doi: 10.1016/j.cma.2005.05.036

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==Abstract==

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A finite element method (FEM) for steady-state convective-diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers is presented. The necessary stabilization of the numerical solution is provided by the Finite Calculus (FIC) approach. The FIC method is based in the solution by the Galerkin FEM of a modified set of governing equations which include characteristic length parameters. It is shown that the FIC balance equation for the multidimensional convection-diffusion problem written in the principal curvature axes of the solution, introduces an orthotropic diffusion which stabilizes the numerical solution both in smooth regions as well in the vicinity of sharp gradients. The dependence of the stabilization terms with the principal curvature directions of the solution makes the method non linear. Details of the iterative scheme to obtain stabilized results are presented together with examples of application which show the efficiency and accuracy of the approach.

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<br/><br/>

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==1 INTRODUCTION==

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It is well known that the standard Galerkin FEM solution of the steady-state convective-diffusive equation is unstable for values of the Peclet number greater than one (see Volume 3 in <span id='citeF-1'></span>[[#cite-1|[1]]]). A number of numerical schemes have been proposed in order to guarantee that the numerical solution is stable, that is, that the solution has a physical meaning. In the first attempts to solve this problem the underdiffusive character of the Galerkin FEM (and the analogous central finite difference scheme) for convective-diffusive problems was corrected by adding “artificial diffusion terms” to the governing equations <span id='citeF-1'></span>[[#cite-1|[1,2]]]. The relationship of this approach with the upwind finite difference method <span id='citeF-2'></span>[[#cite-2|[2]]] lead to the derivation of a variety of Petrov-Galerkin FEM. All these methods can be interpreted as extensions of the standard Galerkin variational form of the FEM by adding residual-based integral terms computed over the element domains. Among the many stabilization methods of this or similar kind we name the Upwind FEM <span id='citeF-3'></span>[[#cite-3|[3,4]]], the Streamline Upwind Petrov-Galerkin (SUPG) method <span id='citeF-5'></span>[[#cite-5|[5-9]]], the Taylor-Galerkin method <span id='citeF-12'></span>[[#cite-12|[10,11]], the generalized Galerkin method <span id='citeF-17'></span>[[#cite-17|[12,13]]], the Galerkin Least Square method and related approaches <span id='citeF-19'></span>[[#cite-19|[14-16]]], the Characteristic Galerkin method <span id='citeF-22'></span>[[#cite-22|[17,18]]], the Characteristic Based Split method <span id='citeF-24'></span>[[#cite-24|[19]]], the  Subgrid Scale method <span id='citeF-25'></span>[[#cite-25|[20-23]]], the Residual Free Bubbles method <span id='citeF-37'></span>[[#cite-37|[24]]], the Discontinuous Enrichment Method <span id='citeF-39'></span>[[#cite-39|[25]]] and the Streamline Upwind with Boundary Terms method <span id='citeF-40'></span>[[#cite-40|[26]]].  Basically all the methods make use of a single stabilization parameter which suffices to stabilize the numerical solution along the velocity (streamline) direction. The computation of the streamline stabilization parameter for multidimensional problems is usually based on extensions of the optimal value of the parameter for the simpler 1D case. Specific attempts to design the stability parameter for multidimensional problems in the context of the Petrov-Galerkin formulation have been recently reported <span id='citeF-41'></span>[[#cite-41|[27-31]]]. In general all the stabilized methods above mentioned yield good results for 1D-type problems where the velocity vector is aligned with the direction of the gradient of the solution. However, the use of a single stabilization parameter is  insufficient to provide stabilized solutions in the vicinity of sharp gradients not aligned with the velocity direction, which may appear at the interior of the domain or at boundary layers. The usual remedy for these situations is to use the so called “shock capturing” or “discontinuity-capturing” schemes <span id='citeF-49'></span>[[#cite-49|[32-36]]]. These methods basically add additional transverse diffusion terms in selected elements in order to correct the undershoots and overshoots yielded by the Petrov-Galerkin solution in the sharp gradient zones. Usually the new shock capturing diffusion terms depend on the gradient of the solution and the scheme becomes non linear.

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The aim of this work is to develop a general finite element formulation which can provide stabilized numerical results for all range of convective-diffusive problems. The new formulation therefore has the necessary intrinsic features to deal with streamline-type instabilities, as well as with the typical undershoots and overshoots in the vicinity of sharp gradients at different angles with the velocity direction. The new formulation thus provides a unified theoretical and computational framework incorporating all the ingredients of the traditional Petrov-Galerkin and shock-capturing methods.

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The formulation proposed is based in the so called Finite Calculus (FIC) approach <span id='citeF-50'></span>[[#cite-50|[37,38]]]. The FIC method is based in expressing the equation of 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. These dimensions are termed ''characteristic length parameters'' and play a key role in the stabilization process.

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The modified governing equations lead to stabilized numerical schemes using whatever numerical method. It is interesting that many of the stabilized FEM can be recovered using the FIC formulation. The FIC method has been successfully applied to problems of convection-diffusion <span id='citeF-50'></span>[[#cite-50|[37-41]]], convection-diffusion-absorption [42,43], incompressible fluid flow <span id='citeF-47'></span>[[#cite-47|[44-47]]] and incompressible solid mechanics [48,49].

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The key to the stabilization in the FIC method is to choose adequately the characteristic length parameters  for each element. For 1D problems the standard optimal  and critical values of the single stabilization parameter can be found, as shown in Section 3. For 2D/3D problems the characteristic length parameters along each space direction are grouped in a characteristic length vector <math display="inline">h</math>. Here the key issue is to choose correctly the direction of this vector. It is interesting that if the direction of <math display="inline">h</math> is taken parallel to that of the velocity, then the resulting FIC stabilized equations ''coincide'' with that of the standard SUPG method <span id='citeF-50'></span>[[#cite-50|[37,38]]].

