Stabilized solution of the multidimensional advection-diffusion-absorption equation using linear finite elements

==Abstract==

A stabilized finite element method (FEM) for the multidimensional 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. For 1D problems the stabilization terms act as a nonlinear additional diffusion governed by a single stabilization parameter. It is shown that for multidimensional problems an orthotropic stabilizing diffusion must be added along the principal directions of  curvature of the solution. A simple iterative algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described. Numerical results for 1D and 2D problems with sharp gradients are presented showing the effectiveness and accuracy of the new stabilized formulation.

==1 INTRODUCTION==

Considerable effort has been spent in recent years to derive finite element methods (FEM) <span id='citeF-1'></span>[[#cite-1|1]] for the solution of the advection-diffusion-reaction equation. In this work we will focus on the so called ''exponential regime'' originated by large absorptive (dissipative) reaction terms. Here the solutions are of the form of real exponential functions.  Numerical schemes  find difficulties to approximating the sharp gradients appearing in the neighborhood of boundary and internal layers due to high Peclet and/or Damköhler numbers. Non physical oscilaltory solution are found with the standard Galerkin FEM unless some stabilization procedure is used.

Stabilized methods to tackle this problem have been based on  streamline-upwind/Petrov-Galerkin (SUPG)  <span id='citeF-2'></span>[[#cite-2|2]], Galerkin/least-squares <span id='citeF-5'></span>[[#cite-5|5]], Subgrid Scale (SGS) <span id='citeF-5'></span>[[#cite-5|5]] and Residual Free Bubbles <span id='citeF-14'></span>[[#cite-14|14]] finite element methods. While a single stabilization parameter suffices to yield stabilized (and even nodally exact results) for the one-dimensional (1D) advection-diffusion and the diffusion-reaction equations (Vol. 3 in <span id='citeF-1'></span>[[#cite-1|1]] and <span id='citeF-8'></span>[[#cite-8|8]]), this is not the case for the diffusion-advection-reaction equation. Here, in general,  ''two stabilization parameters'' are needed in order to ensure a stabilized solution for all range of physical parameters and boundary conditions <span id='citeF-4'></span>[[#cite-4|4]]. As reported in <span id='citeF-12'></span>[[#cite-12|12]] the SUPG, GLS and SGS methods with a single stabilization parameter fail to obtain a stabilized solution for some specific boundary conditions in the exponential regime with negative (absorption) terms when there is a negative streamwise gradient of the solution.

Oñate ''et al.'' [18] have recently presented a stabilized FEM for the advection-diffusion absorption equation based on the use of a single stabilization parameter which has the form of a diffusion term. In [18] the formulation is detailed for 1D problems and only a brief introduction to the multidimensional case is given. This paper extends the ideas presented in [18] and provides evidence of the effectiveness and accuracy of the new formulation to deal with multidimensional advection-diffusion-absorption  problems  with sharp gradients.

The stabilized formulation is based on the standard Galerkin FEM solution of the modified governing differential equations derived via the ''Finite Calculus'' (FIC) method [19&#8211;20]. The FIC equations are obtained by expressing the balance of fluxes in a domain of finite size. This introduces additional stabilizing terms in the differential equations of the infinitessimal theory which are a function of the balance domain dimensions. Although the FIC&#8211;FEM formulation here presented is general, we will restrict its application in this work to linear finite element approximations only.

The Galerkin FIC-FEM formulation  described here introduces naturally an additional nonlinear dissipation term into the discretized equations which is governed by a ''single stabilization parameter''. In the absence of the absorption term the formulation simplifies to the standard Petrov-Galerkin approach for the advection-diffusion problem For the diffusion-absorption case the diffusion-type stabilization term is identical to that recently obtained by Felippa and Oñate using a variational FIC approach  <span id='citeF-15'></span>[[#cite-15|15]]. The general nonlinear form of the stabilization parameter  is a function of the signs of the solution and   its first and second derivatives. This introduces a non-linearity in the solution scheme and a simple iterative algorithm  is described. A simpler constant expression of the stabilization parameter is also presented.

Details of the 1D formulation and its extension to deal with multidimensional problems are given. For the multidimensional case Oñate ''et al.'' <span id='citeF-27'></span>[[#cite-27|27]] have recently shown that a general form of the stabilization parameters can be found by writting the FIC equations along the principal curvature directions of the solution. The resulting FIC-FEM formulation is equivalent in this case (for linear elements) to adding a stabilizing diffusion matrix to the standard infinitessimal equation. The stabilizing diffusion matrix depends on the signs of the solution and its derivatives and on the velocities along the principal curvature directions of the solution. This introduces a nonlinearity in the solution process. We present a simple iterative scheme based in assuming that  the main principal curvature direction at each  point is coincident with the gradient vector direction. In the last part of the paper we present a collection of 1D and 2D examples showing the effectiveness and accuracy of the new FIC-FEM formulation for different values of the advective and absorptive terms.

==2 FIC FORMULATION OF THE 1D STATIONARY ADVECTION-DIFFUSION-ABSORPTION EQUATION==

The governing equation for the 1D stationary advection-diffusion-absorption problem can be written in the FIC formulation as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>r - \underline{{h\over 2} {dr\over dx}}{h\over 2} {dr\over dx}=0\quad \hbox{in } x \in (0,L) </math>
|}
| style="width: 5px;text-align: right;" | (1)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>-u\phi + k {d\phi \over dx} +q^p - \underline{{h\over 2} r}{h\over 2} r=0\quad \hbox{on }\Gamma _q  </math>
|}
| style="width: 5px;text-align: right;" | (2)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\phi -\phi ^p =0 \quad \hbox{on }\Gamma _\phi  </math>
|}
| style="width: 5px;text-align: right;" | (3)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>r =:-u {d\phi \over dx} +{d\over dx} \left(k{d\phi \over dx}\right)- s\phi + Q  </math>
|}
| style="width: 5px;text-align: right;" | (4)
|}

In above equations <math display="inline">\phi </math> is the state variable, <math display="inline">x \in [0,L]</math> is the problem domain, <math display="inline">L</math> is the domain length, <math display="inline">u</math> is the velocity field, <math display="inline">k\ge 0</math> is the diffusion, <math display="inline">s\ge 0</math> is the absorption, dissipation or destruction source parameter, <math display="inline">Q</math> is a constant source term, <math display="inline">q^p</math> and <math display="inline">\phi ^p</math> are the prescribed values of the total flux and the unknown function at the Neumann and Dirichlet boundaries <math display="inline">\Gamma _q</math> and <math display="inline">\Gamma _\phi </math>, respectively and <math display="inline">h</math> is a ''characteristic length'' which plays the role of a stabilization parameter. In the 1D problem <math display="inline">\Gamma _\phi </math> and <math display="inline">\Gamma _q</math> consist of four combinations at the end points of the problem domain.

Eqs.(1) and (2) are obtained by expressing the balance of fluxes in an arbitrary 1D domain of finite size within the problem domain and at the Neumann boundary, respectively. The variations of the transported variables within the balance domain are approximated by Taylor series expansions retaining one order higher terms than in the infinitessimal theory [19,20]. The underlined stabilizing terms in Eqs.(1) and (2) emanate from these higher order expansions. Note that as the characteristic length parameter <math display="inline">h</math> tends to zero the FIC differential equations gradually recover the standard infinitessimal form.

