omega </math>
+| style="text-align: center;" | <math>  \boldsymbol{S}^e_{K\,ab} = \int _{\Omega _e} \rho \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \left(u_l \frac{\partial \boldsymbol{N}_b}{\partial x_l} \right)\, \hbox{d}\Omega </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.244)
 |-
-| style="text-align: center;" | <math>  \boldsymbol{S}^e_{G\,ab\,i} = \int _{\Omega _e} \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \frac{\partial N_b}{\partial x_i} \, \hbox{d}\Omega </math>
+| style="text-align: center;" | <math>  \boldsymbol{S}^e_{G\,ab\,i} = \int _{\Omega _e} \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \frac{\partial N_b}{\partial x_i} \, \hbox{d}\Omega </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.245)
 |-
@@ Line 3,618: / Line 3,618: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.248)
 |-
-| style="text-align: center;" | <math>  \boldsymbol{S}^e_{F\,a} = \int _{\Omega _e} \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \boldsymbol{f} \, \hbox{d}\Omega </math>
+| style="text-align: center;" | <math>  \boldsymbol{S}^e_{F\,a} = \int _{\Omega _e} \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} \left(u_k \frac{\partial \boldsymbol{N}_a}{\partial x_k} \right)^T \boldsymbol{f} \, \hbox{d}\Omega </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.249)
 |-

@@ Line 3,170: / Line 3,170: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>h_u = \max _{\hbox{edges}} \left\lbrace \boldsymbol{e}_u \cdot \boldsymbol{l}_{ab} \right\rbrace \qquad \hbox{with} \quad \boldsymbol{e}_u = \frac{u}{\left\Vert u \right\Vert_{\Omega }} </math>
+| style="text-align: center;" | <math>h_u = \max _{\hbox{edges}} \left\lbrace \boldsymbol{e}_u \cdot \boldsymbol{l}_{ab} \right\rbrace \qquad \hbox{with} \quad \boldsymbol{e}_u = \frac{u}{\left\Vert u \right\Vert_{ }} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.210)
@@ Line 3,196: / Line 3,196: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>h_{g_i} = \max _{edges} \left\lbrace \boldsymbol{e}_{g_i} \cdot \boldsymbol{l}_{ab} \right\rbrace \qquad \hbox{with} \quad \boldsymbol{e}_{g_i} = \frac{\nabla u_i}{\left\Vert \nabla u_i \right\Vert_{\Omega }} \qquad \hbox{No sum on ''i''.} </math>
+| style="text-align: center;" | <math>h_{g_i} = \max _{edges} \left\lbrace \boldsymbol{e}_{g_i} \cdot \boldsymbol{l}_{ab} \right\rbrace \qquad \hbox{with} \quad \boldsymbol{e}_{g_i} = \frac{\nabla u_i}{\left\Vert \nabla u_i \right\Vert_{ }} \qquad \hbox{No sum on ''i''.} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.211)
@@ Line 3,213: / Line 3,213: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\beta _i = \hbox{max} \left\lbrace 1 - \dfrac{u_k \left(\partial u_i / \partial x_k \right)}{\left\Vert u \right\Vert_{\Omega }\left\Vert \nabla u_i \right\Vert_{\Omega }}, \beta _m \right\rbrace \qquad \hbox{No sum on ''i''.}  </math>
+| style="text-align: center;" | <math>\beta _i = \hbox{max} \left\lbrace 1 - \dfrac{u_k \left(\partial u_i / \partial x_k \right)}{\left\Vert u \right\Vert_{ }\left\Vert \nabla u_i \right\Vert_{ }}, \beta _m \right\rbrace \qquad \hbox{No sum on ''i''.}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.212)

@@ Line 3,122: / Line 3,122: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>r^m_i - \beta \frac{h_u}{2 \left\Vert u \right\Vert_{\Omega } } u_j \frac{\partial r^m_i}{\partial x_j} - \left(1-\beta \right)\frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega } } \frac{\partial u_i}{\partial x_k} \frac{\partial r^m_i}{\partial x_k} \qquad \hbox{No sum on ''i''.}  </math>
+| style="text-align: center;" | <math>r^m_i - \beta \frac{h_u}{2 \left\Vert u \right\Vert_{ } } u_j \frac{\partial r^m_i}{\partial x_j} - \left(1-\beta \right)\frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ } } \frac{\partial u_i}{\partial x_k} \frac{\partial r^m_i}{\partial x_k} \qquad \hbox{No sum on ''i''.}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.208)
@@ Line 3,141: / Line 3,141: @@
 | style="text-align: center;" | <math> + \int _\Omega 2\mu \, \frac{\partial w_i}{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3}\, \delta _{ij} \right)\, \hbox{d}\Omega - \int _\Omega \frac{\partial w_i}{\partial x_i} \, p \, \, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - 2\mu \, \frac{\partial }{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3} \, \delta _{ij} \right)- f_i \right)\, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>  + \int _\Omega \beta \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - 2\mu \, \frac{\partial }{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3} \, \delta _{ij} \right)- f_i \right)\, \hbox{d}\Omega  </math>
 |-
 | style="text-align: center;" | <math>  + \int _\Omega \frac{\partial w_i}{\partial x_j} \left(1-\beta \right)D^g_{ik} \frac{\partial u_k}{\partial x_j} \, \hbox{d}\Omega  = </math>