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In previous work of the authors, transverse sharp gradients to the velocity direction were accurately captured  by computing iteratively the direction and modulus of vector <math display="inline">h</math> which minimizes a residual norm <span id='citeF-50'></span>[[#cite-50|[37-41]]]. In <span id='citeF-51'></span>[[#cite-51|[38]]] accurate stabilized solutions for convection-diffusion problems with sharp gradients were obtaining by spliting the characteristic lenght vector <math display="inline">{h}</math> as the sum of two vectors. The first vector is chosen aligned with the velocity direction, hence yielding the standard SUPG terms, while the second vector is taken parallel to the direction of the gradient of the solution.

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In this work a slightly different and more consistent method is proposed. We have found that the key to the general stabilization algorithm for convection-diffusion problems is to express the  FIC balance equation taking as  coordinate axes the principal curvature directions of the solution. These equations contain the necessary additional diffusion to stabilize the numerical solution ''in all situations''. It is interesting that when one of the principal curvature directions coincides with the velocity direction, then the  classical SUPG method is recovered.

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The content of the paper is organized as follows. In the next section, the basic FIC equations for the multidimensional steady-state convective-diffusive problem are given.  The general equation is particularized for the simple 1D problem and the optimal and critical values of the 1D stabilization parameter are obtained. Then the form of the FIC balance equation written in the principal curvature axes is presented. The computation of the stabilization parameters is detailed and the general iterative solution scheme is explained. The formulation is  particularized for linear elements (four node quadrilaterals and three node triangles) where the computation of the curvature directions requires a derivative recovery algorithm. Very good results have been obtained by approximating the main principal curvature direction within the element by the gradient direction at the element center, which is a much more economical approach. A number of examples showing the accuracy and efficiency of the method  are presented. Convergence of the iterative scheme in problems with arbitrary sharp gradients was found in most cases in just two iterations.

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==2 FIC GOVERNING EQUATIONS FOR STEADY-STATE CONVECTION-DIFFUSIVE PROBLEMS==

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The basic FIC equations are obtained by expressing the balance of fluxes in interior and boundary domains of finite size and retaining one order higher terms in the Taylor expansions than those retained in the infinitessimal theory. The resulting governing equations are

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>r - \underline{{1\over 2} {h}^T {\boldsymbol \nabla }r}{1\over 2} {h}^T {\boldsymbol \nabla }r =0 \qquad \hbox{in } \Omega  </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (1)

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with the boundary conditions

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\phi - \phi ^p = 0\qquad \hbox{on}~~\Gamma _\phi </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" |  (2a)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>{\textbf n}^T {D} {\boldsymbol \nabla }\phi +q_n^p - \underline{{1\over 2}{\textbf h}^T {\textbf n}  r}{1\over 2}{h}^T {n}  r= 0\qquad \hbox{on}~~\Gamma _q </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" |  (2b)

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where <math display="inline">\Gamma _\phi </math> and <math display="inline">\Gamma _q</math> are the Dirichlet and Neumann boundaries where the variable <math display="inline">\phi </math> and the outgoing normal diffusive flux are prescribed to values <math display="inline">\phi ^p</math> and <math display="inline">q_n^p</math>, respectively. The modified equation (2b) is obtained by invoking higher order balance of fluxes in a finite domain next to the Neumann boundary <span id='citeF-50'></span>[[#cite-37|[37,38]]]. In above equations

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>r := -{\textbf u}^T {\boldsymbol \nabla } \phi + {\boldsymbol \nabla }^T {D} {\boldsymbol \nabla }\phi + Q </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)

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where <math display="inline">u</math> is the divergence-free velocity vector, <math display="inline">{\textbf D}</math> is the diffusion matrix, <math display="inline">{\boldsymbol \nabla }</math> is the gradient operator, <math display="inline">Q</math> is the external source term and <math display="inline">{\textbf n}</math> is the normal vector. Vector <math display="inline">{\textbf h}</math>  is the ''characteristic length'' vector. For 2D problems <math display="inline">{\textbf h}=[h_x, h_y]^T</math>, where <math display="inline">h_x</math> and <math display="inline">h_y</math> are characteristic distances along the sides of the rectangular domain where higher order balance of fluxes is enforced <span id='citeF-50'></span>[[#cite-37|[37,38]]].

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The underlined terms in Equations (1) and (2b) introduce the necessary stabilization in the numerical solution of the convective-diffusive problem  <span id='citeF-50'></span>[[#cite-37|[37-41]]]. The finite element formulation will be presented next.

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===2.1 Finite element discretization===

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A finite element interpolation of the unknown <math display="inline">\phi </math> can be written as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\phi \simeq \hat \phi =\sum N_i \phi _i </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)

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where <math display="inline">N_i</math> are the shape functions and <math display="inline"> \phi _i</math> are the nodal values of the approximate function <math display="inline">\hat \phi </math> <span id='citeF-1'></span>[[#cite-1|[1]]].

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Application of the Galerkin FE method to Equations (1) and (2b) gives, after integrating by parts the term <math display="inline"> {h}^T{\boldsymbol \nabla }\hat r</math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\int _\Omega N_i \hat r d\Omega - \int _{\Gamma _q} N_i ({\textbf n}^T {\textbf D} {\boldsymbol \nabla }\hat \phi +q_n^p)d\Gamma + \sum \limits _e {1\over 2}  \int _{\Omega ^e} ({\textbf h}^T{\boldsymbol \nabla } N_i + N_i {\boldsymbol \nabla }^T {\textbf h})\hat r d\Omega =0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (5)

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The last integral in Equation (5) has been expressed as a sum of the element contributions to allow for interelement discontinuities in the term <math display="inline">{\boldsymbol \nabla }\hat r</math> of Eq.(1), where <math display="inline">\hat r = r (\hat \phi )</math> is the residual of the FE approximation of the infinitesimal governing equation.

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Note that the residual terms have disappeared from the Neumann boundary <math display="inline">\Gamma _q</math>. This is due to the consistency between the FIC terms in Eqs.(1) and (2b).

