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the paper is concerned with the spectral and computational analysis of effective preconditioners for linear systems arising from finite element approximations to the elliptic convection - diffusion problem @xmath0 with @xmath1 domain of @xmath2 . we consider a model setting in which the structured finite element partition is made by equi - lateral triangles . the interest of such a partition relies on the observation that automatic grid generators tend to construct equi - lateral triangles when the mesh is fine enough . the analysis is performed having in mind two popular preconditioned krylov methods . more precisely , we analyze the performances of the preconditioned conjugate gradient ( pcg ) method in the case of the diffusion problem and of the preconditioned generalized minimal residual ( pgmres ) in the case of the convection - diffusion problem . we define the preconditioner as a combination of a basic ( projected ) toeplitz matrix times diagonal structures . the diagonal part takes into account the variable coefficients in the operator of ( [ eq : modello ] ) , and especially the diffusion coefficient @xmath3 , while the ( projected ) toeplitz part derives from a special approximation of ( [ eq : modello ] ) when setting the diffusion coefficient to @xmath4 and the convective velocity field to @xmath5 . under such assumptions , if the problem is coercive , and the diffusive and convective coefficients are regular enough , then the proposed preconditioned matrix sequences have a strong clustering at unity , the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent , and the eigenvector matrices have a mild conditioning . the obtained results allow to show the optimality of the related preconditioned krylov methods . it is important to stress that interest of such a study relies on the observation that automatic grid generators tend to construct equi - lateral triangles when the mesh is fine enough . numerical tests , both on the model setting and in the non - structured case , show the effectiveness of the proposal and the correctness of the theoretical findings . the outline of the paper is as follows . in section [ sez : fem ] we report a brief description of the fe approximation of convection - diffusion equations and the preconditioner definition . section [ sez : clustering ] is devoted to the spectral analysis of the underlying preconditioned matrix sequences , in the case of structured uniform meshes . in section [ sez : numerical_tests ] , after a preliminary discussion on complexity issues , selected numerical tests illustrate the convergence properties stated in the former section and their extension under weakened assumption or in the case of unstructured meshes . a final section [ sez : conclusions ] deals with perspectives and future works . problem ( [ eq : modello ] ) can be stated in variational form as follows : @xmath6 where @xmath7 is the space of square integrable functions , with square integrable weak derivatives vanishing on @xmath8 . we assume that @xmath1 is a polygonal domain and we make the following hypotheses on the coefficients : @xmath9 the previous assumptions guarantee existence and uniqueness for problem ( [ eq : formulazione_variazionale ] ) and hence the existence and uniqueness of the ( weak ) solution for problem ( [ eq : modello ] ) .    for the sake of simplicity , we restrict ourselves to linear finite element approximation of problem ( [ eq : formulazione_variazionale ] ) . to this end , let @xmath10 be a usual finite element partition of @xmath11 into triangles , with @xmath12 and @xmath13 . let @xmath14 be the space of linear finite elements , i.e. @xmath15 the finite element approximation of problem ( [ eq : formulazione_variazionale ] ) reads : @xmath16 for each internal node @xmath17 of the mesh @xmath18 , let @xmath19 be such that @xmath20 , and @xmath21 if @xmath22 . then , the collection of all @xmath23 s is a base for @xmath24 . we will denote by @xmath25 the number of the internal nodes of @xmath18 , which corresponds to the dimension of @xmath24 . then , we write @xmath26 as @xmath27 and the variational equation ( [ eq : formulazione_variazionale_fe ] ) becomes an algebraic linear system : @xmath28 according to these notations and definitions , the algebraic equations in ( [ eq : modello_discreto ] ) can be rewritten in matrix form as the linear system @xmath29 where @xmath30 and @xmath31 represent the approximation of the diffusive term and approximation of the convective term , respectively . more precisely , we have @xmath32 where suitable quadrature formula are considered in the case of non constant coefficient functions @xmath33 and @xmath34 . as well known , the main drawback in the linear system resolution is due to the asymptotical ill - conditioning ( i.e. very large for large dimensions ) , so that preconditioning is highly recommended . hereafter , we refer to a preconditioning strategy previously analyzed in the case of fd / fe approximations of the diffusion problem @xcite and recently applied to fd / fe approximations @xcite of ( [ eq : modello ] ) with respect to the preconditioned hermitian and skew - hermitian splitting ( phss ) method @xcite . more precisely , the preconditioning matrix sequence @xmath35 is defined as @xmath36 where @xmath37 , i.e. , the suitable scaled main diagonal of @xmath38 and clearly @xmath38 equals @xmath30 . the computational aspects of this preconditioning strategy with respect to krylov methods will be discussed later in section [ sez : complexity_issues ] . here , preliminarily we want to stress as the preconditioner is tuned only with respect to the diffusion matrix @xmath30 : in other words , we are implicity assuming that the convection phenomenon is not dominant , and no stabilization is required in order to avoid spurious oscillations into the solution . moreover , the spectral analysis is performed in the non - hermitian case by referring to the hermitian and skew - hermitian ( hss ) decomposition of @xmath39 ( that can be performed on any single elementary matrix related to @xmath18 by considering the standard assembling procedure ) .    according to the definition , the hss decomposition is given by @xmath40 where @xmath41 since by definition , the diffusion term @xmath30 is a hermitian matrix and does not contribute to the skew - hermitian part of @xmath39 . notice also that @xmath42 if @xmath43 . more in general , the matrix @xmath44 is symmetric and positive definite whenever @xmath45 . indeed , without the condition @xmath46 , the matrix @xmath47 does not have a definite sign in general : in fact , a specific analysis of the involved constants is required in order to guarantee the nonnegativity of the term @xmath47 . moreover , the lemma below allows to obtain further information regarding such a spectral assumption , where @xmath48 indicate both the usual vector norms and the induced matrix norms .    _ @xcite _ [ lemma : normae ] let @xmath49 be the matrix sequence defined according to _ ( [ eq : def_e_nb])_. under the assumptions in _ ( [ eq : ipotesi_coefficienti ] ) _ , then we find @xmath50 with @xmath51 absolute positive constant only depending on @xmath52 and @xmath1 . the claim holds both in the case in which the matrix elements in _ ( [ eq : def_psi ] ) _ are evaluated exactly and whenever a quadrature formula with error @xmath53 is considered for approximating the involved integrals . hereafter , we will denote by @xmath54 , @xmath55 , the matrix sequence associated to a family of meshes @xmath56 , with decreasing finesse parameter @xmath57 . as customary , the whole preconditioning analysis will refer to a matrix sequence instead to a single matrix , since the goal is to quantify the difficulty of the linear system resolution in relation to the accuracy of the chosen approximation scheme . the aim of this section is to analyze the spectral properties of the preconditioned matrix sequences @xmath58 in the case of some special domains @xmath1 partitioned with structured uniform meshes , so that spectral tools derived from toeplitz theory @xcite can be successfully applied . the applicative interest of the considered type of domains will be motivated in the short . indeed , let @xmath59 be a @xmath60variate lebesgue integrable function defined over @xmath61 ^ 2 $ ] . by referring to the fourier coefficients of this function @xmath59 , called generating function , @xmath62 with @xmath63 , one can build the sequence of toeplitz matrices @xmath64 . the matrix @xmath65 is said to be the toeplitz matrix of order @xmath66 , @xmath67 , generated by @xmath59 . now , the spectral properties of the matrix sequence @xmath64 are completely understood and characterized in terms of the underlying generating functions ( see , e.g. , @xcite for more details ) . in the following , with respect to the latter claim , we will refer to the notion of equivalent generating functions . in such a respect , we claim that two nonnegative function @xmath59 and @xmath68 defined over a domain @xmath69 are equivalent if and only if @xmath70 and @xmath71 , where @xmath72 means that there exists a pure positive constant @xmath73 such that @xmath74 almost everywhere on @xmath69 . + ( a ) ( b )    the main motivation of this paper lies in the analysis of the template case reported in figure [ fig : esagono_esagoni].a , where we consider a partition into equilateral triangles . it is self - evident that such a problem represents just an academic example . however , it is a fact that a professional mesh generator will locally produce a partitioning , which is `` asymptotically '' similar to the one reported in the figure .    