Successful applications of the FIC method to a variety of problems in computational mechanics can be found in [19&#8211;30,37].

==3 FINITE ELEMENT FORMULATION==

We will construct a standard finite element discretization <math display="inline">\left\{l^e\right\}</math> of the 1D analysis domain length <math display="inline">L</math> with index <math display="inline">e</math> ranging from 1 to the number of elements <math display="inline">N</math> <span id='citeF-1'></span>[[#cite-1|1]]. The state variable <math display="inline">\phi </math> is approximated by <math display="inline">\bar \phi </math> over the analysis domain. The approximated variable <math display="inline">\bar \phi </math> is interpolated within each element with <math display="inline">n</math> nodes in the standard manner, i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\phi \simeq \bar \phi \quad \hbox{for}\quad x \in [0,L]</math>
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| style="width: 5px;text-align: right;" |  (5a)
|}

with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\bar \phi =\sum \limits _{i=1}^n N_i \phi _i</math>
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| style="width: 5px;text-align: right;" |  (5b)
|}

where <math display="inline">N_i</math> are the element shape functions and <math display="inline">\phi _i</math> are nodal values of the approximate function <math display="inline">\bar \phi </math>. Substituting Eq.(5a) into Eqs.(1) and (2) gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\bar r - {h\over 2} {d\bar r\over dx} =r_\Omega \quad \hbox{in } x\in (0,L) </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>-u\bar \phi + k {d\bar \phi \over dx} +q^p - {h\over 2} \bar r=r_q\quad \hbox{on }\Gamma _q  </math>
|}
| style="width: 5px;text-align: right;" | (6)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\bar \phi -\phi ^p =r_\phi \quad \hbox{on }\Gamma _\phi  </math>
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| style="width: 5px;text-align: right;" | (7)
|}

where <math display="inline">\bar r =r(\bar \phi )</math> and <math display="inline">r_\Omega , r_q</math> and <math display="inline">r_\phi </math> are the residuals of the approximate solution in the problem domain and on the Neumann and Dirichlet boundaries <math display="inline">\Gamma _q</math> and <math display="inline">\Gamma _\phi </math>, respectively.

The weighted residual form of Eqs.(6)&#8211;(8) is written as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\int _L W_i \left(\bar r - {h\over 2} {d\bar r\over dx}\right)dx + \left[\hat W_i \left(-u \bar \phi + k {d\bar \phi \over dx} +q_p - {h\over 2} \bar r \right)\right]_{\Gamma _q} =0 </math>
|}
| style="width: 5px;text-align: right;" | (8)
|}

where <math display="inline">W_i(x)</math> and <math display="inline">\hat W_i</math> are test functions satisfying <math display="inline">W_i =\hat W_i =0</math> on <math display="inline">\Gamma _\phi </math>.

Assuming smooth enough solutions and integrating by parts the term involving <math display="inline">h</math> in the first integral gives for <math display="inline">\hat W_i=-W_i</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\int _L W_i \bar r d x - \left[W_i \left(-u \bar \phi + k {d\bar \phi \over dx} +q^p \right)\right]_{\Gamma _q} +\sum \limits _e \int _{l^e} {h\over 2} {dW_i\over dx} \bar r dx =0</math>
|}
| style="width: 5px;text-align: right;" | (9)
|}

The third term in Eq.(10) is computed as the sum of the integrals over the element interiors, therefore allowing for the space derivatives of <math display="inline">\bar r</math> to be discontinuous. Also in  Eq.(10) <math display="inline">h</math> has been assumed to be constant within each element, (i.e. <math display="inline">\displaystyle{dh\over dx}=0</math> within <math display="inline">l^e</math>).

The  weak form is obtained by integrating by parts the advective and diffusive terms within <math display="inline">\bar r</math> in the first integral of Eq.(10). This gives

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\int _L \left[u {dW_i\over dx} \bar \phi - {dW_i\over dx}k {d\bar \phi \over dx}- W_i s \bar \phi + W_i Q\right]dx -  [W_i q^p]_{\Gamma _q} -  \sum \limits _e \int _{l^e} \left(\beta k {dW_i\over dx} {d\bar \phi \over dx} -{h\over 2} {dW_i\over dx}Q\right)dx =0 </math>
|}
| style="width: 5px;text-align: right;" | (10)
|}

with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\beta = \left[{s\bar \phi \over 2k\bar \phi '}+{u\over 2k}-{\bar \phi ''\over 2\bar \phi '} \right]h </math>
|}
| style="width: 5px;text-align: right;" | (11)
|}

where a prime denotes the derivative with respect to the space coordinate.

Wee see clearly that the last term of Eq.(11) introduces within each element an additional diffusion of value <math display="inline">\beta k</math>.

Substituting expression (5b) into (11) and choosing a Galerkin method with <math display="inline">W_i =N_i</math> within each element gives the discrete system of FE equations written in the standard matrix form as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{K}\bar{\boldsymbol \phi } ={f} </math>
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| style="width: 5px;text-align: right;" | (12)
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where <math display="inline">\bar{\boldsymbol \phi }</math> is the vector of nodal unknowns and the element contributions to matrix '''K''' and vector <math display="inline">f</math> are

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: right;" | <math>K_{ij}^e\!\! </math>
| style="text-align: center;" | <math>=</math>
| <math>\!\! \int _{l^e} \left(-u {dN_i\over dx} N_j + k(1+\beta ) {dN_i\over dx}{dN_j\over dx}+ sN_i N_j\right)dx</math>
| style="width: 5px;text-align: right;" | (13)
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| style="text-align: right;" | <math> f_i^e\!\! </math>
| style="text-align: center;" | <math>=</math>
| <math> \!\! \int _{l^e} \left(N_i + {h\over 2} {dN_i\over dx} \right)Qdx - (N_i q^p)_{\Gamma _q} </math>
| style="width: 5px;text-align: right;" | (14)
|}
|}

The amount of balancing diffusion in Eq.(14) clearly depends on the (nonlinear) stabilization parameter <math display="inline">\beta </math>. The element and critical  values of <math display="inline">\beta </math> are deduced in the next section for linear two node elements.

We note that the integral of the term <math display="inline">\displaystyle{h\over 2} \displaystyle{dN_i\over dx}Q</math> in Eq.(15) vanishes after asssembly when <math display="inline">h </math> and <math display="inline">Q</math> are uniform over a patch of linear elements.