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Integrating by parts the diffusive terms in the first integral of Eq.(5) leads to

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\int _\Omega N_i [{\textbf u}^T \nabla \hat \phi + {\boldsymbol \nabla }^T N_i {\textbf D}{\boldsymbol \nabla }\hat \phi ] d\Omega - \sum \limits _e {1\over 2} \int _{\Omega ^e} ({\textbf h}^T{\boldsymbol \nabla } N_i + N_i {\boldsymbol \nabla }^T {\textbf h})\hat r  d\Omega - \int _\Omega N_i Q d\Omega +\int _{\Gamma _q} N_i q_n^p d\Gamma =0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (6)

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In matrix form

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>{\textbf K}{\textbf a}={\textbf f} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (7)

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Matrix <math>{\textbf K}</math> and vector <math>{\textbf f}</math> are assembled from the element contributions given by

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>K_{ij}^e = \int _{\Omega ^e} [N_i{\textbf u}^T {\boldsymbol \nabla }N_j + {\boldsymbol \nabla }^T N_i ({\textbf D} + {1\over 2}{\textbf h}{\textbf u}^T){\boldsymbol \nabla } N_j + N_i ({\boldsymbol \nabla }^T {\textbf h}) {\textbf u}^T {\boldsymbol \nabla } N_j ]d\Omega -</math>

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| style="text-align: center;" | <math> - {1\over 2} \int _{\Omega ^e} ({\textbf h}^T {\boldsymbol \nabla } N_i + N_i {\boldsymbol \nabla }^T {\textbf h}) {\boldsymbol \nabla }^T ({D} {\boldsymbol \nabla } N_j)d\Omega </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (8.a)

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| style="text-align: center;" | <math> f_i^e = \int _{\Omega ^e} [N_i + {1\over 2} ({\textbf h}^T{\boldsymbol \nabla } N_i +N_i {\boldsymbol \nabla }^T {\textbf h}) ] Qd\Omega - \int _{\Gamma _q^e}N_i q_n^p d\Gamma  </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (8.b)

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Note that the method is equivalent to modifying the original diffusion matrix <math display="inline">D</math> by <math display="inline">{D}_G</math> with

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>{\textbf D}_G = {\textbf D} +\bar {\textbf D} \quad \hbox{and}\quad \bar {\textbf D} = {1\over 2} {\textbf h}{\textbf u}^T </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (9)

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In Eq.(9) <math display="inline">\bar {\textbf D}</math> is the ''balancing diffusion matrix'' introduced by the FIC method.

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We note that the Galerkin/FIC formulation is residual based, i.e. the stabilization term is a function of the FEM residual which progressively vanishes as the numerical results approaches the “exact” solution (see Eq.(5)). This preserves the consistency of the Galerkin method in order to obtain improved convergence rates with finite elements of any interpolation order. In the following we will  restrict the application of the method to linear elements only.

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Note  that when linear elements are used the second integral of Equation (8a) vanishes. The same happens with the term <math display="inline">{\textbf h}^T{\boldsymbol \nabla } N_i Q</math> of the first integral of Equation (8b) when <math display="inline">N_i</math> is linear and <math display="inline">Q</math> is constant. Finally, all terms involving the derivatives  of <math display="inline">\textbf h</math> vanish if <math display="inline">\textbf h</math> is constant over the element. The evaluation of these integrals is mandatory in any other case.

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===2.2 Equivalence with the SUPG method===

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Let us now assume that the direction of vector <math display="inline">h</math> is ''parallel'' to that of the velocity <math display="inline">\textbf u</math>, i.e. <math display="inline">{\textbf h}= h{{\textbf u}\over \vert {\textbf u}\vert }</math> where <math display="inline">h</math> is a characteristic length. Under these conditions, Eq.(5) reads (assuming <math display="inline">{\boldsymbol \nabla }^T{\textbf h}=0</math>)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\int _\Omega N_i \hat r d\Omega - \int _{\Gamma _q} N_i ({\textbf n}^T {D} {\boldsymbol \nabla }\hat \phi +q_n^p)d\Omega{+} \sum \limits _e \int _{\Omega ^e} {h\over 2\vert {\textbf u}\vert  }   {\textbf u}^T{\boldsymbol \nabla } N_i \hat rd\Omega =0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (10)

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Equation (10) coincides precisely with the SUPG method. The ratio <math display="inline">\displaystyle{h\over 2\vert {\textbf u}\vert  }</math> has dimensions of time and it is usually termed element ''intrinsic time'' parameter <math display="inline">\tau </math>.

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The balancing diffusion matrix <math display="inline">\bar {\textbf D}</math> of Eq.(9) is now given by

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\bar {\textbf D} ={h\over 2\vert {\textbf u}\vert }{\textbf u}{\textbf u}^T=\tau {\textbf u}{\textbf u}^T </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (11)

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It can be shown that the definition of <math display="inline">\bar {\textbf D}</math> of Eq.(11) is equivalent to introducing an artificial diffusion of value <math display="inline">{h\vert {\textbf u}\vert \over 2}</math> along the streamlines <span id='citeF-1'></span>[[#cite-1|[1,7,9]]] and this explains the name of the SUPG method. The element characteristic length <math display="inline">h</math> (or the value of <math display="inline">\tau </math>) is computed in practice  by  heuristic linear and non-linear extensions of the optimal expression for the 1D problem <span id='citeF-5'></span>[[#cite-5|[5-9,27-31]]].

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In the following sections we will denote by the standard SUPG method to the stabilization procedure leading to a balancing diffusion matrix <math display="inline">\bar {\textbf D}</math> given by Eq.(11).

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It is important to note that the SUPG expression is a ''particular case'' of the more general FIC formulation. This explains the limitations of the SUPG method to provide stabilized numerical results in the vicinity of sharp gradients of the solution transverse to the flow direction and the need to introduce in these cases additional shock capturing terms (see Section 5). However, in the FIC formulation the direction of <math display="inline">{\textbf h}</math> is arbitrary and not necessarily coincident with that of <math display="inline">{\textbf u}</math>. The components of <math display="inline">{\textbf h}</math>  introduce the necessary stabilization along the streamlines and the transverse directions to the flow. In this manner, the FIC method reproduces the best-features of the so-called stabilized discontinuity-capturing schemes <span id='citeF-49'></span>[[#cite-32|[32-36]]].