our goal is to prove the pcg optimality with respect to the diffusion problem approximation , i.e. , the number of iterations for reaching the solution , within a fixed accuracy , can be bounded from above by a constant independent of the dimension @xmath55 . some additional results about pgmres convergence properties in the case of the convection - diffusion problem approximation are also reported , both in the case of the hexagonal domain @xmath1 with a structured uniform mesh as in figure [ fig : esagono_esagoni].a and in the case @xmath75 with a structured uniform mesh as in figure [ fig : esagono_esagoni].b , so extending previous results proved in @xcite with respect to the phss method @xcite . the spectral analysis makes reference to the following definition .    _ @xcite _ [ def : cluster ] let @xmath76 be a sequence of hermitian matrices of increasing dimensions @xmath77 . the sequence @xmath76 is clustered at @xmath78 in the eigenvalue sense if for any @xmath79 , @xmath80 . the sequence @xmath76 is properly ( or strongly ) clustered at @xmath78 if for any @xmath81 the number of the eigenvalues of @xmath82 not belonging to @xmath83 can be bounded by a pure constant eventually depending on @xmath84 , but not on @xmath77 . in the case of non - hermitian matrices , the same definition can be extended by considering the complex disk @xmath85 , instead of the interval @xmath83 . we start by considering the simpler diffusion problem , i.e. , the analysis concerns the hermitian matrix sequences @xmath86 , @xmath87 . due to the very special choice of the domain @xmath1 , the matrices @xmath87 arising in the case @xmath88 with structured meshes as in figure [ fig : esagono_esagoni].a result to be a proper projection of toeplitz matrices generated by the function @xmath89 , @xmath90 ^ 2 $ ] , related to a bigger parallelogram shaped domain @xmath91 containing @xmath1 ( see figure [ fig : esagono_esagoni_parallelogrammi ] ) , i.e. , @xmath92 , with @xmath93 number of the internal nodes . here , @xmath94 , @xmath95 is a proper projection matrix simply cutting those rows ( columns ) referring to nodes belonging to @xmath91 , but not to @xmath1 . thus , the matrix is full rank , i.e. , @xmath96 , and @xmath97 . in the same way , we can also consider the toeplitz matrix with the same generating function arising when considering a smaller parallelogram shaped domain @xmath98 contained in @xmath1 , i.e. , @xmath99 , with @xmath100 , @xmath101 proper projection matrix and such that @xmath102 , and @xmath103 . these embedding arguments allow to bound , according to the min - max principle ( see @xcite for more details on the use of this proof technique ) , the minimal eigenvalue of the matrices @xmath104 as follows @xmath105    a first technical step in our spectral analysis concerns relationships between this generating function @xmath106 and the more classical generating function @xmath107 arising in the case of fe approximations on a square @xmath75 with friedrichs - keller meshes ( see figure [ fig : esagono_esagoni].b ) , or standard fd discretizations . it can be easily observed that these two function are equivalent , in the sense of the previously reported definition , since we have @xmath108 ^ 2.\ ] ] thus , because @xmath109 is a matrix - valued linear positive operator ( lpo ) for every @xmath77 @xcite , the toeplitz matrix sequences generated by this equivalent functions result to be spectrally equivalent , i.e. , for any @xmath77 @xmath110 furthermore , due to the strict positivity of every @xmath109 ( see @xcite for the precise definition ) , since @xmath111 is not identically zero , we have that @xmath112 is positive definite , and since @xmath113 is not identically zero , we find that @xmath114 is also positive definite . here , we are referring to the standard ordering relation between hermitian matrices , i.e. , the notation @xmath115 , with @xmath116 and @xmath117 hermitian matrices , means that @xmath118 is nonnegative definite . an interesting remark pertains to the fact that the function @xmath106 is the most natural one from the fe point of view , since no contribution are lost owing to the gradient orthogonality as instead in the case related to @xmath59 ; nevertheless its relationships with the function @xmath59 can be fully exploited in performing the spectral analysis . more precisely , from ( [ eq : lmin_proiezione ] ) and ( [ eq : rel_toeplitz ] ) and taking into account ( see e.g. @xcite ) that @xmath119 we deduce @xmath120 following the very same reasoning , we also find that @xmath121 since , by the embedding argument , both @xmath93 and @xmath122 are asymptotic to @xmath77 , it follows that @xmath123 . finally , following the same analysis for the maximal eigenvalue , we find @xmath124 where , by @xcite , we know that @xmath125 and hence the spectral condition number of @xmath104 grows as @xmath126 i.e. @xmath127 where the constant hidden in the previous relation is mild and can be easily estimated . it is worth stressing that the same matrix @xmath128 can also be considered in a more general setting . in fact , the matrix sequence @xmath86 can also be defined as @xmath129 , since again each internal node in @xmath1 is a vertex of the same constant number of triangles . therefore , by referring to projection arguments , the spectral analysis can be equivalently performed both on the matrix sequence @xmath86 and @xmath130 . no matter about this choice , we make use of a second technical step , which is based on standard taylor s expansions . [ lemma : taylor_expansion_esagoni ] let @xmath131 , with @xmath132 and @xmath1 hexagonal domain partitioned as in figure [ fig : esagono_esagoni].a . for any @xmath78 such that @xmath133 there exists a proper @xmath134 such that the taylor s expansions centered at a proper @xmath134 have the form @xmath135 where @xmath136 and @xmath137 are constants independent of @xmath57 . + -1.0 cm    in the cases at hand , the validity of this claim is just a direct check : the key point relies in the symmetry properties induced by the structured uniform nature of the considered meshes , both in the case of @xmath1 and @xmath91 . hereafter , we give some detail with respect to the case of the hexagonal domain partitioned as in figure [ fig : esagono_esagoni].a , see @xcite for the proof in the case @xmath75 . thus , let s consider the taylor s expansion centered at a proper @xmath134 . by calling @xmath138 and similarly denoting the derivatives of the diffusion coefficient , it holds that @xmath139 with @xmath140^t \|$ ] and @xmath141 denoting the barycenter of the triangle @xmath142 . by referring to the assembling procedure , we will choose as center of the related taylor s expansion in the case of figure [ fig : simmetrie].b-[fig : simmetrie].d the point marked by @xmath143 . the symmetric position of the involved triangle barycenters ( marked by @xmath144 ) , with respect to @xmath143 allows to claim ( [ eq : exp.a ] ) . the same holds true for ( [ eq : exp.b ] ) by considering each pertinent point @xmath143 in figure [ fig : simmetrie].a and , analogously , for ( [ eq : exp.c ] ) . [ lemma : taylor_expansion ] _ let s considering the following notation of scaled matrix @xmath145 of a given matrix @xmath116 . under the assumptions of _ lemma [ lemma : taylor_expansion_esagoni ] _ the following representation holds true @xmath146 where @xmath147 and @xmath148 are uniformly bounded in spectral norm and have the same pattern as @xmath149 . we are now ready to prove the pcg optimality . [ teo : clu+se_a_esagoni ] let @xmath86 and @xmath35 be the hermitian positive definite matrix sequences defined according to _ ( [ eq : def_a ] ) _ and _ ( [ eq : def_p ] ) _ in the case of the hexagonal domain partitioned as in figure [ fig : esagono_esagoni].a . under the assumptions in _ ( [ eq : ipotesi_coefficienti ] ) _ , the sequence @xmath150 is properly clustered at @xmath4 . moreover , for any @xmath77 all the eigenvalues of @xmath151 belong to an interval @xmath152 $ ] well separated from zero _ [ spectral equivalence property]_.    since @xmath153 , with @xmath94 such that @xmath96 , and @xmath97 , we have @xmath154^{-1 }   a_n(\a ) \\ % & = &    [ \pi \ , d_n^{\frac{1}{2}}(\a ) \ , \pi^t \ , \pi a_n(1 ) \ , \pi^t \ , \pi \ , d_n^{\frac{1}{2}}(\a ) \ , \pi^t ] ^{-1 } \pi \ , a_n(\a ) \ , \pi^t \\ % & = &    [ \pi \ , \pi^t \ , \pi \ , d_n^{\frac{1}{2}}(\a ) a_n(1 )   d_n^{\frac{1}{2}}(\a ) \ , \pi^t \ , \pi \ , \pi^t ] ^{-1 } \pi \ , a_n(\a ) \ , \pi^t \\ % & = &    [ \pi \ , d_n^{\frac{1}{2}}(\a ) a_n(1 )   d_n^{\frac{1}{2}}(\a ) \ , \pi^t   ] ^{-1 } \pi \ , a_n(\a ) \ , \pi^t \\ % & = &    [ \pi p_n^{-1}(\a ) \ , \pi^t   ] ^{-1 } \pi \ , a_n(\a ) \ , \pi^t.