Inf sup testing of upwind methods

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Met. Engng 000; 48: Inf sup testing of upwind metods Klaus-Jurgen Bate 1; ;, Dena Hendriana 1, Franco Brezzi and Giancarlo Sangalli 1 Department of Mecanical Engineering; Massacusetts Institute of Tecnology; 77 Massacusetts Avenue; Cambridge; MA 0139; U.S.A. Istituto di Analisi Numerica del Consiglio Nazionale delle Ricerce; 7100 Pavia; Italy SUMMARY We propose inf sup testing for nite element metods wit upwinding used to solve convection diusion problems. Te testing evaluates te stability of a metod and compactly displays te numerical beaviour as te convection eects increase. Four discretization scemes are considered: te standard Galerkin procedure, te full upwind metod, te Galerkin least-squares sceme and a ig-order derivative articial diusion metod. Te study sows tat, as expected, te standard Galerkin metod does not pass te inf sup tests, wereas te oter tree metods pass te tests. Of tese metods, te ig-order derivative articial diusion procedure introduces te least amount of articial diusion. Copyrigt? 000 Jon Wiley & Sons, Ltd. KEY WORDS: convection-dominated ow; upwinding; inf sup condition 1. INTRODUCTION Finite element metods perform best in solving elliptic problems. Wen te metods are used for yperbolic problems, diculties are encountered. Here, te one-dimensional convection diusion equation is used, as a model problem, to study tese diculties. Wen te Peclet number is small, te elliptic part of te convection difusion equation is dominant; on te oter and, wen te Peclet number is large, te yperbolic part of te equation is dominant. In solving te convection diusion problem, te nite element metod based on te standard Galerkin formulation gives an excellent solution wen te Peclet number is low but gives articial oscillations in te solution wen te Peclet number is ig. Tese oscillations sow tat te metod is unstable in solving te yperbolic type of problem. Upwind metods ave been developed to overcome tis diculty and various nite element discretizations using upwinding are stable in solving convection diusion problems wit ig Peclet numbers. Te nite element procedure wit upwinding sould be stable and accurate to solve ig Peclet number problems. However, no upwind metod gives as yet totally satisfactory results [1]. Te accuracy is not satisfactory because eiter te results contain oscillations or tey are too diusive. Correspondence to: Klaus-Jurgen Bate, Department of Mecanical Engineering, Massacusetts Institute of Tecnology, 77 Massacusetts Avenue, Cambridge, MA 0139, U.S.A. support@adina.com Received 7 October 1998 Copyrigt? 000 Jon Wiley & Sons, Ltd. Revised 19 August 1999

2 746 K. J. BATHE ET AL. In general, te inf sup condition is a crucial requirement to be satised for te stability of a nite element metod. Te inf sup condition as been extensively used to analyse te stability of nite element formulations in solid mecanics and for Stokes ow [; 3]. Here, we extend te use of te inf sup condition to te stability analysis of nite element formulations for convection diusion problems. Te major diculty in testing an upwind metod for te solution of convection-dominated problems lies in tat te test as to measure te solution errors in te interior of te domain and near te boundary. Wereas te solution is smoot in te interior, it is igly non-smoot near te boundary. Ideally, we would use norms tat can accurately measure errors in te interior and near te boundary. We did not nd a norm tat does so and leads to tractable computations in te numerical evaluation of te inf sup condition. For tis reason, we propose in tis paper a testing wic considers rst te wole domain using te H 1 -norm modied by te Peclet number, and ten considers a reduced domain (disregarding te boundary layer) using te original H 1 -norm. Te testing is employed to study te eectiveness of an upwind metod and is terefore useful in researc to establis more ecient tecniques. Usually, te performance of an upwind metod is evaluated by solving an example problem and evaluating te solution of te problem. If te solution contains some oscillations, te upwind metod is considered not to perform well. Te inf sup testing proposed erein evaluates te performance of an upwind metod in a more compreensive manner tan to just measure te oscillations in te solutions. Te test compactly describes te stability of an upwind metod as te Peclet number and element size are varied. In tis study, we consider four discretization scemes; te standard Galerkin procedure, te full upwind metod, te Galerkin least-squares metod and a ig-order derivative articial diusion metod. First, we briey review te inf sup condition and develop te governing equations of te numerical inf sup testing. Ten we coose a one-dimensional test problem, derive appropriate norm denitions for eac discretization metod and apply te testing to te solution scemes.. THE INF SUP CONDITION AND INF SUP TESTING Consider a general problem in given Hilbert spaces U and W wit a bilinear form a(; ) dened on U W. Te rst argument in te bilinear form a( ; ) is a solution function and te second argument is a weigting function. We dene te following spaces: U = W = u u L } L (Vol);k=1; ; 3; u = g on S k u u L } L (Vol);k=1; ; 3; u =0 on S k Actually, to be precise, U is not a linear space, but an ane manifold tat can be tougt of as obtained by translating te linear space W.