==4 COMPUTATION OF THE STABILIZATION PARAMETER FOR LINEAR ELEMENTS==

The matrix <math display="inline">{K}^e</math> and the vector <math display="inline">{f}^e</math> for two node linear elements are (for constant values of <math display="inline">u,k</math>, <math display="inline">s</math> and <math display="inline">Q</math>)

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{K}^e = - {u\over 2} \left[\begin{matrix}-1 & -1\\ 1 & 1\\\end{matrix}\right]+ {k\over l^e} (1+\beta ^e)  \left[\begin{matrix}1 & -1\\ - 1 & 1\\\end{matrix}\right]+ {sl^e \over 6} \left[\begin{matrix}2 & 1\\ 1 & 2\\\end{matrix}\right]</math>
|}
| style="width: 5px;text-align: right;" |  (16a)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>{f}^e = {Ql^e\over 2}\left\{\begin{matrix}1 - \displaystyle{h^e\over 2}\\ 1+ \displaystyle{h^e\over 2}\\\end{matrix}\right\}\qquad + \quad \hbox{boundary term}</math>
|}
| style="width: 5px;text-align: right;" |  (16a)
|}

In Eqs.(16) index <math display="inline">e</math>denotes element values.

Assuming <math display="inline">Q=0</math>, a typical stencil for elements of equal size <math display="inline">l</math> can be written as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\begin{array}{r}-\gamma (\bar \phi _{i+1} -\bar \phi _{i-1})-(1+\beta ) \bar \phi _{i-1}+2(1+\beta ) \bar \phi _i - (1+\beta ) \bar \phi _{i+1}+\\ + \displaystyle{w\over 6} (\bar \phi _{i-1}+ 4 \bar \phi _i + \bar \phi _{i+1})=0 \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (17)
|}

where for simplicity a constant value of <math display="inline">\beta </math> across the mesh has been assumed. In Eq.(17) <math display="inline">\gamma ={ul\over 2k}</math> and <math display="inline">w={sl^2\over 4}</math> are the Peclet number and a velocity independent dimensionless number, respectively.

From Eq.(17) we deduce

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\beta = \gamma \left({\bar \phi _{i+1} -\bar \phi _{i-1}\over \bar \phi _{i+1}- 2\bar \phi _i+  \bar \phi _{i-1} }\right)+ {w\over 6} \left({\bar \phi _{i-1} + 4 \bar \phi _i+\bar \phi _{i+1}\over \bar \phi _{i+1}- 2\bar \phi _i+ \bar \phi _{i+1} }\right)-1 </math>
|}
| style="width: 5px;text-align: right;" | (18)
|}

In the vecinity of a sharp gradient zone we can take

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\begin{array}{l}\bar \phi _{i+1} -\bar \phi _{i-1}\simeq \bar \phi _{max} S_1\\ \bar \phi _{i+1} -2\bar \phi _i+  \bar \phi _{i-1}=\bar \phi _{max} S_2\\ \bar \phi _i+4 \bar \phi _i + \bar \phi _{i+1}= \bar \phi _{i+1}S_0 \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (19)
|}

where <math display="inline">\bar \phi _{max}</math> is the maximum value of the approximate function <math display="inline">\bar \phi </math> in the patch of elements adjacent to the sharp gradient zone and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>S_0= \hbox{sign } (\bar \phi ) ,\quad S_1 = \hbox{sign } \left({d\bar \phi \over dx}\right),\quad S_2 = \hbox{sign } \left({d^2 \bar \phi \over dx^2}\right) </math>
|}
| style="width: 5px;text-align: right;" | (20)
|}

where sign <math display="inline">\bar{(\cdot )}</math> denotes the sign of the magnitude within the brackets computed at the  patch mid point.

Substituting Eq.(19) into (18) leads to the following expression of the stabilization parameter

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\beta = \left[\left({S_0\over S_2}\right){w\over 6} +\left({S_1\over S_2}\right)  \gamma -1\right] </math>
|}
| style="width: 5px;text-align: right;" | (21)
|}

The ''element stabilization parameter'' <math display="inline">\beta ^e</math> is now defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\begin{array}{l}\beta ^e =\beta \quad \hbox{for } \beta >0\\ \beta ^e =0 \quad \hbox{for } \beta \le 0 \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (22)
|}

where <math display="inline">\beta </math> is given by Eq.(21) and the signs <math display="inline">S_0</math>, <math display="inline">S_1</math> and <math display="inline">S_2</math> are computed now at the element mid-point.

It is clear from above that the computation of the stabilization parameter <math display="inline">\beta ^e</math> requires the knowledge of the sign of the numerical solution <math display="inline">\bar \phi </math> and that of the  first and second derivatives of <math display="inline">\bar \phi </math> within each element. This necessarily leads to an iterative scheme. A simple algorithm which provides a stabilized and accurate solution in just two steps is presented below.

===4.1 Critical stabilization parameter and unstability conditions===

The following constant value of <math display="inline">\beta </math> over the mesh ensures a stabilized solution for all ranges of <math display="inline">\gamma </math> and <math display="inline">w</math>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\displaystyle{\beta \le \beta _c = {w\over 6} +\vert \gamma \vert -1} </math>
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| style="width: 5px;text-align: right;" | (23)
|}

where <math display="inline">\beta _c </math> is the ''critical stabilization parameter''. Note that <math display="inline">\beta _c</math> corresponds to the maximum value of <math display="inline">\beta </math> in Eq.(21) for <math display="inline">{S_0\over S_2}={S_1\over S_2}=1</math>. A mathematical proof of Eq.(23) is given in [18].

Clearly the value of <math display="inline">\beta _c</math> of Eq.(23) is meaningful only if <math display="inline">\beta _c >0</math> and this can be taken as an indicator of an unstable solution. Conversely, a value of <math display="inline">\beta _c \le 0</math> indicates that no stabilization is needed.

===4.2 Iterative solution scheme===

The following two steps iterative scheme is proposed in order to obtain a stabilized and accurate solution.

step step

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.(23). This ensures a stabilized, although sometimes slightly overdiffusive, solution.

==~&nbsp;==

step

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;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>\left({d^2 \bar \phi ^1 \over dx^2}\right)^e= {1\over l^e} \left[\left({d\hat \phi ^1 \over dx}\right)^e_2 - \left({d\hat \phi ^1 \over dx}\right)^e_1\right] </math>
|}
| style="width: 5px;text-align: right;" | (24)
|}

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.(24); 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>.

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 Eq.(22).

In all the 1D examples solved the above two steps have sufficed to obtain a converged stabilized and accurate solution [18]. 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 elements adjacent to high gradient zones.

==5 EXTENSION TO MULTI-DIMENSIONAL PROBLEMS==

Consider the general steady-state advection-diffusion-reaction equation written for the zero constant source case (<math display="inline">Q=0</math>) as

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
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| style="text-align: center;" | <math>r(\phi ): =- {u}^T {\boldsymbol \nabla } \phi + {\boldsymbol \nabla }^T {D}{\boldsymbol \nabla }\phi - s\phi =0 </math>
|}
| style="width: 5px;text-align: right;" | (25)
|}

For 2D problems

<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{u}=[u,v]^T\quad ,\quad {\boldsymbol \nabla }=\left[{\partial  \over \partial x},{\partial  \over \partial y}\right]^T\quad ,\quad {D} =k \left[\begin{matrix}1 &0\\ 0&1\\\end{matrix}\right] </math>
|}
| style="width: 5px;text-align: right;" | (26)
|}

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-25|25]]) is written as

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>r - \underline{{1\over 2} {h}^T {\boldsymbol \nabla }r}{1\over 2} {h}^T {\boldsymbol \nabla }r=0 </math>
|}
| style="width: 5px;text-align: right;" | (27)
|}

where <math display="inline">r</math> is the original infinitessimal differential equation as defined in Eq.([[#eq-25|25]]).