@@ Line 3,014: / Line 3,014: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{h_{g_i}} = h_{g_i} \frac{\nabla u_i}{\left\Vert \nabla u_i \right\Vert_{\Omega }} \qquad \hbox{No sum on ''i''.}  </math>
+| style="text-align: center;" | <math>\boldsymbol{h_{g_i}} = h_{g_i} \frac{\nabla u_i}{\left\Vert \nabla u_i \right\Vert_{ }} \qquad \hbox{No sum on ''i''.}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.197)
@@ Line 3,027: / Line 3,027: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>r^m_i - \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega } } \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \qquad \hbox{No sum on ''i''.}  </math>
+| style="text-align: center;" | <math>r^m_i - \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ } } \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \qquad \hbox{No sum on ''i''.}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.198)
@@ Line 3,041: / Line 3,041: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _\Omega w_i \left(r^m_i - \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega } } \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \right)\, \hbox{d}\Omega =  </math>
+| style="text-align: center;" | <math>\int _\Omega w_i \left(r^m_i - \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ } } \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \right)\, \hbox{d}\Omega =  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.199)
 |-
-| style="text-align: center;" | <math>   \int _\Omega w_i \, r^m_i \, \hbox{d}\Omega - \int _\Omega w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \, \hbox{d}\Omega = 0  </math>
+| style="text-align: center;" | <math>   \int _\Omega w_i \, r^m_i \, \hbox{d}\Omega - \int _\Omega w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \, \hbox{d}\Omega = 0  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.200)
 |}
@@ Line 3,057: / Line 3,057: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>- \int _\Omega w_i  \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \, \hbox{d}\Omega = </math>
+| style="text-align: center;" | <math>- \int _\Omega w_i  \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j} \frac{\partial r^m_i}{\partial x_j} \, \hbox{d}\Omega = </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.201)
 |-
-| style="text-align: center;" | <math>  \int _\Omega \frac{\partial }{\partial x_j} \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j} \right)r^m_i  - \int _\Omega \frac{\partial }{\partial x_j} \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j}  r^m_i \right)\, \hbox{d}\Omega = </math>
+| style="text-align: center;" | <math>  \int _\Omega \frac{\partial }{\partial x_j} \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j} \right)r^m_i  - \int _\Omega \frac{\partial }{\partial x_j} \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j}  r^m_i \right)\, \hbox{d}\Omega = </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.202)
 |-
-| style="text-align: center;" | <math>  \int _\Omega \frac{\partial w_i}{\partial x_j} \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j} \, r^m_i \, \, \hbox{d}\Omega +  \int _\Omega w_i \, \frac{\partial }{\partial x_j} \left(\frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j} \right)r^m_i \, \hbox{d}\Omega - </math>
+| style="text-align: center;" | <math>  \int _\Omega \frac{\partial w_i}{\partial x_j} \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j} \, r^m_i \, \, \hbox{d}\Omega +  \int _\Omega w_i \, \frac{\partial }{\partial x_j} \left(\frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j} \right)r^m_i \, \hbox{d}\Omega - </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.203)
 |-
-| style="text-align: center;" | <math>  \int _\Omega \frac{\partial }{\partial x_j}  \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \frac{\partial u_i}{\partial x_j}  r^m_i \right)\, \hbox{d}\Omega   </math>
+| style="text-align: center;" | <math>  \int _\Omega \frac{\partial }{\partial x_j}  \left(w_i \frac{h_{g_i}}{2 \left\Vert \nabla u_i \right\Vert_{ }} \frac{\partial u_i}{\partial x_j}  r^m_i \right)\, \hbox{d}\Omega   </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.204)
 |}
@@ Line 3,079: / Line 3,079: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _\Omega \frac{\partial w_i}{\partial x_j} \left(\frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \delta _{ik} \right)\frac{\partial u_k}{\partial x_j} \, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>\int _\Omega \frac{\partial w_i}{\partial x_j} \left(\frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{ }} \delta _{ik} \right)\frac{\partial u_k}{\partial x_j} \, \hbox{d}\Omega  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.205)
@@ Line 3,092: / Line 3,092: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>D^g_{ij} = \frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{\Omega }} \delta _{ij} \qquad \hbox{No sum on ''i''}  </math>
+| style="text-align: center;" | <math>D^g_{ij} = \frac{h_{g_i} r^m_i}{2 \left\Vert \nabla u_i \right\Vert_{ }} \delta _{ij} \qquad \hbox{No sum on ''i''}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.206)