3 INF SUP TESTING OF UPWIND METHODS 747 were g is te Diriclet boundary condition function applied on S u and L (Vol) is te space of square integrable functions in te volume, Vol, of te body considered, L (Vol) = u u is dened in Vol and Vol u dvol = u L (Vol) + } Given a linear functional b( ) from W to R, we ave for te continuous problem: Find U suc tat a(; )=b( ) W (1) wit b( )=(f; ), were f is te forcing term. Te nite-dimensional subspaces of U and W are dened as follows: U = u u L } L (Vol);k=1; ; 3; u Q n (Vol (m) ); u = g on S k W = u u L } L (Vol);k=1; ; 3; u Q n (Vol (m) ); u =0 on S k were Q n (Vol (m) ) denotes te nt-order polynomial function in element m. An approximate solution of Equation (1) is obtained by solving te following nite-dimensional problem: Find U suc tat a( ; )=b( ) W () wit b( )=(f; ). Let us introduce a norm S for measuring te size of te solution functions and a norm T for measuring te size of te weigting functions. In general, we ave te following relation [ 4]: ( S 6 1+ k ) m inf S (3) U were k m is obtained from te continuity equation of te continuous space a(; )6k m S T ; W (4) Te continuity equation simply states tat te bilinear form a(; ) beaves normally. Also, is obtained from te inf sup condition of te nite-dimensional spaces inf W a( ; ) sup 0 (5) W S T See footnote ; now U is actually an ane manifold.

4 748 K. J. BATHE ET AL. To prove tat inequalities (4) and (5) imply inequality (3), consider te following derivation. From inequality (5) wit =, for any U we ave a( ; ) S 6 sup W T Using te triangle inequality we tus ave a( ; )+a( ; ) = sup W T a( ; ) = sup W T k m S T 6 sup W T = k m S S 6 S + S 6 k m S + S ( = 1+ k ) m S wic proves inequality (3). Here k m is given by te problem considered (and as an upper bound by te given pysics), and sould be independent of critical pysical constants (tat would make 0), te mes parameter and te solution of te problem. Note tat we use te inequality relations given in References [ 4] wit dierent norms still to be selected for te solution and weigting functions. For te moment, let us assume tat we ave identied appropriate norms and proceed wit te evaluation of te inf sup value. Te value of cannot easily be obtained analytically, especially wen we consider a sequence of irregular meses. Here, we evaluate te inf sup expression (inequality (5)) using a numerical metod tat is similar to te metod given in References [; 5]. We now need to consider te non-symmetric bilinear form a( ; ). In matrix form, te general Equation () can be written as Find x R n suc tat Ax = b (6) were in general A is an n n non-symmetric matrix and b R n. Inequality (5) becomes, for a given mes, inf W sup T AW (W T SW) 1= ( T T ) 1= = n 0 (7)

5 INF SUP TESTING OF UPWIND METHODS 749 were S and T are symmetric matrices of te norm operators S and T ; W and are vectors tat contain te nodal values of and and is to be independent of and te aforementioned pysical quantities. To evaluate te left-and side of inequality (7), let us dene and Hence, (W; )= (W; ^)= We use te Caucy Scwarz inequality wit te norm denition: T AW (W T SW) 1= ( T T ) 1= (8) T = L T L; ^ = L (9) ^TL T AW (W T SW) 1= (^T^) 1= (10) ^TL T AW 6 ^ L T AW (11) ( N v = vi i=1 ) 1= =(v T v) 1= We note tat in relation (11) equality olds for ^ = L T AW; ence Noting tat sup we consider te following eigenproblem: Terefore, (W; )= (WT A T L 1 L T AW) 1= (W T SW) 1= (1) A T L 1 L T A = A T (L T L) 1 A = A T T 1 A (13) (A T T 1 A)x = Sx (14) inf W sup (W; )= 1= min (15) were min is te smallest eigenvalue of eigenproblem (14). Hence, for a given formulation, pysical constants and nite-dimensional spaces, te value of n is equal to 1= min. In te inf sup testing, we would terefore consider a sequence of meses and measure min.if tis eigenvalue does not tend to zero, te solution metod is stable and optimal in te discretization errors measured in te norm used in Equation (3). Te testing is performed like in te inf sup test for te incompressible problem proposed in References [; 5].

