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@@ Line 2,025: / Line 2,025: @@
 are respectively the velocity vector, the gradient vector and the diffusivity matrix, respectively. For simplicity we have assumed an isotropic diffusion matrix.
-The FIC form of Eq.([[#eq-40|(40)]]) is written
+The FIC form of Eq.([[#eq-40|40]]) is written
 <span id="eq-42"></span>
@@ Line 2,038: / Line 2,038: @@
 |}
-where <math display="inline">r</math> is the original differential equation as defined in Eq.([[#eq-40|(40)]]).
+where <math display="inline">r</math> is the original differential equation as defined in Eq.([[#eq-40|40]]).
 The Dirichlet and boundary conditions of the FIC formulation are
@@ Line 2,066: / Line 2,066: @@
 where <math display="inline">n</math> is the normal vector to the boundary where the normal flux is prescribed. As usual index <math display="inline">p</math> denotes the prescribed values.
-In Eqs.([[#eq-42|(42)]]) and ([[#eq-44|(44)]]) <math display="inline">{\boldsymbol h}=[h_x,h_y]^T</math> is the characteristic vector of the FIC formulation which components play the role of stabilization parameters. The underlined terms in Eqs.([[#eq-42|(42)]]) and ([[#eq-44|(44)]]) introduce the necessary stability in the numerical solution.
+In Eqs.([[#eq-42|42]]) and ([[#eq-44|44]]) <math display="inline">{\boldsymbol h}=[h_x,h_y]^T</math> is the characteristic vector of the FIC formulation which components play the role of stabilization parameters. The underlined terms in Eqs.([[#eq-42|42]]) and ([[#eq-44|44]]) introduce the necessary stability in the numerical solution.
 As shown in <span id='citeF-23'></span>[[#cite-23|[23]]] the components of the characteristic length '''h''' allow to obtain stabilized finite element solutions in the presence of strong gradients of <math display="inline">\phi </math> near the boundaries (boundary layers) and within the analysis domain (internal layers). If vector '''h'''  is taken to be parallel to the velocity '''u''' the standard SUPG method is recovered <span id='citeF-20'></span>[[#cite-20|[20]]]. Keeping the more general form of  '''h''' allows to reproduce the best features of the so called shock capturing or transverse dissipation schemes.
@@ Line 2,146: / Line 2,146: @@
 where <math display="inline">u_\xi </math>, <math display="inline">u_\eta </math> are the velocity components along each of the principal curvature directions and <math display="inline">l_\xi </math> and <math display="inline">l_\eta </math> are the maximum values of the projection of the element sides along the principal directions <math display="inline">\xi </math> and <math display="inline">\eta </math>, respectively.
-The values of <math display="inline"> k^s_\xi </math> and <math display="inline">k^s_\eta </math> in Eq.([[#eq-49|(49)]]) correspond to the ''critical stabilization parameters'' for the absorptive limit, as defined for the 1D problem. A non linear form of these parameters depending on the sign of the solution derivatives along the principal curvature directions can be obtained <span id='citeF-34'></span>[[#cite-34|[23,34]]].
+The values of <math display="inline"> k^s_\xi </math> and <math display="inline">k^s_\eta </math> in Eq.([[#eq-49|49]]) correspond to the ''critical stabilization parameters'' for the absorptive limit, as defined for the 1D problem. A non linear form of these parameters depending on the sign of the solution derivatives along the principal curvature directions can be obtained <span id='citeF-34'></span>[[#cite-34|[23,34]]].
-The derivation of the finite element equations follows the standard Galerkin  procedure applied to Eq.([[#eq-43|(43)]]&#8211;[[#eq-45|(45)]]). For details see <span id='citeF-23'></span>[[#cite-23|[23,34]]].
+The derivation of the finite element equations follows the standard Galerkin  procedure applied to Eq.([[#eq-43|43]]&#8211;[[#eq-45|45]]). For details see <span id='citeF-23'></span>[[#cite-23|[23,34]]].
 The stabilized numerical results presented in the next section have been found by the following two steps algorithm.<br/>
@@ Line 2,158: / Line 2,158: @@
 '''Step 2'''. Compute  the principal curvature directions of the solution at the element center. As mentioned above we have chosen the main principal direction <math display="inline">\xi </math> to be parallel to the gradient vector <math display="inline">{\boldsymbol \nabla }\phi </math> for simplicity. The direction <math display="inline">\eta </math> is chosen orthogonal to <math display="inline">\xi </math>.
-Compute <math display="inline"> \bar {\boldsymbol D}'</math>, <math display="inline">{\boldsymbol T}</math> and <math display="inline">\bar{\boldsymbol D}</math> from Eqs.([[#eq-46|(46)]]&#8211;[[#eq-48|(48)]]). Solve for <math display="inline">{}^2\bar{\boldsymbol \phi }</math>.<br/>
+Compute <math display="inline"> \bar {\boldsymbol D}'</math>, <math display="inline">{\boldsymbol T}</math> and <math display="inline">\bar{\boldsymbol D}</math> from Eqs.([[#eq-46|46]]&#8211;[[#eq-48|48]]). Solve for <math display="inline">{}^2\bar{\boldsymbol \phi }</math>.<br/>
 Excellent results are obtained with this simple two steps algorithm for problems with boundary layers orthogonal to the velocity vector as shown  in the examples presented in the next section. A refined version of the algorithm applicable to problems with internal layers is described in <span id='citeF-34'></span>[[#cite-34|[34]]].