The Dirichlet and boundary conditions of the FIC formulation are

<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\phi - \phi ^p =0 \quad \hbox{on}\quad \Gamma _\phi   </math>
|}
| style="width: 5px;text-align: right;" | (28)
|}

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>- {u}^T {n}\phi + {n}^T {D} {\boldsymbol \nabla } \phi + q^p - \underline{{1\over 2} {h}^T {n}r}{1\over 2} {h}^T {n}r=0 \quad \hbox{on}\quad \Gamma _q </math>
|}
| style="width: 5px;text-align: right;" | (29)
|}

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-27|27]]) and ([[#eq-29|29]]) <math display="inline">{h}=[h_x,h_y]^T</math> is the characteristic vector of the 2D FIC formulation which components play the role of stabilization parameters. The underlined terms in Eqs.([[#eq-27|27]]) and ([[#eq-29|29]]) introduce the necessary stability in the numerical solution [19,20,21].

If vector '''h'''  is taken to be parallel to the velocity '''u''' the standard SUPG method is recovered [18&#8211;23]. The more general form of '''h''' allows 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) <span id='citeF-27'></span>[[#cite-27|27]].  The FIC formulation therefore reproduces the best features of the so called shock-capturing or transverse-dissipation schemes <span id='citeF-2'></span>[[#cite-2|2]].

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_447243531_2415_Fig1.png|600px|]]
|- style="text-align: center; font-size: 75%;padding:10px;"
| colspan="1" | '''Figure 1'''. Global axes (<math>x,y</math>) and principal curvature axes (<math>\xi ,\eta </math>)
|}

Let us write down the FIC balance equation  in the principal curvature axes of the solution <math display="inline">\xi ,\eta </math> (Figure 1). The FIC balance equation is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: right;" | 
| style="text-align: center;" | 
| <math>-u_\xi {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+k \left({\partial ^2\phi \over \partial \xi ^2}+ {\partial ^2\phi \over \partial \eta ^2}\right)-s\phi - {h_\xi \over 2} {\partial  \over \partial \xi }\left[-u_\xi {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+ k \left({\partial ^2\phi \over \partial \xi ^2}+ {\partial ^2\phi \over \partial \eta ^2}\right)-s\phi \right]</math>
|-
| style="text-align: right;" | 
| style="text-align: center;" | 
| <math> - {h_\eta \over 2} {\partial  \over \partial \eta } \left[-u_\xi {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+ k \left({\partial ^2\phi \over \partial \xi ^2}+ {\partial ^2\phi \over \partial \eta ^2}\right)-s\phi \right]=0 </math>
|}
| style="width: 5px;text-align: right;" | (30)
|}

where <math display="inline">u_\xi , u_\eta </math> are the velocities along the principal axes of curvature <math display="inline">\xi </math> and <math display="inline">\eta </math>, respectively.

As <math display="inline">\xi </math> and <math display="inline">\eta </math> are the principal curvature axes of the solution then

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\partial ^2\phi \over \partial \xi \partial \eta }= {\partial ^2\phi \over \partial \eta \partial \xi }=0 </math>
|}
| style="width: 5px;text-align: right;" | (31)
|}

Introducing this simplification into Eq.(30) we can rewrite this equation as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{l} -u_\xi \displaystyle {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+\left(k + {u_\xi h_\xi \over 2}+ {sh_\xi \over 2} {\partial \phi \over \partial \xi } \left({\partial ^2\phi \over \partial \xi ^2} \right)^{-1}  \right){\partial ^2\phi \over \partial \xi ^2} +\\ \displaystyle + \left(k + {u_\eta h_\eta \over 2}+ {sh_\eta \over 2} {\partial \phi \over \partial \eta } \left({\partial ^2\phi \over \partial \eta ^2} \right)^{-1}\right){\partial ^2\phi \over \partial \eta ^2} - s\phi - k \left({h_\xi \over 2}{\partial ^3 \phi \over \partial \xi ^3}+  {h_\eta \over 2} {\partial ^3 \phi \over \partial \eta ^3}\right)=0 \end{array}</math>
|}
| style="width: 5px;text-align: right;" |  (32a)
|}

or

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math> -u_\xi {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+ k(1+\beta _\xi ) {\partial ^2\phi \over \partial \xi ^2} + k (1+\beta _\eta ) {\partial ^2\phi \over \partial \eta ^2} - s\phi -k \left({h_\xi \over 2}{\partial ^3 \phi \over \partial \xi ^3}+  {h_\eta \over 2} {\partial ^3 \phi \over \partial \eta ^3}\right)=0</math>
|}
| style="width: 5px;text-align: right;" |  (32b)
|}

We can see clearly from Eq.(33) that the FIC governing equations introduce orthotropic diffusion parameters  of values <math display="inline">{\beta _\xi k }</math> and <math display="inline"> { \beta _\eta k}</math> along the <math display="inline">\xi </math> and <math display="inline">\eta </math> axes, respectively with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\beta _\xi = {u_\xi h_\xi \over 2k} + {sh_\xi \over 2k} {\partial \phi  \over \partial \xi } \left({\partial ^2\phi \over \partial \xi ^2} \right)^{-1},\quad \beta _\eta ={u_\xi h_\xi \over 2k} + {sh_\eta \over 2k} {\partial \phi  \over \partial \eta } \left({\partial ^2\phi \over \partial \eta ^2} \right)^{-1} </math>
|}
| style="width: 5px;text-align: right;" |  (33)
|}

Also note that the last term of Eq.(32b) will vanish after discretization for linear elements.

Eq.(32b) can be rewritten in matrix form (neglecting the last term) as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>- {u}^{\prime T} {\boldsymbol \nabla }^\prime \phi + {\boldsymbol \nabla }^{\prime T} ({D}+\bar {D}^\prime ) {\boldsymbol \nabla }^{\prime }\phi - s\phi =0 </math>
|}
|}

where <math display="inline">{u}^\prime =[u_\xi , u_\eta ]^T</math>, <math display="inline">{\boldsymbol \nabla }^\prime = \left[{\partial \over \partial \xi }, {\partial \over \partial \eta }\right]^T</math>, <math display="inline">D</math> is the “physical” isotropic diffusion matrix and <math display="inline">\bar {D}'</math> is the balancing diffusion matrix in the local axes <math display="inline">\xi </math> and <math display="inline">\eta </math>. The form of this matrix is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar {D}^\prime = \left[\begin{array}{cc}\beta _\xi k& 0\\ 0 &\beta _\eta k \end{array}\right] </math>
|}
| style="width: 5px;text-align: right;" | (34)
|}

The velocities  along the principal curvature axes <math display="inline">u_\xi </math> and <math display="inline">u_\eta </math> can be obtained by projecting the cartesian velocities into the principal curvature axes <math display="inline">\xi </math> and <math display="inline">\eta </math> as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{u}'=\left\{\begin{array}{c}u_\xi \\ u_\eta \end{array}\right\}= {T} {u}  \quad \hbox{with}\quad {T}= \left[\begin{array}{cc}c_\alpha & s_\alpha \\ - s_\alpha & c_\alpha \end{array}\right]\quad ,\quad {u}=\left\{\begin{array}{c}u\\v\end{array}\right\} </math>
|}
| style="width: 5px;text-align: right;" | (35)
|}

where <math display="inline">c_\alpha =\cos \alpha </math>, <math display="inline">s_\alpha =\sin \alpha </math> and <math display="inline">\alpha </math> is the angle which the <math display="inline">\xi </math> axis forms with the <math display="inline">x</math> axis (Figure 1). Note that as the solution is continuous the principal curvature directions <math display="inline">\xi </math> and <math display="inline">\eta </math> are orthogonal.