6 750 K. J. BATHE ET AL. Te key point is tat appropriate norms must be selected, for wic te matrices A; T and S in Equation (14) are calculated. Te requirement for te S-norm is tat S sould be bounded in order for te inequality (3) to make sense, and te norm sould be strong enoug to measure te errors in te solution. Clearly, te H 1 -norm cannot be used because H 1 as Pe, were Pe is te Peclet number, Pe = vl=, wit v te caracteristic velocity, L te caracteristic lengt and te diusivity of te uid. Hence, we must modify tis norm and we propose two ways to proceed. In te rst approac, we use a modied H 1 -norm by introducing te Peclet number suc tat te norm beaves well even wen Pe. For example, for te full upwind metod we use for a one-dimensional problem (see Section 3.1) S = Pe ( ) (16) We refer to testing using tis approac as testing wit a modied H 1 -norm. We sall see tat for certain spatial discretizations, owever, te norm in Equation (16) does not measure te accuracy of solution suciently well wen coarse meses are used. Te diculty in using te H 1 -norm stems from te eect of te boundary layers. Hence, our second approac is to simply not include te boundary layer region in te norm and use te true H 1 -norm in te rest of te domain. In tis case, we cannot claim tat Equation (3) is applicable and we are not using Equation (4), but we simply measure te stability of te solution using f s = inf L ( ;f) (17) L were ( ;f) is a pair of (solution, forcing term) as in Equation (), and L denotes te L -norm not including te boundary layer region. Te stability of te solution sceme is clearly not aected by te spatial boundary conditions (te eect of wic could be subtracted as usual []), and ence we use f L = sup W b( ) L a( ; ) = sup W L (18) and s = inf W a( ; ) sup W L L (19) Now comparing Equation (19) wit te expressions in Equations (5) and (15), we realize tat te same eigenvalue problem in Equation (14) sould be solved for te smallest eigenvalue min to obtain for a given discretization te value of s. Tat is, denoting by s n te value of s for a given formulation, pysical constants and nite-dimensional spaces, te value of s n is equal to 1= min.

7 INF SUP TESTING OF UPWIND METHODS 751 Te inf sup testing is performed as in te modied H 1 -norm testing, but using Equation (19), we refer to te procedure as testing wit te H 1 -norm excluding te boundary layer. 3. MODEL PROBLEM, NORMS AND MATRICES FOR THE INF SUP TESTING In tis section, we apply te inf sup testing derived in Section for upwind metods to a convection diusion problem. Te selection of te norm denitions used for eac upwind metod is described. Consider te non-dimensionalized convection diusion problem in one dimension (described in Figure 1) wit te governing equation 1 d Pe + d = 0 in 0 x 1 (0) were is te temperature, Pe is te Peclet number, Pe = vl= were L; v; are te domain lengt, te given uid ow velocity and te termal diusivity. Te boundary conditions are (0) = 0 and (1)=1 In tis specic case, sould be independent of Pe and te mes parameter. Here, we consider te case wen te convective term is dominating, Pe 1, and its limit case wen Pe. Te exact solution for te problem is = exp(pe x) 1 exp(pe) 1 For te Galerkin metod, te full upwinding and te Galerkin least-squares metod [; 6], we discretize te domain uniformly using linear elements. Terefore, we ave te spaces U = W = u u L L (Vol); u Q 1 (Vol (m) ); u = g on S u u u L L (Vol); u Q 1 (Vol (m) ); u =0 on S u were Q 1 (Vol (m) ) denotes te linear function in element m. Figure 1. Domain and boundary conditions for te test problem.