← Older revision		Revision as of 14:19, 17 October 2018
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		+	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 195, pp. 3926–3946, 2006<br />
		+	doi: 10.1016/j.cma.2005.07.020


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	A stabilized finite element method (FEM) for the steady state advection-diffusion-absorption equation is presented. The stabilized formulation is based on the modified governing differential equations derived via the Finite Calculus (FIC) method. The basis of the method is detailed for the 1D problem. It is shown that the stabilization terms act as a non linear additional diffusion governed by a single stabilization parameter. A critical constant value of this parameter ensuring a stabilized solution using two node linear elements is given. More accurate numerical results can be obtained by using a simple expression of the non linear stabilization parameter depending on the signs of the numerical solution and of its derivatives. A straightforward two steps algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. The extension to the multi-dimensional case is briefly described. Numerical results for 1D and 2D problems are presented showing the efficiency and accuracy of the new stabilized formulation.		A stabilized finite element method (FEM) for the steady state advection-diffusion-absorption equation is presented. The stabilized formulation is based on the modified governing differential equations derived via the Finite Calculus (FIC) method. The basis of the method is detailed for the 1D problem. It is shown that the stabilization terms act as a non linear additional diffusion governed by a single stabilization parameter. A critical constant value of this parameter ensuring a stabilized solution using two node linear elements is given. More accurate numerical results can be obtained by using a simple expression of the non linear stabilization parameter depending on the signs of the numerical solution and of its derivatives. A straightforward two steps algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. The extension to the multi-dimensional case is briefly described. Numerical results for 1D and 2D problems are presented showing the efficiency and accuracy of the new stabilized formulation.
−

	==1 INTRODUCTION==		==1 INTRODUCTION==

@@ Line 1,988: / Line 1,988: @@
 <li>The results obtained with <math display="inline">\beta ^e</math>  are less diffusive than those obtained with <math display="inline">\beta _c</math> and quite accurate for all ranges of <math display="inline">\gamma </math> and <math display="inline">w</math>.  </li>
-<li>The FIC algorithm provides a stabilized solution for Dirichlet problems when there is a negative streamwise gradient of the solution. This is an advantage versus SUPG, GLS and SGS methods using a single stabilization parameter which fail for these type of problems <span id='citeF-12'></span>[[#cite-12|[[12,13]]]].  </li>
+<li>The FIC algorithm provides a stabilized solution for Dirichlet problems when there is a negative streamwise gradient of the solution. This is an advantage versus SUPG, GLS and SGS methods using a single stabilization parameter which fail for these type of problems <span id='citeF-12'></span>[[#cite-12|[12,13]]].  </li>
 </ol>