The values of <math display="inline">\beta _\xi </math> and <math display="inline">\beta _\eta </math> are  computed by considering the solution of two uncoupled 1D problems along the <math display="inline">\xi </math> and <math display="inline">\eta </math> directions. This gives from Eqs.(21) and (22)

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\beta _\xi = \left[\left({S_0\over S_{\xi _2}}\right){w_\xi \over 6} +  \left({S_{\xi _1}\over S_{\xi _2}}\right)\gamma _\xi -1\right]\quad ,\quad \gamma _\xi = {u_\xi l_\xi \over 2k} \quad ,\quad  w_\xi = {sl^2_\xi \over k}</math>
|}
| style="width: 5px;text-align: right;" |  (37a)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\beta _\eta = \left[\left({S_0\over S_{\eta _2}}\right){w_\eta \over 6} +  \left({S_{\eta _1}\over S_{\eta _2}}\right)\gamma _\eta -1\right]\quad ,\quad \gamma _\eta = {u_\eta l_\eta \over 2k} \quad ,\quad  w_\eta = {sl^2_\eta \over k} </math>
|}
| style="width: 5px;text-align: right;" |  (37b)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}S_0 =\hbox{sign }(\bar \phi ) \quad ,\quad S_{\xi _1}= \hbox{sign } \left(\displaystyle{\partial \bar \phi \over \partial \xi }\right)\quad ,\quad  S_{\xi _2}= \hbox{sign } \left(\displaystyle{\partial ^2 \bar \phi \over \partial \xi ^2}\right)\\ S_{\eta _1}= \hbox{sign } \left(\displaystyle{\partial \bar \phi \over \partial \eta }\right)\quad ,\quad  S_{\eta _2}= \hbox{sign } \left(\displaystyle{\partial ^2 \bar \phi \over \partial \eta ^2}\right) \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (38)
|}

and <math display="inline">\bar \phi </math> is as usual the approximate solution.

The lengths <math display="inline">l_\xi </math> and <math display="inline">l_\eta </math> are taken as the maximum projection of the velocities <math display="inline">u_\xi </math> and <math display="inline">u_\eta </math> along the element sides (for triangles) and the element diagonals (for quadrilaterals), i.e.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>l_i =\max ({d}_j^T {u}_i)\quad ,\quad i=\xi ,\eta </math>
|}
| style="width: 5px;text-align: right;" |  (39a)
|}

with

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{ll}j=1,2,3 \hbox{ (for triangles) and }\\ j=1,2 \hbox{ (for quadrilaterals)}\end{array}</math>
|}
| style="width: 5px;text-align: right;" |  (39b)
|}

In Eq.(39a) <math display="inline">{u}_\xi </math> and <math display="inline">{u}_\eta </math> contain the global components of the  velocity vectors <math display="inline">\vec u_\xi </math> and <math display="inline">\vec u_\eta </math>, respectively. For triangles <math display="inline">{d}_j</math> are the element side vectors, whereas for quadrilaterals <math display="inline">{d}_j</math> are the element diagonal vectors <span id='citeF-27'></span>[[#cite-27|27]].

The next step is to transform Eq.(34) to global axes <math display="inline">x,y</math>. The resulting equation is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>-{u}^T {\boldsymbol \nabla }\phi +{\boldsymbol \nabla }^T {D}_G {\boldsymbol \nabla }\phi -s\phi=0 </math>
|}
|}

where the global diffusion matrix <math display="inline">{D}_G</math> is given by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{D}_G={D}+\bar{D}</math>
|}
| style="width: 5px;text-align: right;" | (41a)
|}

where the global balancing diffusion matrix <math display="inline">\bar{D}</math> is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar {D} = {T}^T \bar {D}^\prime {T}</math>
|}
| style="width: 5px;text-align: right;" | (41b)
|}

===Remark===

Similarly as for the 1D problems, a negative value of the parameters <math display="inline">\beta _\xi </math> and <math display="inline">\beta _\eta </math> indicates that no stabilization is needed along the directions <math display="inline">\xi </math> and <math display="inline">\eta </math>, respectively. A zero value of the corresponding stabilization parameter is chosen in this case.

===Remark===

The expressions of <math display="inline">\beta _\xi </math> and <math display="inline">\beta _\eta </math> in Eq.(37) can be simplified to

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}\beta _\xi \simeq \beta _{\xi _c} =\left[\displaystyle{w_\xi \over 6} +|\gamma _\xi | -1\right]\\ \beta _\eta \simeq \beta _{\eta _c} =\left[\displaystyle{w_\eta \over 6} +|\gamma _\eta | -1\right] \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (42)
|}

This avoids the computation of the sign of the solution and of its first and second derivatives. The expressions of <math display="inline">\beta _{\xi _c}</math> and <math display="inline">\beta _{\eta _c}</math> in Eq.(42) are equivalent to that of the 1D critical stabilization parameter <math display="inline">\beta _c</math> of Eq.(23). The main difference is that the computation of the local directions <math display="inline">\xi </math> and <math display="inline">\eta </math> is still mandatory in the multidimensional case and, therefore, the nonlinearity of the process can not be avoided here.

===5.1 Computation of the principal curvature axes for linear elements===

Excellent results have been obtained in our work ''by approximating the main  curvature direction <math>\vec \xi </math> by the direction of the gradient vector'' <math display="inline">{\boldsymbol \nabla } \phi </math>.

This simplification allows us to estimate the direction <math display="inline">\vec {\xi }</math> in a very economical manner as the gradient vector <math display="inline">{\boldsymbol \nabla } \bar \phi </math> can be directly computed at any point of a linear element. Direction <math display="inline">\vec {\eta }</math> is taken orthogonal to that of <math display="inline">\vec {\xi }</math> in an anti-clockwise sense.

For linear triangles <math display="inline">{\boldsymbol \nabla }\bar \phi </math> is constant within the element. For four node quadrilaterals <math display="inline">{\boldsymbol \nabla } \bar \phi </math> varies linearly. We have assumed in this case that the direction of <math display="inline">\vec \xi </math> is constant within the element and equal to the direction of vector <math display="inline">{\boldsymbol \nabla } \bar \phi </math> computed at the element center.