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==3 COMPUTATION OF THE CHARACTERISTIC LENGTH FOR THE 1D CONVECTION-DIFFUSION EQUATION==

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The computation of the characteristic length is a crucial step as its value affect to the stability and accuracy of the numerical solution.

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Before we face the general multidimensional problem, some based concepts of the computation of the characteristic length for the 1D convection-diffusion equation are briefly given.

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The FIC equation for the 1D steady-state convection-diffusion problem is written as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>r - {h\over 2}\frac{d r}{dx}=0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (12)

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with

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>r:=-u \phi'  +k \phi''{+}Q </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (13)

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where primes denote differentiation with respect to the independent space variable <math display="inline">x</math>.

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The element matrices of Eqs.(8) are now written for ''two node linear elements'' as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>K_{ij}^e = \int _{l^e} \left(N_i u N_j^\prime + N_i^\prime \left(k+{hu\over 2}\right)N_j^\prime \right)dx</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (14.a)

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| style="text-align: center;" | <math> f_i^e = \int _{l^e} \left(N_i + {h\over 2} N_j^\prime \right)Qdx - [N_i q^p]_{x_L} </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (14.b)

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|}

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where <math display="inline">l^e</math> is the element length and <math display="inline">x_L</math> is the coordinate of the end point of the 1D domain where the outgoing flux is prescribed to a value <math display="inline">q^p</math>. In Eq.(14a) the space derivatives of <math display="inline">h</math> have been neglected as <math display="inline">h</math> is assumed to be constant within each element.

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Note that the  FIC method introduces a diffusion of value <math display="inline">{hu\over 2}</math>. This term is analogous to that provided by artificial diffusion and upwinding methods <span id='citeF-1'></span>[[#cite-1|[1,2]]].

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The characteristic length parameter <math display="inline">h</math> for each element can  be made proportional to the element length as <math display="inline">h= \alpha l^e</math> where <math display="inline">\alpha </math> is an element stabilization parameter. A typical stencil for a mesh of uniform size 1D elements of length <math display="inline">l^e =l</math> is written as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>- u {\phi _{i+1} - \phi _{i-1}\over 2} + \left(k+\alpha {ul\over 2}\right)\left(   {\phi _{i+1} - 2\phi _i + \phi _{i-1}\over l}  \right)=0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (15)

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Injecting into Eq.(15) the exact analytical solution <math display="inline">\phi _i = A+Be^{{u\over k}x_i}</math> where <math display="inline">A</math> and <math display="inline">B</math> are contants, gives the optimal value of <math display="inline">\alpha =\alpha _{opt}</math> yielding the exact solution at the nodes with

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\alpha _{opt} =\coth \gamma - {1\over \gamma } </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (16)

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where <math display="inline">\gamma = {ul\over 2k}</math> is the element Peclet number.

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A critical value of <math display="inline">\alpha </math> ensuring physically correct results can be found by writing the stencil (15) for a two element mesh with nodes 1, 2 and 3 and Dirichlet boundary conditions <math display="inline">\phi _1=0</math> and <math display="inline">\phi _3 =\bar \phi </math>. The resulting equation reads

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>-u {\bar \phi \over 2} + \left(k+\alpha {ul\over 2}\right){\bar \phi - 2\phi _2\over l}=0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (17)

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An unphysical (unstable) solution will imply <math display="inline">\phi _2<0</math>. This is avoided if the following  value of <math display="inline">\alpha \ge \alpha _c</math> is chosen with

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{array}{l}\alpha _c = \left(1- {1\over \vert \gamma \vert }\right)\hbox{sign} (u) \quad ,\quad \hbox{for } \vert \gamma \vert \ge 1\\ \alpha _c =0  \quad \hbox{for } \vert \gamma \vert < 1 \end{array} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (18)

|}

​

where <math display="inline">\alpha _c</math> is termed the critical stabilization parameter. It can be verified that  that <math display="inline">\alpha _c\simeq \alpha _{opt}</math> for <math display="inline">\alpha >2</math>. Note that both <math display="inline">\alpha </math> and <math display="inline">\alpha _c</math> are constant within each element, as initially assumed.

​

Above well know results from the finite element and finite difference literature will be useful for the general multidimensional case described next.

​

 '''Remark 1'''. The stabilizing diffusion of Eq.(15) can be expressed in terms of the equivalent intrinsic time parameter <math display="inline">\tau </math> by replacing the term <math display="inline">\alpha {ul\over 2}</math> by <math display="inline">\tau u^2</math> where <math display="inline">\tau = \alpha {l\over 2u}</math>. The optimal and critical values of <math display="inline">\tau </math> are therefore obtained by dividing by <math display="inline">{l\over 2u}</math> the expressions of <math display="inline">\alpha _{opt}</math> and <math display="inline">\alpha _c</math> of Eqs.(16) and (18), respectively.

​

==4 COMPUTATION OF THE CHARACTERISTIC LENGTH VECTOR==

​

We present here a general procedure to compute the characteristic length vector <math display="inline">h</math> for convection-diffusion problems. For the sake of preciseness the method is explained for 2D problems although it is equally applicable to 3D problems.

​

<div id='img-1'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

|-

|[[Image:Draft_Samper_426828278-Figure1.png|351px|Global axes (x,y) and principal curvature axes (ξ,η)]]

|- style="text-align: center; font-size: 75%;"

| 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). For simplicity we consider the 2D sourceless case (<math display="inline">Q=0</math>) with an isotropic diffusion defined by a constant diffusion parameter <math display="inline">k</math>.  The FIC balance equation is

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| 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)- {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)\right]</math>

|-

| 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)\right]=0 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (19)

|}

​

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;width: 100%;" 

|-

| 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;white-space: nowrap;" | (20)

|}

​

Introducing this simplification into Eq.(19) we can rewrite this equation as

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math> -u_\xi {\partial \phi  \over \partial \xi }-u_\eta {\partial \phi  \over \partial \eta }+\left(k + {u_\xi h_\xi \over 2}\right){\partial ^2\phi \over \partial \xi ^2} + \left(k + {u_\eta h_\eta \over 2}\right){\partial ^2\phi \over \partial \eta ^2}  - {k\over 2} \left(h_\xi {\partial ^3 \phi \over \partial \xi ^3}+  h_\eta  {\partial ^3 \phi \over \partial \eta ^3}\right)=0</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (21)

|}

​

​

We can see clearly from Eq.(21) that the FIC governing equations introduce orthotopic diffusion parameters  of values <math display="inline">{u_\xi h_\xi \over 2}</math> and <math display="inline"> {u_\eta h_\eta \over 2}</math> along the <math display="inline">\xi </math> and <math display="inline">\eta </math> axes, respectively. Also note that the last term of Eq.(21) will vanish after discretization for linear elements.