@@ Line 2,171: / Line 2,171: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_267560997_1174_fig9.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [2,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=1 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
+|[[File:Draft_Samper_267560997_1174_fig9.png|600px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [2,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=1 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 9:''' 2D advection-conduction-absorption problem over a square domain of size equal to 8 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=8</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=8</math>. <math>{\boldsymbol u} = [2,0]^T</math>, <math>k=1</math>, <math>s=20</math>, <math>w=20</math>, <math>\gamma _x=1</math> and <math>\gamma _y=0</math>. Galerkin and FIC solutions obtained with a mesh of <math>8 \times 8</math> four node square elements.
@@ Line 2,183: / Line 2,183: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_267560997_7137_Fig10.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [20,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=10 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
+|[[File:Draft_Samper_267560997_7137_Fig10.png|600px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [20,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=10 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 10:''' 2D advection-conduction-absorption problem over a square domain of size equal to 8 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=8</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=8</math>. <math>{\boldsymbol u} = [20,0]^T</math>, <math>k=1</math>, <math>s=20</math>, <math>w=20</math>, <math>\gamma _x=10</math> and <math>\gamma _y=0</math>. Galerkin and FIC solutions obtained with a mesh of <math>8 \times 8</math> four node square elements.
@@ Line 2,197: / Line 2,197: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_267560997_7278_Fig11.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,1×10⁻³]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>= 5×10⁻⁴. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
+|[[File:Draft_Samper_267560997_7278_Fig11.png|600px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,1×10⁻³]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>= 5×10⁻⁴. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 11:''' 2D advection-conduction-absorption problem over a square domain of size equal to 10 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=10</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=10</math>. <math>{\boldsymbol u} = [0,1\times 10^{-3}]^T</math>, <math>k=1</math>, <math>s=12</math>, <math>w=12</math>, <math>\gamma _x=0</math> and <math>\gamma _y= 5\times 10^{-4}</math>. Galerkin and FIC solutions obtained with a mesh of <math>10 \times 10</math> four node square elements.
@@ Line 2,211: / Line 2,211: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_267560997_4095_Fig12.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,5]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>=2.5. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
+|[[File:Draft_Samper_267560997_4095_Fig12.png|600px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,5]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>=2.5. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 12:''' 2D advection-conduction-absorption problem over a square domain of size equal to 10 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=10</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=10</math>. <math>{\boldsymbol u} = [0,5]^T</math>, <math>k=1</math>, <math>s=12</math>, <math>w=12</math>, <math>\gamma _x=0</math> and <math>\gamma _y=2.5</math>. Galerkin and FIC solutions obtained with a mesh of <math>10 \times 10</math> four node square elements.