@@ Line 2,933: / Line 2,933: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\boldsymbol{h_u} = h_u \frac{u}{\left\Vert u \right\Vert_{\Omega } }  </math>
+| style="text-align: center;" | <math>\boldsymbol{h_u} = h_u \frac{u}{\left\Vert u \right\Vert_{ } }  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.192)
 |}
-where <math display="inline">h_u</math> is the projected length of a given element along the direction of flow, defined by the unit vector <math display="inline">u/\left\Vert u \right\Vert_{\Omega }</math>. Using this expression, we can rewrite Eq.&nbsp;[[#eq-3.191|3.191]] as
+where <math display="inline">h_u</math> is the projected length of a given element along the direction of flow, defined by the unit vector <math display="inline">u/\left\Vert u \right\Vert_{ }</math>. Using this expression, we can rewrite Eq.&nbsp;[[#eq-3.191|3.191]] as
 <span id="eq-3.193"></span>
@@ Line 2,946: / Line 2,946: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>r^m_i - \frac{h_u}{2 \left\Vert u \right\Vert_{\Omega } } u_j \frac{\partial r^m_i}{\partial x_j}  </math>
+| style="text-align: center;" | <math>r^m_i - \frac{h_u}{2 \left\Vert u \right\Vert_{ } } u_j \frac{\partial r^m_i}{\partial x_j}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.193)
@@ Line 2,959: / Line 2,959: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _\Omega w_i \, r^m_i - w_i \, \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial r^m_i}{\partial x_k} \, \hbox{d}\Omega = 0  </math>
+| style="text-align: center;" | <math>\int _\Omega w_i \, r^m_i - w_i \, \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial r^m_i}{\partial x_k} \, \hbox{d}\Omega = 0  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.194)
@@ Line 2,972: / Line 2,972: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\int _\Omega w_i \, r^m_i \, \hbox{d}\Omega + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \, r^m_i \, \hbox{d}\Omega - \sum _{(e)} \int _{\Gamma _e} \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} w_i  \left(u_k n_k \right)r^m_i \, \hbox{d}\Gamma = 0  </math>
+| style="text-align: center;" | <math>\int _\Omega w_i \, r^m_i \, \hbox{d}\Omega + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \, r^m_i \, \hbox{d}\Omega - \sum _{(e)} \int _{\Gamma _e} \frac{h_u}{2\left\Vert u \right\Vert_{ }} w_i  \left(u_k n_k \right)r^m_i \, \hbox{d}\Gamma = 0  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.195)
@@ Line 2,989: / Line 2,989: @@
 | style="text-align: center;" | <math> + \int _\Omega 2\mu \, \frac{\partial w_i}{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3}\, \delta _{ij} \right)\, \hbox{d}\Omega - \int _\Omega \frac{\partial w_i}{\partial x_i} \, p \, \, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>  + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\rho \partial _t u_i + \rho \left(\dfrac{1}{2} u_k \frac{\partial u_i}{\partial x_k} + \dfrac{1}{2} \frac{\partial }{\partial x_k} \left(u_i u_k \right)\right)\right)\, \hbox{d}\Omega  </math>
 |-
-| style="text-align: center;" | <math>  + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{\Omega }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - 2\mu \, \frac{\partial }{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3} \, \delta _{ij} \right)- f_i \right)\, \hbox{d}\Omega  = </math>
+| style="text-align: center;" | <math>  + \int _\Omega \frac{h_u}{2\left\Vert u \right\Vert_{ }} u_k \frac{\partial w_i}{\partial x_k} \left(\frac{\partial p}{\partial x_i} - 2\mu \, \frac{\partial }{\partial x_j} \left(\varepsilon _{ij} - \frac{\varepsilon _{kk}}{3} \, \delta _{ij} \right)- f_i \right)\, \hbox{d}\Omega  = </math>
 |-
 | style="text-align: center;" | <math>  \int _\Omega w_i \, f_i \, \, \hbox{d}\Omega + \int _{\Gamma _N} w_i \, t_i \, \, \hbox{d}\Gamma  - \int _{\Gamma _N} \dfrac{1}{2} \rho w_i \left(u_k n_k \right)u_i \, \hbox{d}\Gamma   </math>