The computation of the signs of the second derivative of <math display="inline">\bar \phi </math> in Eq.(38) involves the following steps: 1) recovery of the cartesian first derivative field at the nodes using a nodal averaging procedure; 2) computation of the second derivative tensor at the element center and 3) transformation of this tensor to the local axes <math display="inline">\xi </math> and <math display="inline">\eta </math>.

We note that in problems where positive values of <math display="inline">\bar \phi </math> are prescribed at the Dirichlet boundary, the signs of <math display="inline">S_{\xi _2}</math>, <math display="inline">S_{\eta _2}</math> are positive due to the convexity of the numerical solution.

As mentioned above the dependence of the balancing diffusion matrix <math display="inline">\bar{D}</math> with the principal  curvature directions <math display="inline">\vec \xi </math> and <math display="inline">\vec {\eta }</math> introduces a nonlinearity in the solution process. A simple and effective iterative algorithm is described next.

===5.2 General iterative scheme===

A stabilized numerical solution can be found by the following algorithm.

'''Step 1''' . For elements in the interior of the domain choose <math display="inline">{}^1{\boldsymbol \xi } ={u}</math>, i.e. the gradient direction is taken coincident with the velocity direction. If <math display="inline">{u}=0</math> then <math display="inline">{}^1{\boldsymbol \xi }</math> is taken coincident with the global <math display="inline">{x}</math> axis.

In elements adjacent to a boundary choose <math display="inline">{}^1{\boldsymbol \xi } ={n }</math> where '''n ''' is the normal to the boundary.

Compute <math display="inline">{}^1{\boldsymbol \eta }, {}^1\bar  {D}'</math>, <math display="inline">{}^1\bar  {D}</math> and <math display="inline">{}^1{D}_G</math> using the expressions of <math display="inline">\beta _\xi </math> and <math display="inline">\beta _\eta </math> from Eq.(42).

Solve for <math display="inline">{}^1\bar{\boldsymbol \phi }</math>.

Verify that the solution is stable. This implies that there are not undershoots or overshoots in the numerical results with respect to the expected physical values. If the numerical solution is unstable, then go to step 2.

'''Step 2''' . For all elements, compute at the element center <math display="inline">{}^2{\boldsymbol \xi } = {\boldsymbol \nabla }^1\bar \phi </math>. Then compute <math display="inline">{}^2{\boldsymbol \eta } , {}^2\bar{D}'</math>, <math display="inline">{}^2\bar  {D}</math> and <math display="inline">{}^2{D}_G</math> using the expressions of <math display="inline">\beta _\xi </math> and <math display="inline">\beta _\eta </math> from Eqs.(37).

Solve for <math display="inline">{}^2\bar{\boldsymbol \phi }</math>.

'''Step 3''' . Estimate the convergence of the process. We have chosen the following convergence norm

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\Vert \phi \Vert = {1\over N\bar \phi _{max}} \left[\sum \limits _{j=1}^n \left({}^{i}\bar \phi _j - {}^{i-1}\bar \phi _j \right)^2 \right]^{1/2} \le \varepsilon  </math>
|}
| style="width: 5px;text-align: right;" | (43)
|}

where <math display="inline">N</math> is the total number of nodes in the mesh and <math display="inline">\phi _{max}</math> is the maximum prescribed value at the Dirichlet boundary (if <math display="inline">\bar \phi _{max}=0</math> then <math display="inline">\bar  \phi _{max}=1</math>). In above steps the left upper indices denote the iteration number.

In the examples shown in the next section <math display="inline">\varepsilon =10^{-3}</math> has been taken.

If condition (43) is not satisfied, repeat steps 2 and 3 until convergence.

===Remark===

For the advective-diffusive problems (i.e. <math display="inline">s=0</math>) the  expression of the balancing diffusion matrix in the interior elements for step 1 coincides with the standard (linear) SUPG form <span id='citeF-27'></span>[[#cite-27|27]].

===Remark===

An  alternative solution scheme is to use a time relaxation technique based in the solution of a pseudo transient problem with a forward Euler scheme and a diagonal mass matrix.

==6 1D NUMERICAL EXAMPLES==

The examples presented in this section are solved in a 1D domain discretized with eight two-node linear elements of unit size. The examples are solved with the same Dirichlet conditions <math display="inline">\phi _1^p =8</math> and <math display="inline">\phi _9^p=3</math> and two different values of <math display="inline">\gamma </math> and <math display="inline">w</math> (<math display="inline">\gamma =1, w=20</math> and <math display="inline">\gamma =10, w=20</math>). We note that for both cases the instability condition (<math display="inline">\beta _c>0</math>) is violated and, hence, the Galerkin solution should yield non-physical results.

Figures 2 and 3 show the numerical results obtained with the standard  Galerkin method (<math display="inline">\beta =0</math>) and using the element (<math display="inline">\beta ^e</math>) and critical (<math display="inline">\beta _c</math>) values of the stabilization parameter  given by Eqs.(22) and (23), respectively. The exact analytical solution is also shown for comparison.

Table 1 shows the nodal values of the results of the example of Figure 3 for comparison with the 2D solutions presented in the next section.

The following conclusions are drawn from the 1D results.

<ol>

<li>The Galerkin solution (<math display="inline">\beta =0</math>) is unstable for both problems, as expected. </li>

<li>The solution obtained with the critical value <math display="inline">\beta _c</math> is always stable, although it yields slightly overdiffusive results in both cases. </li>

<li>The results obtained with <math display="inline">\beta ^e</math>  are less diffusive and more accurate than those obtained with <math display="inline">\beta _c</math>. The explanation is that the sign of the ratio <math display="inline">{S_1/S_2}</math> is negative in the region close to the left end point of the domain. This naturally reduces the value of the stabilizing diffusion parameter <math display="inline">\beta </math> in Eq.(21) with respect to that of <math display="inline">\beta _c</math> in Eq.(23) where the sign effect is not relevant.  </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 in some cases for these type of problems <span id='citeF-12'></span>[[#cite-12|12]]. </li>

</ol>

Above conclusions have been confirmed in the solution of a wider collection of 1D problems presented in [18].
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_447243531-Fig2.png|450px|]]
|- style="text-align: center;"
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 2'''. <math>\phi _1^p=8, \phi _9^p =3, \gamma =1</math> and <math>\omega =20</math>. FIC results for a mesh of 8 linear elements obtained for <math>\beta =0</math> (Galerkin), <math>\beta ^e</math> and <math>\beta _c</math>. Comparison with the analytical solution.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_447243531-Fig6.png|450px|ϕ₁<sup>p</sup>=8, ϕ₉<sup>p</sup>=3, γ=10 and ω=20. FIC results for a mesh of 8 linear elements obtained for β=0 (Galerkin), β<sup>e</sup> and β<sub>c</sub>. Comparison with the analytical solution.]]
|- style="text-align: center;"
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 3'''.  <math>\phi _1^p =8, \phi _9^p =3, \gamma =10</math> and <math>\omega =20</math>. FIC results for a mesh of 8 linear elements obtained for <math>\beta =0</math> (Galerkin), <math>\beta ^e</math> and <math>\beta _c</math>. Comparison with the analytical solution.
|}

==7 2D EXAMPLES==

The analysis domain in the  first two 2D examples presented is a square of size 8 units. The problems are solved first with relatively coarse meshes of <math display="inline">8\times 8</math> four node bi-linear square elements and <math display="inline">8\times 8\times 2</math> linear triangles.