@@ Line 2,171: / Line 2,171: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[File:Draft_Samper_267560997_1174_fig9.png|400px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [2,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=1 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
+|[[File:Draft_Samper_267560997_1174_fig9.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [2,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=1 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 9:''' 2D advection-conduction-absorption problem over a square domain of size equal to 8 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=8</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=8</math>. <math>{\boldsymbol u} = [2,0]^T</math>, <math>k=1</math>, <math>s=20</math>, <math>w=20</math>, <math>\gamma _x=1</math> and <math>\gamma _y=0</math>. Galerkin and FIC solutions obtained with a mesh of <math>8 \times 8</math> four node square elements.
@@ Line 2,183: / Line 2,183: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_267560997-test-Figure_10.png|400px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [20,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=10 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
+|[[File:Draft_Samper_267560997_7137_Fig10.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 8 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=8, qₙ=0 at y=0 and y=8. u = [20,0]<sup>T</sup>, k=1, s=20, w=20, γₓ=10 and γ<sub>y</sub>=0. Galerkin and FIC solutions obtained with a mesh of 8 ×8 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 10:''' 2D advection-conduction-absorption problem over a square domain of size equal to 8 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=8</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=8</math>. <math>{\boldsymbol u} = [20,0]^T</math>, <math>k=1</math>, <math>s=20</math>, <math>w=20</math>, <math>\gamma _x=10</math> and <math>\gamma _y=0</math>. Galerkin and FIC solutions obtained with a mesh of <math>8 \times 8</math> four node square elements.
@@ Line 2,197: / Line 2,197: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_267560997-test-Figure_11.png|400px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,1×10⁻³]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>= 5×10⁻⁴. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
+|[[File:Draft_Samper_267560997_7278_Fig11.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,1×10⁻³]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>= 5×10⁻⁴. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 11:''' 2D advection-conduction-absorption problem over a square domain of size equal to 10 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=10</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=10</math>. <math>{\boldsymbol u} = [0,1\times 10^{-3}]^T</math>, <math>k=1</math>, <math>s=12</math>, <math>w=12</math>, <math>\gamma _x=0</math> and <math>\gamma _y= 5\times 10^{-4}</math>. Galerkin and FIC solutions obtained with a mesh of <math>10 \times 10</math> four node square elements.
@@ Line 2,211: / Line 2,211: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_267560997-test-Figure12.png|400px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,5]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>=2.5. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
+|[[File:Draft_Samper_267560997_4095_Fig12.png|500px|2D advection-conduction-absorption problem over a square domain of size equal to 10 units. ϕ<sup>p</sup>=8 at x=0, ϕ<sup>p</sup>=3 at x=10, qₙ=0 at y=0 and y=10. u = [0,5]<sup>T</sup>, k=1, s=12, w=12, γₓ=0 and γ<sub>y</sub>=2.5. Galerkin and FIC solutions obtained with a mesh of 10 ×10 four node square elements.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 12:''' 2D advection-conduction-absorption problem over a square domain of size equal to 10 units. <math>\phi ^p =8</math> at <math>x=0</math>, <math>\phi ^p =3</math> at <math>x=10</math>, <math>q_n =0</math> at <math>y=0</math> and <math>y=10</math>. <math>{\boldsymbol u} = [0,5]^T</math>, <math>k=1</math>, <math>s=12</math>, <math>w=12</math>, <math>\gamma _x=0</math> and <math>\gamma _y=2.5</math>. Galerkin and FIC solutions obtained with a mesh of <math>10 \times 10</math> four node square elements.

← Older revision	Revision as of 10:15, 12 November 2018
(No difference)

@@ Line 468: / Line 468: @@
 step
-'''Step 1.'''
+'''Step 1.''' Compute a first stabilized solution <math display="inline">\bar{\boldsymbol \phi }^1</math> using the critical value <math display="inline">\beta ^e = \beta _c</math> given by Eq.(28). This ensures a stabilized, although sometimes slightly overdiffusive solution.
-* Compute a first stabilized solution <math display="inline">\bar{\boldsymbol \phi }^1</math> using the critical value <math display="inline">\beta ^e = \beta _c</math> given by Eq.(28). This ensures a stabilized, although sometimes slightly overdiffusive solution.
-−
-−
 '''Step 2.'''
-* ''Step 2.1''. Compute the signs of the first and second derivatives of <math display="inline">\bar{\boldsymbol \phi }^1</math> within each element. The second derivative field is obtained as follows
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 490: / Line 488: @@
 where <math display="inline">({\hat \cdot })_i^e</math> denotes averaged values of the first derivative at node <math display="inline">i</math> of element <math display="inline">e</math>. At the boundary nodes the constant value of the derivative of <math display="inline">\bar \phi </math> within the element is taken in Eq.(30); i.e. <math display="inline">(\hat{\cdot })_i^e = \left({d\bar \phi \over dx}\right)^{(e)} = {\bar \phi _2 - \bar \phi _1 \over l^e}</math>.
-* ''Step 2.2''. Compute the enhanced stabilized solution <math display="inline">{\boldsymbol \phi }^2</math> using the element value of <math display="inline">\beta ^e</math> given by Eqs.(25) and (26).
+''Step 2.2''. Compute the enhanced stabilized solution <math display="inline">{\boldsymbol \phi }^2</math> using the element value of <math display="inline">\beta ^e</math> given by Eqs.(25) and (26).
 In all the examples solved the above two steps have sufficied to obtain a converged stabilized solution. The reason of this is that  the signs of the first and second derivative fields typically do not change any further after the second step over the relevant regions of the analysis domain (i.e. over regions adjacent to high gradient zones). We note that the stabilized solution is also remarkably accurate as shown in the examples presented in a next section.