\end{aligned}\ ] ] since @xmath128 is full rank , it is evident that the spectral behavior of @xmath155^{-1}\pi a_n(\a ) \pi^t\ ] ] is in principle better than the one of @xmath156 , to which we can address the spectral analysis . thus , the proof technique refers to @xcite and the very key step is given by the relations outlined in ( [ eq : lmin_toeplitz_bottom ] ) and ( [ eq : lmin_toeplitz_top ] ) . it is worth stressing that the previous claims can also be proved by considering the sequence @xmath150 , instead of @xmath157 , simply by directly referring to the quoted asymptotical expansions . the interest may concern the analysis of the same uniform mesh on more general domains . moreover , the same spectral properties has been proved in the case of uniform structured meshes as in figure [ fig : esagono_esagoni].b in @xcite . the natural extension of the claim in theorem [ teo : clu+se_a_esagoni ] , in the case of the matrix sequence @xmath158 with @xmath159 , can be proved under the additional assumptions of lemma [ lemma : normae ] , in perfect agreement with theorem 5.3 in @xcite . [ teo : clu+se_rea_esagoni ] let @xmath158 and @xmath35 be the hermitian positive definite matrix sequences defined according to _ ( [ eq : def_rea ] ) _ and _ ( [ eq : def_p ] ) _ in the case of the hexagonal domain partitioned as in figure [ fig : esagono_esagoni].a . under the assumptions in _ ( [ eq : ipotesi_coefficienti ] ) _ , the sequence @xmath160 is properly clustered at @xmath4 . moreover , for any @xmath77 all the eigenvalues of @xmath161 belong to an interval @xmath152 $ ] well separated from zero _ [ spectral equivalence property]_.    the claim holds both in the case in which the matrix elements in _ ( [ eq : def_theta ] ) _ and _ ( [ eq : def_psi ] ) _ are evaluated exactly and whenever a quadrature formula with error @xmath53 is considered to approximate the involved integrals . the proof can be done verbatim as in @xcite , since the key point is proved by referring to relations in ( [ eq : lmin_toeplitz_bottom ] ) and ( [ eq : lmin_toeplitz_top ] ) . [ teo : clu+sb_ima ] let @xmath162 and @xmath35 be the hermitian matrix sequences defined according to _ ( [ eq : def_ima ] ) _ and _ ( [ eq : def_p ] ) _ in the case of the hexagonal domain partitioned as in figure [ fig : esagono_esagoni].a . under the assumptions in _ ( [ eq : ipotesi_coefficienti ] ) _ , the sequence @xmath163 is spectrally bounded and properly clustered at @xmath5 with respect to the eigenvalues . the claim holds both in the case in which the matrix elements in _ ( [ eq : def_psi ] ) _ are evaluated exactly and whenever a quadrature formula with error @xmath53 is considered to approximate the involved integrals . this result has been proved in @xcite with respect to friedrichs - keller triangulations , but it can easily be extended to the case of matrix sequences arising in the case of a fe partitioning as in figure [ fig : esagono_esagoni].a by using the same arguments .    on the basis of these two splitted spectral results , we can easily obtain the spectral description of the whole preconditioned matrix sequence @xmath164 , according to the theorem below . the proof technique refers to an analogous result proved in @xcite with respect to fd discretizations of convection - diffusion equations . [ teo : clua ] let @xmath165 and @xmath35 be the matrix sequences defined according to _ ( [ eq : def_a ] ) _ and _ ( [ eq : def_p ] ) _ , both in the case of the hexagonal domain @xmath1 with a structured uniform mesh as in figure [ fig : esagono_esagoni].a and in the case @xmath75 with a structured uniform mesh as in figure [ fig : esagono_esagoni].b    under the assumptions in _ ( [ eq : ipotesi_coefficienti ] ) _ , the sequence @xmath164 is properly clustered at @xmath166 with respect to the eigenvalues . in addition , these eigenvalues all belong to a uniformly bounded rectangle with positive real part , well separated from zero . the claim holds both in the case in which the matrix elements in _ ( [ eq : def_theta ] ) _ and _ ( [ eq : def_psi ] ) _ are evaluated exactly and whenever a quadrature formula with error @xmath53 is considered to approximate the involved integrals . a localization result for the eigenvalues of the sequence @xmath167 can be stated simply by referring to the properties of the field of values of the preconditioned matrix : in fact , for any @xmath17 @xmath168 where @xmath169 since for any @xmath17 @xmath170 the claimed localization result is obtained as a consequence of those previously proved in theorems [ teo : clu+se_rea_esagoni ] and [ teo : clu+sb_ima ] . moreover , the same localization results together with the claimed clustering properties allow to apply theorem 4.3 in @xcite and the proof is concluded . the subsequent result , giving estimates on the condition number of the eigenvector matrix of the preconditioned structure , is of interest in the study of the convergence speed of the gmres . in particular a logarithmic growth in number of iteration is associated to a polynomial bound in the spectral condition number : such a bound is precisely given below . [ lemmma : eigenvectors ] let @xmath3 and @xmath52 constant functions and let @xmath165 and @xmath35 be the matrix sequences defined as in theorem [ teo : clua ]    @xmath171 can be diagonalized as @xmath172 where the matrix of the eigenvectors @xmath173 can be chosen such that @xmath174 .    