​

Eq.(21) 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;width: 100%;" 

|-

| style="text-align: center;" | <math>- {\textbf u}^{\prime T} {\boldsymbol \nabla }^\prime \phi + {\boldsymbol \nabla }^{\prime T} ({\textbf D}+\bar {\textbf D}^\prime ) {\boldsymbol \nabla }^{\prime }\phi =0 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (22)

|}

​

where <math display="inline">{\textbf 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">\textbf D</math> is the “physical” isotropic diffusion matrix and <math display="inline">\bar {\textbf 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 these matrices is

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>{\textbf D}= \left[\begin{array}{cc}k &0\\0&k\end{array}\right]\qquad \hbox{and} \qquad \bar {\textbf D}^\prime = \left[\begin{array}{cc}\displaystyle{u_\xi h_\xi \over 2} & 0\\ 0 & \displaystyle{u_\eta h_\eta \over 2}\end{array}\right] </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (23)

|}

​

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;width: 100%;" 

|-

| style="text-align: center;" | <math>{\textbf u}'=\left\{\begin{array}{c}u_\xi \\ u_\eta \end{array}\right\}= {\textbf T} {\textbf u}  \quad \hbox{with}\quad {\textbf T}= \left[\begin{array}{cc}c_\alpha & s_\alpha \\ - s_\alpha & c_\alpha \end{array}\right]\quad ,\quad {\textbf u}=\left\{\begin{array}{c}u\\v\end{array}\right\} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (24)

|}

​

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 characteristic length distances <math display="inline">h_\xi </math> and <math display="inline">h_\eta </math> are defined as

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>h_\xi =\alpha _\xi l_\xi \quad \hbox{and}\quad h_\eta =\alpha _\eta l_\eta  </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (25)

|}

​

where <math display="inline">l_\xi </math> and <math display="inline">l_\eta </math> are typical element dimensions along the <math display="inline">\xi </math> and <math display="inline">\eta </math> axes, respectively and <math display="inline">\alpha _\xi </math> and <math display="inline">\alpha _\eta </math> are the corresponding stabilization parameters.

​

The values of <math display="inline">\alpha _\xi </math> and <math display="inline">\alpha _\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 Eq.(16)

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\alpha _\xi = \coth \gamma _\xi - {1\over \gamma _\xi } \quad ,\quad \gamma _\xi = {u_\xi l_\xi \over 2k}</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (26.a)

|-

| style="text-align: center;" | <math> \alpha _\eta = \coth \gamma _\eta - {1\over \gamma _\eta } \quad ,\quad \gamma _\eta = {u_\eta l_\eta \over 2k} </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (26.b)

|}

|}

​

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;width: 100%;" 

|-

| 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;white-space: nowrap;" | (27.a)

|}

​

with

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| 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;white-space: nowrap;" | (27.b)

|}

​

In Eq.(27a) <math display="inline">{\textbf u}_\xi </math> and <math display="inline">{\textbf 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">{\textbf d}_j</math> are the element sides vectors, whereas for quadrilaterals <math display="inline">{\textbf d}_j</math> are the element diagonals vectors.

​

The next step is to transform Eq.(22) to global axes <math display="inline">x,y</math>. The resulting equation is written as

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>-{\textbf u}^T {\boldsymbol \nabla }\phi +{\boldsymbol \nabla }^T {\textbf D}_G {\boldsymbol \nabla }\phi=0 </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (28)

|}

​

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

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>{\textbf D}_G={\textbf D}+\bar{\textbf D}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (29a)

|}

​

The isotropic diffusion matrix <math display="inline">\textbf D</math> is given by Eq.(23) and  the global balancing diffusion matrix <math display="inline">\bar{\textbf D}</math> is

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\bar {\textbf D} = {\textbf T}^T \bar {\textbf D}^\prime {\textbf T}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (29b)

|}

​

where the transformation matrix '''T''' is given in Eq.(24).

​

'''Remark 2'''. We can write the local balancing diffusion matrix <math display="inline">\bar{\textbf D}'</math> of Eq.(23) for <math display="inline">k_\xi{-}k_\eta >0</math>  as<br />

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\bar{\textbf D}' ={1\over 2}\left[\begin{array}{cc} (k_\xi -k_\eta ) &0\\ 0&0\end{array}\right]+ {1\over 2}\left[\begin{array}{cc} k_\eta  &0\\ 0&k_\eta \end{array}\right]=\bar{\textbf D}_\xi + \bar{\textbf D}_{iso}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (30a)

|}

​

where

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>k_\xi ={u_\xi h_\xi \over 2}\quad ,\quad k_\eta ={u_\eta h_\eta \over 2}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (30b)

|}

​

If <math display="inline">k_\eta </math> is greater than <math display="inline"> k_\xi </math> a similar split is performed with <math display="inline">(k_\eta - k_\xi )</math> in the position 22 of matrix <math display="inline">\bar{\textbf D}_\xi </math> and <math display="inline">k_\eta </math> is replaced by <math display="inline">k_\xi </math>  in <math display="inline">\bar{\textbf D}_{iso}</math>.