8 75 K. J. BATHE ET AL. For te ig-order derivative articial diusion metod, we discretize te domain uniformly using quadratic elements [1; 7]. Hence, we ave te spaces U = u u L L (Vol); u Q (Vol (m) ); u = g on S u W = u u L L (Vol); u Q (Vol (m) ); u =0 on S u were Q (Vol (m) ) denotes te quadratic function in element m Modied H 1 -norm testing We derive in tis section te norms and matrices for te modied H 1 -norm testing Standard Galerkin metod. Te standard Galerkin metod for te convection diusion Equation (0) is []: Find U suc tat ( d 1 d Pe + ) d =0 W (1) were te integration sign sall denote from now on te integration over te uid domain. Te norm denitions are determined by te continuity equation in te continuous space. Hence, we ave ( 1 Pe + ) ( ) 1= ( 1 6 Pe Pe [ ] 1= [ ( 1 6 Pe Pe +Pe ) 1= ) ] 1= +Pe terefore k m =1 S = Pe ( ) 1 T = +Pe Pe Te norm value of te exact solution in te S-norm is 1 S = 0 Pe = exp( Pe) 1 (exp(pe) 1)

9 INF SUP TESTING OF UPWIND METHODS 753 Terefore, as Pe, S 1 and te norm value is bounded. Te element matrices of te standard Galerkin metod for te inf sup test are terefore ( 1 A = Pe HT ;xh ;x + H T H ;x were H is te vector containing te interpolation functions. ) () S = Pe HT ;xh ;x (3) ( ) 1 T = Pe HT ;xh ;x +PeH T H (4) Full upwind metod. Using te same solution and weigting function spaces as for te standard Galerkin metod, te full upwind metod for te convection diusion Equation (0) is []: Find U suc tat ( d 1 Pe + ) d + } d =0 W (5) were is te normalized element lengt (using L = 1). Te continuity equation of te full upwind metod in te continuous space is ( 1 Pe + ) } [ ] 1= [( ) Pe Pe + Pe + 4 [ ] 1= [ ( ) 1 6 Pe Pe + Pe + 4 +Pe ] 1= } ] 1= +Pe Terefore, we ave k m =1 S = Pe ( ) 1 T = Pe + Pe + 4 +Pe }

10 754 K. J. BATHE ET AL. Te element matrices of te full upwind metod for te inf sup test are ( 1 A = S = Pe + ) H T;xH ;x + H T H ;x } (6) Pe HT ;xh ;x (7) ) } H;xH T ;x +PeH T H (8) ( 1 T = Pe + Pe Galerkin least-squares metod. Using te same solution and weigting function spaces as for te standard Galerkin metod, te Galerkin least-squares formulation for te convection diusion equation is [; 6]: Find U suc tat ( ) d 1 Pe + d + } d =0 W (9) wit = ( ) Pe cot 1 (30) Pe Tis value of gives te nodally exact solution. Substituting Equation (30) into Equation (9), we ave [ ( )} ] d Pe cot d + d = 0 (31) Te continuity equation of te Galerkin least-squares metod in te continuous space is Terefore, we ave [ ( )} ] Pe cot + [ ] 1= [ ( ) ] 1= Pe Pe 6 Pe 4 cot +Pe [ ] 1= [ ( ( ) ) ] 1= Pe Pe 6 Pe 4 cot +Pe k m =1 S = Pe ( ) Pe Pe T = 4 cot +Pe }

11 INF SUP TESTING OF UPWIND METHODS 755 Te element matrices of te Galerkin least-squares metod for te inf sup test are ( ) 1 A = Pe + H T;xH } ;x + H T H ;x (3) S = Pe HT ;xh ;x (33) Pe T = 4 cot ( Pe ) } H;xH T ;x +PeH T H (34) Hig-order derivative articial diusion metod. Te ig-order derivative articial diusion metod for te convection diusion equation is [1; 7] Find U suc tat ( ) d 1 d Pe + d + d 1 m l 9 m For a regular mes, =dr = =, so we ave ( d 1 d Pe + d + d 3 7 m l m 3 d dr =0 W (35) d ) = 0 (36) Te continuity equation of te ig-order derivative articial diusion metod in te continuous space is m l m 6 m ( 1 Pe + + l m [ 6 and we ave m ) 3 7 ( 3 7 Pe 3 + ) 1= ( 3 Pe 3 Pe 3 7 l m ( 3 7 Pe 3 + ) ] 1= [ Pe 3 m +Pe +Pe 3 ) 1= l m ( 3 Pe 3 7 ) ] 1= +Pe +Pe 3 k m =1 S = m T = m l m l m ( 3 7 Pe 3 + ) Pe 3 ( 3 Pe 3 ) +Pe +Pe 3 7