The boundary conditions for both examples are <math display="inline">\phi ^p =8</math> and <math display="inline">\phi ^p =3</math> at the boundaries <math display="inline">x=0</math> and <math display="inline">x=8</math>, respectively and zero normal flux at <math display="inline">y=0</math> and <math display="inline">y=8</math>. This reproduces the condition of the two 1D examples solved in the previous section. The first example is analized for <math display="inline">{u} = [2,0]^T</math>, <math display="inline">k=1</math> and <math display="inline">s=20</math> giving <math display="inline">w=20</math>, <math display="inline">\gamma _x=1</math> and <math display="inline">\gamma _y=0</math> which corresponds to the first 1D example (Figure 2). The correct solution for this problem has a boundary layer in the vecinity of the  two sides at <math display="inline">x=0</math> and <math display="inline">x=8</math> where <math display="inline">\phi </math> is prescribed (Figure 4). The numerical results obtained with the standard  Galerkin solution  are oscillatory as expected. The stabilized FIC formulation elliminates the oscillations and yields the correct physical solution. Good results are obtained for both meshes of linear rectangles and triangles (Figures 4 and 5).

Results labelled as FIC-1 and FIC-2 in the figures correspond to those obtained in the first and second iteration of the algorithm presented in Section 5.2, respectively. We note that the FIC-1 results agree precisely with those obtained in the 1D case for <math display="inline">\beta =\beta _c</math>, whereas the FIC-2 results agree with the more accurate 1D values obtained with the element stabilization parameter <math display="inline">\beta ^e</math> (see Figure 2).

The second example is similar to the first one with <math display="inline">{u} =[20,0]^T</math>, <math display="inline">k=1</math> and <math display="inline">s=20</math> giving <math display="inline">w=20</math>, <math display="inline">\gamma _x =10</math> and <math display="inline">\gamma _y =0</math>. These values correspond to the second 1D problem of the previous section (Figure 3). The Galerkin solution is again oscillatory, whereas the FIC results are physically sound (Figures 6 and 7). Once more  the FIC-1 and FIC-2 results  are in good agreement with the 1D values shown in Figure 3 for <math display="inline">\beta _c</math> and <math display="inline">\beta _e</math>, respectively for both meshes of square and triangular elements. The coincidence of the 1D and 2D results for this problem can be clearly seen in Table 1.

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|
[[File:Draft_Samper_447243531_5062_Fig4.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 4'''. 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>{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 for a mesh of <math>8 \times 8</math> four node square elements.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_5330_Fig5.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 5'''. Solution of problem of Figure 4 with a mesh of <math>8 \times 8\times 2</math> linear triangles.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_5509_Fig6.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 6'''. 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>{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 for a mesh of <math>8 \times 8</math> four node square elements.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_4066_Fig7.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 7'''. Solution of problem of Figure 5 with a mesh of <math>8 \times 8\times 2</math> linear triangles.
|}

<br/>


{| class="wikitable" style="text-align: center; margin: 1em auto;"
|+ Table. 1 Comparison of 1D and 2D solutions for the advection-diffusion-absorption problem of Figure 3  (<math>\gamma _x =10</math>, <math>w=20</math>)
|- style="border-top: 2px solid;"
| colspan='5' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | '''1D'''
| colspan='4' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | '''2D (nodes along line A-A')'''
|- style="border-top: 2px solid;"
| colspan='5' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | Figure 3
| colspan='2' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | 4 node quads. (Fig. 6)
| colspan='2' style="border-left: 2px solid;border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | 3 node triangles (Fig. 7)
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  Node 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\bar \phi (\beta =0</math>)
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\bar \phi (\beta ^e</math>)
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\bar \phi (\beta _c</math>)
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\phi </math>(exact)
| style="border-left: 2px solid;border-right: 2px solid;" | FIC-1
| style="border-left: 2px solid;border-right: 2px solid;" | FIC-2
| style="border-left: 2px solid;border-right: 2px solid;" | FIC-1
| style="border-left: 2px solid;border-right: 2px solid;" | FIC-2
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  1 
| style="border-left: 2px solid;border-right: 2px solid;" | 8,00 
| style="border-left: 2px solid;border-right: 2px solid;" | 8 
| style="border-left: 2px solid;border-right: 2px solid;" | 8 
| style="border-left: 2px solid;border-right: 2px solid;" | 8
| style="border-left: 2px solid;border-right: 2px solid;" | 8
| style="border-left: 2px solid;border-right: 2px solid;" | 8
| style="border-left: 2px solid;border-right: 2px solid;" | 8
| style="border-left: 2px solid;border-right: 2px solid;" | 8
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  2 
| style="border-left: 2px solid;border-right: 2px solid;" | 2,94 
| style="border-left: 2px solid;border-right: 2px solid;" | 3,06 
| style="border-left: 2px solid;border-right: 2px solid;" | 4 
| style="border-left: 2px solid;border-right: 2px solid;" | 3,08 
| style="border-left: 2px solid;border-right: 2px solid;" | 3,99 
| style="border-left: 2px solid;border-right: 2px solid;" | 3,057  
| style="border-left: 2px solid;border-right: 2px solid;" | 4,0 
| style="border-left: 2px solid;border-right: 2px solid;" | 3,059 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  3 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,32 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,17 
| style="border-left: 2px solid;border-right: 2px solid;" | 2 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,19 
| style="border-left: 2px solid;border-right: 2px solid;" | 2,00
| style="border-left: 2px solid;border-right: 2px solid;" | 1,170 
| style="border-left: 2px solid;border-right: 2px solid;" | 2,0 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,167
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  4 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,80 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,447 
| style="border-left: 2px solid;border-right: 2px solid;" | 1 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,457 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,00
| style="border-left: 2px solid;border-right: 2px solid;" | 0,448 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,0 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,452
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  5 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,599 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,172 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,5 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,176 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,49 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,172
| style="border-left: 2px solid;border-right: 2px solid;" | 0,499 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,166
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  6 
| style="border-left: 2px solid;border-right: 2px solid;" | -0,633 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0646 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,25 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0677 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,248
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0648 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,2501
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0681
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  7 
| style="border-left: 2px solid;border-right: 2px solid;" | 1,16 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0264 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,125 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0261 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,125
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0255 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,1250
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0257
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  8 
| style="border-left: 2px solid;border-right: 2px solid;" | -1,83 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0073
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0625
| style="border-left: 2px solid;border-right: 2px solid;" | 0,01
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0615 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0101 
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0624
| style="border-left: 2px solid;border-right: 2px solid;" | 0,0072
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" |  9 
| style="border-left: 2px solid;border-right: 2px solid;" | 3
| style="border-left: 2px solid;border-right: 2px solid;" | 3 
| style="border-left: 2px solid;border-right: 2px solid;" | 3 
| style="border-left: 2px solid;border-right: 2px solid;" | 3
| style="border-left: 2px solid;border-right: 2px solid;" | 3
| style="border-left: 2px solid;border-right: 2px solid;" | 3 
| style="border-left: 2px solid;border-right: 2px solid;" | 3 
| style="border-left: 2px solid;border-right: 2px solid;" | 3

|}

Note that, similarly to the 1D case, the FIC-2 results are more accurate (less diffusive) than those obtained in the first iteration (FIC-1). This is due to the more precise evaluation of <math display="inline">\beta _s</math> and <math display="inline">\beta _\eta </math> in Eqs.(37) accounting for the correct sign of all the terms.