​

Eq.(30a) shows that the diffusivity matrix <math display="inline">\bar{\textbf D}'</math> is equivalent to the sum of a balancing diffusion along the principal curvature direction <math display="inline">\vec {\xi }</math> (or <math display="inline">\vec {\eta }</math> if <math display="inline">k_\eta > k_\xi </math>) and an isotropic diffusion matrix <math display="inline">\bar{\textbf D}_{iso}</math>

​

The global balancing diffusion matrix of Eq.(29a) can therefore be written as

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\bar {\textbf D} = \bar{\textbf D}_{iso} + {\textbf T}^T\bar{\textbf D}_\xi {\textbf T} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (31)

|}

​

Clearly, if the principal curvature direction <math display="inline">\vec {\xi }</math> is parallel to the velocity direction, then <math display="inline">u_\eta =0</math>, <math display="inline">k_\eta =0</math>, <math display="inline"> \bar{\textbf D}_{iso}={0}</math> and

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\bar{\textbf D}= \bar{\textbf D}_\xi =\left[\begin{array}{cc}{u_\xi h_\xi \over 2} &0\\0&0\end{array}\right] </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (32)

|}

​

where <math display="inline">u_\xi = \vert {u}\vert </math> and <math display="inline">h_\xi </math> is computed by Eq.(26a). Note that the method coincides with the standard SUPG approach in this case.

​

'''Remark 3'''. The global balance diffusion matrix <math display="inline">\bar  {\textbf D}</math> can be also computed from the expression of vector <math display="inline">\textbf h</math> in global axes as

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\bar  {\textbf D} ={1\over 2} {\textbf h}^T {\textbf u}\quad \hbox{with }{\textbf h}=\left\{\begin{array}{c}h_x\\h_y \end{array}\right\}  = {\textbf T}^T {\textbf h}' </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (33)

|}

​

where <math display="inline">{\textbf h}'=[h_\xi , h_\eta ]^T</math> and <math display="inline">{\textbf T}</math> is the transformation matrix of Eq.(24). The proof of this equivalence is given in the Appendix. In our computation however we have chosen to compute the balancing diffusion matrix <math display="inline">\bar  {\textbf D}</math> via Eqs.(23)&#8211;(29).

​

===4.1 Computation of the principal curvature axes for linear elements===

​

The principal curvature axes <math display="inline">\xi ,\eta </math> can be estimated as the eigenvectors of the curvature matrix

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>{\textbf C}= \left[\begin{array}{cc}\phi _x^{''} & \phi ^{''}_{xy}\\ \phi ^{''}_{xy} & \phi _y^{''}  \end{array}\right]\quad ,\quad \hbox{where } \phi _x^{''}={\partial ^2\phi \over \partial x^2}, \hbox{etc.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (34)

|}

​

In general matrix '''C''' varies within each element.

​

Computation of the second derivatives field for linear elements is not straightforward and it requires the nodal recovery of the first derivative field. This can be performed by using a simple nodal averaging procedure or more sophisticated superconvergence patch recovery techniques for the first derivative field [50,51].

​

Excellent results have been obtained by the authors in this work ''by approximating the principal 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 } \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 } \phi </math> is constant within the element. For four node quadrilaterals <math display="inline">{\boldsymbol \nabla } \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 that of vector <math display="inline">{\boldsymbol \nabla } \phi </math> computed at the element center.

​

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

​

===4.2 General iterative scheme===

​

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

​

'''Step 0'''  (SUPG step). At each integration point choose <math display="inline">{}^0{\boldsymbol \xi } ={\textbf u}</math>, i.e. the gradient direction is taken coincident with the velocity direction. Compute <math display="inline">{}^0{\boldsymbol \eta }, {}^1\bar  {\textbf D}'</math>, <math display="inline">{}^0\bar  {\textbf D}</math> and <math display="inline">{}^0{\textbf D}_G</math> from Eqs.(23&#8211;29). ''The  expression of the balancing diffusion matrix coincides  now precisely with the standard (linear) SUPG form''.

​

Solve for <math display="inline">{}^0\bar{\boldsymbol \phi }</math>.

​

Verify that the solution is stable. This can be performed by verifying that there are not undershoots or overshoots in the numerical results with respect to the expected physical values. If the SUPG solution is unstable, then implement the following iterative scheme.

​

For each iteration: 

​

'''Step 1'''  Compute at the element center. <math display="inline">{}^1{\boldsymbol \xi } = {\boldsymbol \nabla }^0\bar \phi </math>. Then compute <math display="inline">{}^1{\boldsymbol \eta } , {}^1\bar  {\textbf D}'</math>, <math display="inline">{}^1\bar  {\textbf D}</math> and <math display="inline">{}^1{\textbf D}_G</math>.

​

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

​

'''Step 2'''  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;width: 100%;" 

|-

| 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;white-space: nowrap;" | (35)

|}

​

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 in Eq.(35).

​

If condition (35) is not satisfied, start a new iteration and repeat steps 1 and 2 until convergence. Indexes 0 and 1 are replaced now by <math display="inline">i-1</math> and <math display="inline">i</math>, respectively.

​

'''Remark 4'''. The convergence of the iterative scheme of the previous section obviously depends on the “quality” of the standard SUPG solution obtained in Step 0. Clearly when boundary layers or  sharp transverse internal layers dominate the solution, the non linearity of the processes increases. The authors have found that the convergence of the scheme can be substantially improved in some problems if the following expression for the stabilizing dissipation matrix is used in Step 1

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>{\textbf D}_G = \beta ^i \bar{\textbf D}_G + (1-\beta ) {}^{i-1}\bar{\textbf D}_G  </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (36)

|}

​

where <math display="inline">\beta </math> is a relaxation factor such that <math display="inline">0\le \beta \le 1</math>.

​

Convergence of the iterative scheme was achieved in just two iteration in Examples 6.1&#8211;6.3 for values of <math display="inline">\beta </math> between 0.8 and 1. The best solution in Example 6.4 was found in five iterations for <math display="inline">\beta =0.3</math>. This is due to the higher non linearity of the process induced by the transverse boundary layers in this case.

​

'''Remark 5'''. The direct iterative scheme presented in Section 4.2 is obviously not the only option to solve the non linear equations resulting from the dependence of the stabilizing diffusion matrix with the principal curvatures of the solution.

​

A simple and economical alternative 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. This technique was used by Codina in <span id='citeF-11'></span>[[#cite-9|[9,33]]] for solving a similar class of convection-diffusion problems.