12 756 K. J. BATHE ET AL. Te norm value of te exact solution in te S-norm is Terefore, as Pe, S = 1 0 ( 3 7 Pe 3 + ) Pe 3 = 3 (exp( Pe) 1) (exp( Pe) 1) + 144(exp(Pe) 1) Pe (exp(pe) 1) S wic is bounded for small element lengt. Te element matrices of te ig-order derivative articial diusion metod for te inf sup test are ( ) 1 A = Pe HT ;xh ;x + H T 3 H ;x + 7 HT ;xxh ;xx (37) S = Pe 3 HT ;xh ;x + T = (Pe H;xH T ;x +Pe 3 H T H) Pe 3 HT ;xxh ;xx (38) 3 Pe 3 7 HT ;xxh ;xx (39) 3.. H 1 -norm testing excluding boundary layer For te H 1 -norm testing excluding te boundary layer, te same element matrices A as for te modied H 1 -norm testing are used (see Section 3.1), but te matrices representing te norms are simpler. Namely, in eac case te element matrices are (see Equation (19)) S = T = H T ;x H ;x (40) H T H (41) Of course, an important point is tat te elements in te boundary layer must not be included in te assemblage of te complete system matrix S. Tis matrix, terefore, as zero rows and columns corresponding to te degrees of freedom in te boundary layer. Eac suc zero row and column results into an innite eigenvalue, wic owever does not aect our result tat min sould be computed, see Equation (15) []. We also note tat wit tis coice of te S-norm for te continuous problem S 0 (4)

13 INF SUP TESTING OF UPWIND METHODS 757 Figure. Inf sup value curves as te mes is coarsened wit Pe = 100 for te modied H 1 -norm testing. and inf sup 1 0 ((1=Pe) + ) S T 1 (43) as Pe tends to, provided te inmum is taken only over te functions H 1 suc tat (1=Pe) + belongs to L (0; 1). Indeed, for every xed smoot te sup equals (1=Pe) + L = S. Always for smoot tis quotient tends (as Pe tends to ) to L = S, wic is always bigger tan or equal to 1, but equals 1 wenever vanises identically in te interval (1 ; 1). Tis continuous inf sup property justies te use of te H 1 -norm testing excluding te boundary layer for discrete problems. 4. INF SUP TEST RESULTS We consider te model problem of Section 3 and perform te inf sup tests described in Section. Figures and 3 sow te results using te modied H 1 -norm test. In Figure, te Peclet number of te problem is 100 and te number of elements is increased. In Figure 3, te element lengt is (number of elements = 16) and te Peclet number is increased. Figure sows tat as te mes is made coarser, te inf sup value corresponding to te standard Galerkin metod decreases. Tis trend indicates tat te metod does not pass te inf sup test wic means tat te metod does not satisfy te inf sup condition (Equation (5)). Te metod is predicted to be unstable wen we use too coarse a mes. Tis instability is displayed by oscillations in te temperature solution. Figure also sows tat as te mes is made ner, te inf sup value corresponding to te standard Galerkin metod approaces a xed value. Of course, as known, te metod is stable wen te element Peclet number.