Figures 8&#8211;11 show results for the two 2D problems above described solved now with relatively coarse unstructured meshes of linear triangles and quadrilaterals. The effectiveness and accuracy of the FIC iterative scheme is again noticeable in all cases. Note the agreement of the FIC-2 results of Figures 10 and 11 with the exact solution for the equivalent 1D problem of Figure 3.


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_4598_Fig8.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 8'''. Solution of problem of Figure 4 with an unstructured mesh of 209 four node bi-linear quadrilaterals
|}


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_3146_Fig9.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;" |'''Figure 9'''.  Solution of problem of Figure 4 with an unstructured mesh of 176 three node linear triangles
|}


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_4133_Fig10.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;" |'''Figure 10'''.  Solution of problem of Figure 6 with an unstructured mesh of 209 four node bi-linear quadrilaterals
|}


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_5097_Fig11.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 11'''.  Solution of problem of Figure 6 with an unstructured mesh of 176 three node triangles
|}


Figure 12 presents the solution of a similar problem where the values of <math display="inline">\phi </math> are prescribed at the four boundaries. The solution domain has now 10 units and the problem is solved first with a mesh of <math display="inline">10\times 10</math> four node square elements. Details of the physical parameters are given in Figure 12. Excellent results are again obtained with the FIC scheme. Similar good results are obtained with a structured mesh of linear triangles (Figure 13) as well as with  non structured meshes of linear quadrilateral and triangles (Figures 14 and 15).

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_9361_Fig12.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 12'''.  2D advection-diffusion-absorption problem over a square domain of size equal to 10 units. <math>\phi ^p=50</math> along <math>x=0</math> and <math>y=10</math> and <math>\phi ^p=100</math> along <math>x=10</math> and <math>y=0</math>, <math>{u}= [0,15]^T</math>, <math>k=1</math>, <math>s=12</math>, <math>w=12</math>, <math>\gamma _x=0</math> and <math>\gamma _y= 7.5</math>. Galerkin and FIC solutions for a mesh of <math>10 \times 10</math> four node bi-linear square elements.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_6440_Fig13.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 13'''.  Solution of the problem of Figure 12 with an unstructured mesh of 432 four node bi-linear quadrilaterals
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_3226_Fig14.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 14'''.  Solution of problem of Figure 12 with an structured mesh of <math>10\times 10\times 2</math>  three node linear triangles
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_6545_Fig15.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 15'''.  Solution of problem of Figure 12 with an unstructured mesh of 780 three node triangles
|}


The effectiveness of the FIC scheme for a diffusive-absorptive problem with Dirichlet boundary conditions is shown in Figure 16. The results shown have been obtained with structured meshes of linear quadrilateral and triangles. Note that the four boundary layers  are well captured in the first step of the iterative solution. Similar good results have also been obtained with unstructured meshes not shown here.


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_4306_Fig16.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 16'''.   Diffusive-absorptive problem over a square domain of size equal to 10 units. <math>\phi ^p=50</math> over <math>x=0</math> and <math>y=10</math> and <math>\phi ^p=100</math> over <math>x=10</math> and <math>y=0</math>, <math>{u}= [0,0]^T</math>, <math>k=1</math>, <math>s=20</math>, <math>w=20</math>, <math>\gamma _x=0</math> and <math>\gamma _y= 0</math>. Galerkin and FIC solutions obtained with structured  meshes of four node quadrilaterals and linear triangles.
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_6269_Fig16cont.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 16'''.  (cont.)
|}


The final example is a standard benchmark problem of advection-diffusion where sharp layers appear at both the boundary and the interior of the domain. The problem is the advective-diffusive transport of <math display="inline">\phi </math> in a square domain with non uniform Dirichlet conditions, downwards diagonal velocity and zero source terms (i.e. <math display="inline">Q=0</math> and <math display="inline">s=0</math>). Figure 17 displays the SUPG solution and FIC results obtained after two iterations using a structured mesh of <math display="inline">20\times 20</math> linear four node square elements. It is remarkable that the FIC results capture the sharp gradient zones at the boundaries where <math display="inline">\phi </math> is prescribed to zero and at the interior of the domain and elliminate all the spurious oscillations present in the SUPG method.

Similar good results obtained with the FIC method for a wide range of advective-diffusive problems are presented in <span id='citeF-27'></span>[[#cite-27|27]]. Recent applications of the FIC method to incompressible fluid flow problems are reported in [37].


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[File:Draft_Samper_447243531_2098_Fig17.png]]
|- 
| colspan="1" style="text-align: center; font-size: 75%;padding:10px;"| '''Figure 17'''.  Square domain with non uniform Dirichlet conditions, downwards diagonal velocity and zero source. SUPG and FIC solutions obtained with a structured mesh of <math>20\times 20</math> linear four node square elements
|}

==8 CONCLUSIONS==

The FIC-FEM formulation presented allows to obtain a stabilized and accurate solution for the advection-diffusion-absorption equation. For the 1D problem the formulation is equivalent to adding a non-linear diffusion term to the standard discretized equations which is  governed by a single stabilization parameter. The use of the constant critical value of the 1D stabilization parameter  provides a stabilized numerical solution in ''a single step''. A more accurate (less diffusive) solution can be obtained using the two step iterative scheme proposed.

The equivalence of the FIC  method with a nonlinear stabilizing diffusion term extends naturally to multidimensional problems using structured and unstructured meshes. The key step is to express the governing equations of the FIC formulation in the principal curvature directions of the solution. The resulting FIC equation is equivalent to adding a nonlinear diffusion matrix to the infinitessimal governing equations. The solution process becomes non linear and a simple iterative algorithm has been presented. The approximation of the main principal curvature direction by that of the gradient vector simplifies the computations in the iterative scheme. Excellent results have been obtained for all the 2D problems solved in just two iterations for structured and nonstructured meshes.

It is remarkable that, similarly to the 1D case, good stabilized results are obtained in the first iteration of the scheme proposed (FIC-1 results) and this may be sufficient for many practical cases. More accurate (less diffusive) results are obtained by performing a  second iteration at a relatively small additional computational cost.

==ACKNOWLEDGEMENTS==

The authors also thank Profs. C. Felippa and S.R. Idelsohn for many useful discussions.

This work has been sponsored by the ''Ministerio de Educación y Ciencia of Spain''.  Plan Nacional, Project numbers: DPI2001-2193, BIA2003-09078-C02-01, and DPI2004-07410-C03-02.

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