​

==5 EQUIVALENCE WITH SUPG AND SHOCK-CAPTURING TECHNIQUES==

​

As mentioned earlier, the SUPG formulation based on the use of a stabilization term of the form shown in Eq.(10) has severe limitations to provide stabilized solutions (i.e. without overshoots and undershoots) in the vicinity of sharp gradients of the solution transverse to the flow direction. The usual remedy in these cases is to introduce an additional stabilization term (the shock capturing term).

​

In esence all shock capturing formulations are equivalent to an split of the stabilization term in two parts. One part is taken equal to the standard  linear SUPG term. This term suffices to stabilize the effect of the gradient of the solution in the direction of the velocity. The second part is introduced in order to account for  gradients of the solution along a direction not aligned with the velocity. This second term (the shock capturing or transverse diffusion term) usually involves the velocity projection along the gradient direction. This term introduces therefore a non linearity in the solution process. The usual approach is to start by computing the linear SUPG solution, check if there are sharp transverse gradients which deserve further stabilization and, if so, introduce the shock capturing term and iterate until a stabilized solution is found in the whole domain.

​

Despite their effectiveness, most shock capturing methods lack of a general theoretical framework and can be viewed as “ad hoc” extensions of the SUPG method. A serious drawback of these methods is that, in some cases, the amount of transverse diffusion can not be properly controlled and    hence the final solution is either underdiffusive or overdiffusive.

​

The general stabilized formulation we propose here includes the best features of the SUPG and shock capturing techniques. The iterative scheme in fact follows the lines of the standard shock capturing methods <span id='citeF-49'></span>[[#cite-32|[32-36]]]. The innovation in the proposed formulation  lays in the definition of the shock capturing terms, which are deduced form the FIC governing equations written in the principal curvature directions. Numerical results obtained in all examples show that the method preserves the quality of the numerical solution in all situations by introducing the necessary  streamline and transverse dissipation.

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A similar shock capturing method of this kind was proposed by Oñate <span id='citeF-51'></span>[[#cite-38|[38]]] in the context of the FIC approach. There the characteristic length vector was split as the sum of a  vector along the velocity direction (the linear SUPG term) and a vector in the direction of the solution gradient (the non linear schock capturing term). Good results were obtained with this method for a variety of advection-diffusion problems <span id='citeF-51'></span>[[#cite-38|[38]]]. The formulation we present here is more general as it encompasses the SUPG method and the shock capturing strategy in a unified approach.

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We have shown in Remark 2 that the proposed approach coincides with the standard SUPG method (with no discontinuity capturing) when the principal direction <math display="inline">\vec {\xi }</math> is parallel to the velocity direction. This analogy is discussed further next.

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Let us split the characteristic vector <math display="inline">h</math> as the sum of two vectors along the velocity and the gradient directions respectively as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>{\textbf h}= h_s {{\textbf u}\over \vert {\textbf u}\vert } +h_t {{\boldsymbol \nabla }\hat \phi \over \vert {\boldsymbol \nabla } \hat \phi \vert } </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (37)

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where <math display="inline">h_s</math> and <math display="inline">h_t</math> are the characteristic lengths   along the velocity and gradient directions, respectively. When <math display="inline">h_s</math> is linear (i.e., independent of the solution) the first term in the r.h.s of Eq.(37) is the classical SUPG term. The second term depending on the solution gradient is the shock-capturing term.

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Eq.(37) allows to solve for <math display="inline">h_s</math> and <math display="inline">h_t</math> in terms of the components of <math display="inline">{u}, {\boldsymbol \nabla } \hat \phi </math> and <math display="inline">{h}</math> as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_s = {h_x \hat \phi _y^\prime - h_y \hat \phi _x^\prime \over u \hat \phi _y^\prime - v \hat \phi _x^\prime }\vert {\textbf u}\vert </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (38.a)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_t = {h_y u - h_x  v\over u \hat \phi _y^\prime - v \hat \phi _x^\prime }\vert {\boldsymbol \nabla }\hat \phi \vert </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (38.b)

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where <math display="inline">h_x</math> and <math display="inline">h_y</math> are computed from <math display="inline">h_\xi </math> and <math display="inline">h_\eta </math> by Eq.(33).

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Eqs.(38) clearly show that in the general case the expressions of <math display="inline">h_s</math> and <math display="inline">h_t</math> are a function of the numerical solution <math display="inline">\hat \phi </math>. Hence, for arbitrary orientations of the principal curvature direction <math display="inline">\vec {\xi }</math>, the stabilized method proposed can be interpreted as the combination of a ''non linear SUPG term and a non linear shock capturing term''. The introduction of this gradient-dependent non linearity in the SUPG term is one of the distinct features of the new formulation.

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When the principal curvature direction <math display="inline">\vec{\xi } </math>  is parallel to the velocity direction then <math display="inline">h_\eta =0</math> and

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>c_\alpha = {u\over \vert {\textbf u}\vert }, \quad s_\alpha ={v\over \vert {\textbf v}\vert }, \quad h_x = {uh_\xi  \over \vert {\textbf u}\vert }, \quad h_y ={vh_\xi  \over \vert {\textbf u}\vert }</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" |  (39a)

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Hence

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_s =h_\xi \quad \hbox{and } h_t=0</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" |  (39b)

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and the method coincides in this case with the standard linear SUPG approach, as expected.

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A case of practical interest is when the directions of <math display="inline">\textbf u</math> and <math display="inline">{\boldsymbol \nabla }\phi </math> are ''orthogonal''. This occurs for some specific distributions of the velocity and the solution fields or in the vicinity of sharp layers when the mesh is fine enough, or it has some specific orientation. In these cases we have

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_\xi = 0\quad ,\quad h_\eta \not =0 \,\, \hbox{(it is computed by Eqs.(25)--(27))}\,\, ,\,\, c_\alpha = - {v\over \vert {\textbf u}\vert }\quad \hbox{and} \quad s_\alpha ={u\over \vert {\textbf u}\vert } </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (40)

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Hence from Eq.(33)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_x = - h_\eta   {u\over \vert {\textbf u}\vert } \quad ,\quad  h_y = - h_\eta  {v\over \vert {\textbf u}\vert } </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (41.a)

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and from Eqs.(38)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>h_s = h_\eta \quad \hbox{ and }h_t =0 </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (41.b)

|}

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Once again, the linear SUPG method is  recovered and the stabilized formulation does not introduce any additional transverse dissipation, as should be expected.