@@ Line 1,984: / Line 1,984: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\rho \, \boldsymbol{a} \cdot \nabla u_s - 2\mu \nabla \cdot \nabla ^s u_s\sim u_s\left[\left(c_1 \frac{\mu }{h^2} \right)^2 + \left(c_2 \frac{\rho \left\Vert \boldsymbol{a} \right\Vert_{\Omega }}{h} \right)^2 \right]^{1/2}  </math>
+| style="text-align: center;" | <math>\rho \, \boldsymbol{a} \cdot \nabla u_s - 2\mu \nabla \cdot \nabla ^s u_s\sim u_s\left[\left(c_1 \frac{\mu }{h^2} \right)^2 + \left(c_2 \frac{\rho \left\Vert \boldsymbol{a} \right\Vert_{ }}{h} \right)^2 \right]^{1/2}  </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.152)
 |-
-| style="text-align: center;" | <math>   \left\Vert \nabla p_s \right\Vert_{\Omega } \sim p_s\, \frac{1}{h}  \qquad  \nabla \cdot u_s\sim \frac{1}{h}\left\Vert u_s \right\Vert_{\Omega }   </math>
+| style="text-align: center;" | <math>   \left\Vert \nabla p_s \right\Vert_{ } \sim p_s\, \frac{1}{h}  \qquad  \nabla \cdot u_s\sim \frac{1}{h}\left\Vert u_s \right\Vert_{ }   </math>
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.153)
 |}
@@ Line 1,999: / Line 1,999: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\left[\left(c_1 \frac{\mu }{h^2} \right)^2 + \left(c_2 \frac{\rho \left\Vert \boldsymbol{a} \right\Vert_{\Omega }}{h} \right)^2 \right]^{1/2} \nabla \cdot u_s+ \nabla \cdot \nabla p_s= 0 </math>
+| style="text-align: center;" | <math>\left[\left(c_1 \frac{\mu }{h^2} \right)^2 + \left(c_2 \frac{\rho \left\Vert \boldsymbol{a} \right\Vert_{ }}{h} \right)^2 \right]^{1/2} \nabla \cdot u_s+ \nabla \cdot \nabla p_s= 0 </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.154)
@@ Line 2,036: / Line 2,036: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{1}{\tau _p} p_s\approx \nabla \cdot u_s= {R^c \left(u_h\right)}- \delta _h \qquad \tau _p = \frac{h^2}{c_1 \tau _u} = \mu + \frac{c_2 \left\Vert \boldsymbol{a} \right\Vert_{\Omega } h}{c_1} </math>
+| style="text-align: center;" | <math>\frac{1}{\tau _p} p_s\approx \nabla \cdot u_s= {R^c \left(u_h\right)}- \delta _h \qquad \tau _p = \frac{h^2}{c_1 \tau _u} = \mu + \frac{c_2 \left\Vert \boldsymbol{a} \right\Vert_{ } h}{c_1} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.157)

@@ Line 1,931: / Line 1,931: @@
 ::
 :'''end'''
-:    Set <math display="inline">n = n+1</math>, <math display="inline">t = t + \Delta t</math>    Set <math display="inline">u_h^{n+1,\,0} = u_s^n</math>, <math display="inline">p_h^{n+1,\,0} = p_s^n</math>    '''while''' <math>\left\Vert  \boldsymbol{R}\left(\boldsymbol{U}^{n+1,\,i},\boldsymbol{P}^{n+1,\,i}\right) \right\Vert_{\Omega } \leq \hbox{tolerance}</math> '''do'''
+:    Set <math display="inline">n = n+1</math>, <math display="inline">t = t + \Delta t</math>    Set <math display="inline">u_h^{n+1,\,0} = u_s^n</math>, <math display="inline">p_h^{n+1,\,0} = p_s^n</math>    '''while''' <math>\left\Vert  \boldsymbol{R}\left(\boldsymbol{U}^{n+1,\,i},\boldsymbol{P}^{n+1,\,i}\right) \right\Vert_{ } \leq \hbox{tolerance}</math> '''do'''
 ::
 :'''end'''

@@ Line 926: / Line 926: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\overline{\boldsymbol{u}}\left(\boldsymbol{x},t\right)= \int _{-\Delta /2}^{\Delta /2} \int _{-\Delta /2}^{\Delta /2} \int _{-\Delta /2}^{\Delta /2} G\left(\boldsymbol{x}-\boldsymbol{\chi }\right)u\left(\boldsymbol{\chi },t\right)\, \hbox{d}\Omega  </math>
+| style="text-align: center;" | <math>\overline{\boldsymbol{u}}\left(\boldsymbol{x},t\right)= \int _{-\Delta /2}^{\Delta /2} \int _{-\Delta /2}^{\Delta /2} \int _{-\Delta /2}^{\Delta /2} G\left(\boldsymbol{x}-\boldsymbol{\chi }\right)u\left(\boldsymbol{\chi },t\right)\, \hbox{d}\boldsymbol \chi  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.61)