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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.		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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	==1 INTRODUCTION==		==1 INTRODUCTION==

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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.		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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	==1 INTRODUCTION==		==1 INTRODUCTION==

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		+	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 195 (13-16), pp. 1793–1825, 2006<br />
		+	doi: 10.1016/j.cma.2005.05.036
		+

	==Abstract==		==Abstract==

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 The authors thank S. Badia, C. Felippa, R. Löhner, R.L. Taylor and O.C. Zienkiewicz for many useful discussions.
-==Appendix A==
+==APPENDIX A==
 ===Computation of the balancing diffusion matrix <math>\overline{\textbf D}</math> in local and global axes===

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 The authors thank S. Badia, C. Felippa, R. Löhner, R.L. Taylor and O.C. Zienkiewicz for many useful discussions.
-==Appendix A. Computation of the balancing diffusion matrix <math>\overline{\textbf D}</math> in local and global axes==
+==Appendix A==
 The balancing diffusion matrix <math display="inline">\bar {\textbf D}</math> can be computed by transforming the local matrix <math display="inline">\bar{\textbf D}'</math> of Eq.(23) to global axes, or by using directly Eq.(33). We will prove next that both expressions yield the same system of governing differential equations, i.e. we will prove that

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 <div id="cite-6"></div>
-'''[6]''' Hughes T.J.R and Tezduyar, T.E. Finite element method for first order hyperbolic systems with particular emphasis on the compressible Euler equations. ''Comput. Meth. Appl. Mech. Engng.'', '''45''', 217&#8211;284, 1984.
+'''[[#citeF-6|[6]]]''' Hughes T.J.R and Tezduyar, T.E. Finite element method for first order hyperbolic systems with particular emphasis on the compressible Euler equations. ''Comput. Meth. Appl. Mech. Engng.'', '''45''', 217&#8211;284, 1984.
 <div id="cite-7"></div>
@@ Line 1,194: / Line 1,194: @@
 <div id="cite-20"></div>
-'''[20]''' Hughes T.J.R. Multiscale phenomena: Green functions,  subgrid scale models, bubbles and the origins of stabilized methods.  ''Comput. Methods Appl. Mech. Engrg'' 1995, '''127''', 387&#8211;401, 1995.
+'''[[#citeF-20|[20]]]''' Hughes T.J.R. Multiscale phenomena: Green functions,  subgrid scale models, bubbles and the origins of stabilized methods.  ''Comput. Methods Appl. Mech. Engrg'' 1995, '''127''', 387&#8211;401, 1995.
 <div id="cite-21"></div>
-'''[21]''' Brezzi F., Franca L.P., Hughes T.J.R. and  Russo A. <math>b=\int  g</math>. ''Comput. Methods Appl. Mech. Engrg.'', '''145''', 329&#8211;339, 1997.
+'''[[#citeF-21|[21]]]''' Brezzi F., Franca L.P., Hughes T.J.R. and  Russo A. <math>b=\int  g</math>. ''Comput. Methods Appl. Mech. Engrg.'', '''145''', 329&#8211;339, 1997.
 <div id="cite-22"></div>

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 ==1 INTRODUCTION==
-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]]. 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]], the Streamline Upwind Petrov-Galerkin (SUPG) method <span id='citeF-5'></span>[[#cite-5|5]], the Taylor-Galerkin method <span id='citeF-12'></span>[[#cite-12|12]], the generalized Galerkin method <span id='citeF-17'></span>[[#cite-17|17]], the Galerkin Least Square method and related approaches <span id='citeF-19'></span>[[#cite-19|19]], the Characteristic Galerkin method <span id='citeF-22'></span>[[#cite-22|22]], the Characteristic Based Split method <span id='citeF-24'></span>[[#cite-24|24]], the  Subgrid Scale method <span id='citeF-25'></span>[[#cite-25|25]], the Residual Free Bubbles method <span id='citeF-37'></span>[[#cite-37|37]], the Discontinuous Enrichment Method <span id='citeF-39'></span>[[#cite-39|39]] and the Streamline Upwind with Boundary Terms method <span id='citeF-40'></span>[[#cite-40|40]].  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|41]]. 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|49]]. 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.
+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.
 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.
-The formulation proposed is based in the so called Finite Calculus (FIC) approach <span id='citeF-50'></span>[[#cite-50|50]]. 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.
+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.
-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|50]], convection-diffusion-absorption [42,43], incompressible fluid flow <span id='citeF-47'></span>[[#cite-47|47]] and incompressible solid mechanics [48,49].
+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].
-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|50]].
+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]]].
-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|50]]. In <span id='citeF-51'></span>[[#cite-51|51]] 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.
+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.
 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.