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'''Remark 6'''. From above arguments we conclude that the simplified expression <math display="inline">{\textbf h}=h_s {{\textbf u} \over \vert {\textbf u}\vert }</math> with <math display="inline">h_s</math> being a non linear function of the solution leads to a non linear SUPG term. This has a similarity with the non linear SUPG formulation reported in <span id='citeF-28'></span>[[#cite-28|[28,31]]]. We have found that in general this assumption does not suffice to stabilize the solution in presence of high gradients transverse to the solution. In these cases the introduction of the term involving <math display="inline">h_t</math> in Eq.(37) is essential in order to obtain an improved solution.

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==6 EXAMPLES==

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The efficiency of the algorithm is verified in a number of examples of application including sharp gradients at the interior of the domain as well as in boundary layers. The examples show the ability of the algorithm to provide a stabilized and accurate solution in all cases in very few iterations. All units in the examples are  in the International System.

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In the examples solved a solution is consider ''unstable'' if <math display="inline">(1+\bar \varepsilon ) \phi ^p_{max} \le \bar \phi \le (1+\bar \varepsilon ) \phi ^p_{min}</math> where <math display="inline">\phi ^p_{max}</math> and <math display="inline">\phi ^p_{min}</math> are the maximum and minimum prescribed values of <math display="inline">\bar \phi </math> at the Dirichlet boundaries. The parameter <math display="inline">\bar \epsilon </math> has been taken equal to <math display="inline">10^{-2}</math>. This condition basically ensures that the solution has no overshoots or undershoots with respect to the prescribed boundary values.

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In all examples presented the classical SUPG results (with no discontinuity capturing)  are compared with those obtained with the FIC iterative scheme. Examples 6.1&#8211;6.3 were solved for a value of <math display="inline">\beta =1</math> in Eq.(36). A converged and accurate solution was obtained in all cases in just two iterations. The convergence process was quite insensitive to the value of <math display="inline">\beta </math> for values of <math display="inline">\beta </math> ranging between 0.8 and 1.0. We note that good stabilized results were also obtained in these examples using just ''one iteration''. However, a slight increase in the accuracy of the results was obtained using two iterations, while no significant increase in accuracy was reached by iterating any further.

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The FIC solution reported for Example 6.4 was obtained in five iterations for a value of <math display="inline">\beta =0.3</math> in Eq.(36).

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We finally note that most of the examples presented have been successfully solved in the past using different shock capturing techniques <span id='citeF-49'></span>[[#cite-32|[32-36]]]. In this paper the FIC results are compared with values obtained using the standard (linear) SUPG formulation. The basic aim of the examples presented here is to show that the FIC formulation integrates naturally the best features of both the SUPG and the shock capturing formulations.

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===6.1 Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source===

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Figure 2 shows the geometry of the analysis domain and the boundary conditions. The domain is a square of side equal to 10. Zero values of the variable <math display="inline">\phi </math> are prescribed at the inlet boundaries <math display="inline">x=0</math> and <math display="inline">y=0</math>, whereas <math display="inline">\phi =10</math> is prescribed at the outlet boundaries <math display="inline">x=10</math> and <math display="inline">y=10</math>. The isotropic diffusion <math display="inline">k=0.01</math> and the velocity vectors are taken parallel to the diagonal of the domain with <math display="inline">{u}={3\sqrt 2\over 2}[1,1]^T</math>. The problem has been solved with the structured mesh of <math display="inline">10\times 10</math> four node rectangles shown in the figure. The Peclet number along the velocity direction is <math display="inline">\gamma =300</math>.

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<div id='img-2'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

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|[[Image:Draft_Samper_426828278-Figure2.png|551px|Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source. SUPG and FIC solutions obtained with a structured mesh of 10×10 linear four node square elements]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 2:''' Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source. SUPG and FIC solutions obtained with a structured mesh of <math>10\times 10</math> linear four node square elements

|}

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Results in Figure 2 show contours of the numerical solution for the SUPG and the FIC iterative scheme proposed. FIC<math display="inline">^2</math> in the figures denotes results obtained after two iterations of the scheme described in Section 4.2. A value of <math display="inline">\beta =1.0</math> in Eq.(36) has been chosen. Note that the SUPG method yields incorrect negative values of <math display="inline">\phi </math> in the vicinity of the upper right corner, as expected, whereas the FIC<math display="inline">^2</math> results are physically correct over the domain. A plot of the distribution of the solution along the horizontal center line and the diagonal line is shown. The difference between the FIC and SUPG results along the diagonal line is clearly seen.

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Figure 3 shows results  for the same problem solved with a structurerd mesh of <math display="inline">2\times 10 \times 10</math> three  node triangular elements. The effect of the stabilization process in more visible in this case as the SUPG solution has more undershoots than for four node square elements.

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<div id='img-3'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

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|[[Image:Draft_Samper_426828278-Figure3.png|551px|Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source. SUPG and FIC solutions obtained with a structured mesh of 2×10×10 linear three node triangles]]

|- style="text-align: center; font-size: 75%;"

| colspan="1" | '''Figure 3:''' Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source. SUPG and FIC solutions obtained with a structured mesh of <math>2\times 10\times 10</math> linear three node triangles

|}

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The convergence of the iterative process using square and triangular elements is plotted in Figure 4 where the evolution of the values of <math display="inline">\bar \phi </math> at three different points within the analysis domain is plotted. The evolution of the norm of Eq.(35) with the number of iterations is also shown. Note that a converged solution (<math display="inline">\varepsilon <10^{-3}</math>) is obtained in just two iterations.

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<div id='img-4'></div>

{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"

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|[[Image:Draft_Samper_426828278-Figure4.png|551px|Square domain with uniform Dirichlet conditions, upward diagonal velocity and zero source. Convergence of the FIC solution for square and triangular elements]]