under the quoted assumption it holds that @xmath175 and @xmath31 is skew - symmetric . thus , we have @xmath176 with @xmath177 skew - symmetric matrix . therefore , the matrix @xmath178 is a normal normal , so that @xmath179 , where @xmath180 is diagonal and @xmath181 is unitary . consequently , @xmath182 with @xmath183 eigenvector matrix . due to the fact that @xmath181 is unitary , we have @xmath184 and finally @xmath185 by invoking the key relation ( [ eq : cond - an1 ] ) . the section is divided into two parts . in the first we briefly discuss the complexity features of our preconditioning proposals . in the second part we report and critically discuss few numerical experiments in which the meshes are both structured and unstructured , while the regularity features required by the theoretical analysis are somehow relaxed . first of all , we report some remarks about the computational costs of the proposed iterative procedure . the main idea is that such a technique is easily applicable . in fact , a krylov method is considered , so simply requiring a matrix vector routine for sparse matrices and a solver for the chosen preconditioner . + therefore , since the preconditioner is defined as @xmath186 , where @xmath187 , the solution of the linear system in ( [ eq : modello_discreto ] ) with matrix @xmath39 is reduced to computations involving diagonals and the matrix @xmath188 ( @xmath189 and @xmath188 for the diffusion problem , respectively ) . as well known , whenever the domain is partitioned by considering a uniform structured mesh this latter task can be efficiently performed by means of fast poisson solvers , among which we can list those based on the cyclic reduction idea ( see e.g. @xcite ) and several specialized algebraic or geometric multigrid methods ( see e.g. @xcite ) . in addition , the latter can be efficiently considered also in more general mesh settings . the underlying idea is that the main effort in devising efficient algorithms must be devoted only to this simpler problem with constant coefficient , instead of the general one . now , the effectiveness of the proposed method is measured by referring to the optimality definition below .    _ @xcite _ [ def : opt ] let @xmath190 be a given sequence of linear systems of increasing dimensions . an iterative method is _ optimal _ if    1 . the arithmetic cost of each iteration is at most proportional to the complexity of a matrix - vector product with matrix @xmath191 , 2 . the number of iterations for reaching the solution within a fixed accuracy can be bounded from above by a constant independent of @xmath192 . in such a respect theorem [ teo : clu+se_a_esagoni ] proves the optimality of the pcg method : the iterations number for reaching the solution within a fixed accuracy can be bounded from above by a constant independent of the dimension @xmath55 and the arithmetic cost of each iteration is at most proportional to the complexity of a matrix - vector product with matrix @xmath38 . moreover , theorem [ teo : clua ] proves that all the eigenvalues of the preconditioned matrix belong in a complex rectangle @xmath193 , \ \mathrm{im}(z ) \in [ -\hat{d},\hat{d}]\}$ ] , with @xmath194 , @xmath195 independent of the dimension @xmath55 . it is worth stressing that the existence of a proper eigenvalue cluster and the aforementioned localization results in the preconditioned spectrum can be very important for fast convergence of preconditioned gmres iterations ( see , e.g. , @xcite ) . finally , we want to give notice that the pcg / pgmres numerical performances do not get worse in the case of unstructured meshes , despite the lack of a rigorous proof .      before analyzing in detail selected numerical results , we wish to give technical information on the performed numerical experiments . we apply the pcg or the pgmres method , in the symmetric and non - symmetric case , respectively , with the preconditioning strategy described in section [ sez : fem ] , to fe approximations of the problem ( [ eq : modello ] ) . whenever required , the involved integrals have been approximated by means of the barycentric quadrature rule ( the approximation by means of the nodal quadrature rule gives rise to similar results , indeed both are exact when applied to linear functions ) . the domains of integration @xmath1 are those reported in figure [ fig : esagono_esagoni ] and we