14 758 K. J. BATHE ET AL. Figure 3. Inf sup value curves as Pe is increased wit =0:065 for te modied H 1 -norm testing. Figure sows tat as te mes is coarsened, te inf sup values corresponding to te full upwind metod, te Galerkin least-squares metod and te ig-order derivative articial diusion metod are bounded from below. Tis indicates tat tese metods pass te inf sup test and are predicted to be stable. Note tat as te mes is made ner, all curves approac te value of te Galerkin metod. Te inf sup values corresponding to te full upwind metod are iger tan tose of te oter curves. Tis indicates tat te metod is te most diusive. Te ig-order derivative upwind metod is stable and yields te smallest articial diusion. Comparing te slopes of te inf sup value curves in te coarse meses, we observe tat te Galerkin metod as te largest (absolute value) slope. Tis corresponds to te igest convergence rate of te metod (being of second order). Figure 3 sows te inf sup values as te Peclet number increases. Te results in Figure 3 lead to te same conclusions as obtained from Figure. In tis study, we ave used an even number of elements to discretize te domain. If an odd number of elements is used, te inf sup value corresponding to te standard Galerkin metod is bounded from below as we coarsen te mes, or as te Peclet number increases. Tis is because te metod is stable wen an odd number of elements is used, altoug igly inaccurate in te interior domain wen te mes is coarse. Te nite element solution for a given Peclet number using a coarse mes is a saw toot solution for wic Equation (3) is still satised. However, wen te mes is ne, te rigt-and side in Equation (3) is small and te saw toot response is not satisfying Equation (3) and terefore not a solution. Hence, an even number of elements sould be used for te one-dimensional problem in tis inf sup test. Te reason wy te saw-toot solution is not identied as a igly inaccurate solution lies in te norms used. Te H 1 -norm modied by te Peclet number includes te boundary layer but to include it, te norm contains te factor (1=Pe) 1=. Te result is tat te norm does not provide a

15 INF SUP TESTING OF UPWIND METHODS 759 Figure 4. Inf sup value curves as te mes is coarsened wit Pe = 100 for te H 1 -norm testing excluding boundary layer. suciently ard measure for te errors in te numerical saw-toot solution wen a coarse mes is used. Indeed te S-norm of a basis function wit value 1 at one internal node and 0 at te oter nodes is =(Pe ) 1=, wic is small for coarse grids and large Peclet number. We next apply te H 1 -norm testing excluding te boundary layer. Figure 4 presents te results for Pe = 100 as we coarsen te mes. We observe tat te curves for te tree stable metods considered are bounded from below; on te oter and, in te case of te standard Galerkin metod te inf sup value, measured on te value of one, decreases as te mes becomes coarse. Te same observations are valid for Figure 5, were we consider a xed mes of 18 elements (19 in te case of te Galerkin metod wit an odd number of elements) and te Peclet number is increased. Furter teoretical and numerical results using te H 1 -norm testing excluding te boundary layer are given in Reference [8]. 5. CONCLUDING REMARKS Our objective in tis paper was to develop an inf sup testing procedure for measuring te eectiveness of stabilization metods used in te nite element solution of convection-dominated ows. We rst reviewed te inf sup condition for te problem area considered and ten developed te numerical testing procedure. To demonstrate te tecnique, we applied te testing to a one-dimensional model problem wen various well-known nite element discretization tecniques are used.

16 760 K. J. BATHE ET AL. Figure 5. Inf sup value curves as Pe is increased wit =0:0556 for te H 1 -norm testing excluding boundary layer ( =0:056 for Galerkin metod wit odd number of elements). Te procedures developed in tis paper are quite general, but te eectiveness of te inf sup testing depends on te norms used. Te diculty wit convection-dominated ow problems is tat te solution is smoot in te interior of te domain, but can be igly non-smoot near te boundary. Te norm used for te solution function sould ideally be able to measure equally well any errors in te smoot and non-smoot parts of te solution. We ave not succeeded as yet to identify an ideal suc norm tat can also be employed eectively in te computations. Hence, wile we ave used adequate norms to perform te inf sup testing, we leave te searc for more eective norms for furter researc. REFERENCES 1. Hendriana D, Bate KJ. On upwind metods for parabolic nite elements in incompressible ows. International Journal for Numerical Metods in Engineering 000; 47: Bate KJ. Finite Element Procedures. Prentice Hall: Englewood Clis, NJ, Brezzi F, Fortin M. Mixed and Hybrid Finite Element Metods. Springer: Berlin, Brezzi F, Bate KJ. A discourse on te stability conditions for mixed nite element formulations. Computer Metods in Applied Mecanics and Engineering 1990; 8: Capelle D, Bate KJ. Te inf sup test. Computers and Structures 1993; 47: Huges TJR, Franca LP, Hulbert GM. A new nite element formulation for computational uid dynamics: VIII. te Galerkin=least-squares metod for advective-diusive equations. Computer Metods in Applied Mecanics and Engineering 1989; 73: Hendriana D, Bate KJ. On a parabolic quadrilateral nite element for compressible ows. Computer Metods in Applied Mecanics and Engineering, in press. 8. Sangalli G. Numerical evaluation of nite element metods for convection diusion problems. Calcolo, to appear.