assume dirichlet boundary conditions . all the reported numerical experiments are performed in matlab , by employing the available ` pcg ` and ` gmres ` library functions ; the iterative solvers start with zero initial guess and the stopping criterion @xmath196 is considered . the case of unstructured meshes is also discussed and compared , together with various types of regularity in the diffusion coefficient . in fact , we consider the case of a coefficient function satisfying the assumptions ( [ eq : ipotesi_coefficienti ] ) . more precisely , the second columns in table [ tab : it - es123esagono_triangoliequilateri_m5 ] report the number of iterations required to achieve the convergence for increasing values of the coefficient matrix size @xmath55 when considering the fe approximation with structured uniform meshes as in figure [ fig : esagono_esagoni ] and with template function @xmath197 , @xmath198^t$].the subsequent meshes are obtained by means of a progressive refinement procedure , consisting in adding new nodes corresponding to the middle point of each edge . the numerical experiments plainly confirm the previous theoretical analysis in section [ sez : clustering ] : the convergence behavior does not depend on the coefficient matrix dimension @xmath55 .    as anticipated , despite the lack of corresponding theoretical results , we want to test the convergence behavior also in the case in which the regularity assumption on @xmath199 in theorems [ teo : clu+se_a_esagoni ] and [ teo : clua ] are not satisfied . the analysis is motivated by favorable known numerical results in the case of fd approximations ( see , for instance , @xcite ) or fe approximation with only the diffusion term @xcite . more precisely , we consider as template the @xmath200 function @xmath201 , the @xmath202 function @xmath203 , with @xmath204 or @xmath205 .    the number of required iterations is listed in the remaining columns in table [ tab : it - es123esagono_triangoliequilateri_m5 ] . notice that also in these cases with a @xmath200 or @xmath202 diffusion function , the iteration count does not depend on the coefficient matrix dimension @xmath55 . the same results are reported in table [ tab : it - es123quadrato_triangolirettangoli_m1 ] with respect to structured uniform meshes on the domain @xmath75 . furthermore , we want to test our proposal in the case of some unstructured meshes generated by triangle @xcite with a progressive refinement procedure . the first meshes in the considered mesh sequences are reported in figures [ fig : mesh_m6newnew ] and [ fig : mesh_m4 ] , respectively . tables [ tab : it - es123esagono_unstructured_m6newnew ] and [ tab : it - es123quadrato_unstructured_m4 ] report the number of required iterations in the case of the previous template functions . negligible differences in the iteration trends are observed for increasing dimensions @xmath77 . all these remarks are in perfect agreement with the outliers analysis of the matrices @xmath206 , with respect to a cluster at @xmath207 ( see some examples in figure [ fig : outlier_analysis ] ) .    to conclude the section we take into consideration the more realistic setting in which the meshes are generated by a specialized automatic procedure . according to this point of view , we have applied a spectral approximation of the matrix sequence @xmath208 in terms of product of low - cost matrix structures , i.e. , @xmath104 and @xmath209 , that carry the `` structural '' content of the variational problem and the specific `` informative '' content contained in the diffusion coefficient function @xmath210 , respectively . however , the matrix @xmath104 can be still not easy to handle . therefore , we go beyond in such idea by defining a preconditioner @xmath211 in which the matrix @xmath104 related to the unstructured mesh is replaced by a suitable projection of the toeplitz matrix @xmath212 , @xmath89 , @xmath90 ^ 2 $ ] ( see section [ sez : diffusion_equation ] ) . the numerical results are reported in table [ tab : it_perturbed_mesh ] . as expected , the number of iterations is a constant with respect to the dimension when the preconditioner @xmath213 is applied . nevertheless , for @xmath77 large enough , the same seems to be true also for the preconditioner @xmath211 . indeed , for increasing dimensions @xmath77 , the unstructured partitioning is more and more similar to the one given by equilateral triangles sketched in fig . [ fig : esagono_esagoni].a . a theoretical ground supporting these observation is still missing and would be worth in our opinion to be studied and developed . @xmath77 & @xmath214 & @xmath215 & @xmath216 + 37 & 3 & 4 & 5 + 169 & 3 & 4 & 5 + 721 & 3 & 4 & 4 + 2977 & 3 & 4 & 4 + 12097 & 3 & 4 & 4 + 48769 & 3 & 4 & 4 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 37 & 4 & 5 & 5 + 169 & 4 & 5 & 5 + 721 & 4 & 5 & 5 + 2977 & 4 & 5 & 5 + 12097 & 4 & 5 & 5 + 48769 & 4 & 5 & 5 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 81 & 3 & 4 & 4 + 361 & 3 & 4 & 5 + 1521 & 3 & 4 & 5 + 6241 & 3 & 4 & 5 + 25281 & 3 & 4 & 5 + 128881 & 3 & 4 & 5 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 81 & 4 & 5 & 5 + 3611 & 4 & 5 & 5 + 1521 & 4 & 5 & 5 + 6241 & 4 & 5 & 5 + 25281 & 4 & 5 & 5 + 128881 & 4 & 5 & 5 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 28 & 5 & 5 & 5 + 73 & 4 & 4 & 5 + 265 & 4 & 4 & 5 + 1175 & 4 & 4 & 5 + 4732 & 4 & 4 & 5 + 19288 & 4 & 4 & 5 + 76110 & 4 & 4 & 4 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 28 & 4 & 5 & 5 + 73 & 4 & 5 & 5 + 265 & 4 & 5 & 5 + 1175 & 4 & 5 & 5 + 4732 & 4 & 5 & 5 + 19288 & 4 & 5 & 5 + 76110 & 4 & 5 & 5 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 24 & 4 & 5 & 5 + 109 & 4 & 5 & 5 + 465 & 4 & 5 & 5 + 1921 & 4 & 5 & 5 + 7809 & 4 & 5 & 5 + 31489 & 4 & 5 & 5 +     @xmath77 & @xmath214 & @xmath215 & @xmath216 + 24 & 4 & 5 & 5 + 109 & 4 & 5 & 5 + 465 & 4 & 5 & 5 + 1921 & 4 & 5 & 5 + 7809 & 4 & 5 & 5 + 31489 & 4 & 5 & 5 +    [ tab : it - es123esagono_perturbed ]     @xmath77 & @xmath217 & @xmath218 + 37 & 4 & 8 + 169 & 4 & 10 + 721 & 4 & 11 + 2977 & 4 & 11 + 12095 & 4 & 13 + 48769 & 4 & 16 +     @xmath77 & @xmath217 & @xmath218 + 37 & 4 & 8 + 169 & 4 & 9 + 721 & 4 & 9 + 2977 & 4 & 10 + 12095 & 4 & 11 + 48769 & 4 & 13 +    -0.3 cm + -4.5 cm -0.5 cm -3.5 cm    -0.5 cm -0.5 cm + -0.5 cm -0.5 cm + -0.5 cm -0.5 cm as emphasized in the introduction it is clear that the problem in figure [ fig : esagono_esagoni].a is just an academic example , due to the perfect structure made by equi - lateral triangles . however , it is a fact that a professional mesh generator will produce a partitioning , which `` asymptotically '' , that is a for a mesh fine enough , tends to the one in figure [ fig : esagono_esagoni].a . the latter fact has a practical important counterpart , since the academic preconditioner @xmath219 is optimal for the real case with nonconstant coefficients and with the partitioning in b ) . a theoretical ground supporting these observations is still missing and would be worth in our opinion to be studied and developed in three directions : a ) giving a formal notion of convergence of a partitioning to a structured one , b ) proving spectral and convergence results in the case of an asymptotically structured partitioning , c ) extending the spectral analysis in the case of weak regularity assumptions . other possible developments include the case of higher order finite element spaces : it would be intriguing to find an expression of the underlying toeplitz symbol as a function of the different parameters in the considered finite element space ( degrees of freedom , polynomial degree , geometry of the mesh ) , and this could be done uniformly in @xmath220 dimension , i.e. for equation ( [ eq : modello ] ) with @xmath221 , @xmath222 . 10 o. axelsson , m. neytcheva , _ the algebraic multilevel iteration methods  theory and applications . _ in _ proceedings of the second international colloquium on numerical analysis _ ( plovdiv , 1993 ) , 1323 , vsp , 1994 . z. bai , g.h . golub , m.k . ng , _ hermitian and skew - hermitian splitting methods for non - hermitian positive definite linear systems . _ siam j. matrix anal . 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the paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a finite element approximation to diffusion - dominated convection - diffusion equations . 
 we consider a model setting in which the structured finite element partition is made by equi - lateral triangles . under such assumptions , 
 if the problem is coercive , and the diffusive and convective coefficients are regular enough , then the proposed preconditioned matrix sequences exhibit a strong clustering at unity , the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent , and the eigenvector matrices have a mild conditioning . 
 the obtained results allow to show the optimality of the related preconditioned krylov methods . 
 the interest of such a study relies on the observation that automatic grid generators tend to construct equi - lateral triangles when the mesh is fine enough . 
 numerical tests , both on the model setting and in the non - structured case , show the effectiveness of the proposal and the correctness of the theoretical findings .    matrix sequences , clustering , preconditioning , non - hermitian matrix , krylov methods , finite element approximations    65f10 , 65n22 , 15a18 , 15a12 , 47b65