Error estimation and adjoint based refinement for an adjoint consistent DG discretisation of the compressible Euler equations

Transcription

1 Int. J. Comting Science and Matematics, Vol., Nos. /3/4, Error estimation and adjoint based refinement for an adjoint consistent DG discretisation of te comressible Eler eqations R. Hartmann Institte of Aerodynamics and Flow Tecnology, German Aerosace Center (DLR), Branscweig, Germany Institte of Scientific Comting, TU Branscweig, Germany Abstract: Adjoint consistency in addition to consistency is te key reqirement for discontinos Galerkin discretisations to be of otimal order in L as well as measred in terms of target fnctionals. We rovide a general framework for analysing adjoint consistency and introdce consistent modifications of target fnctionals. Tis framework is ten sed to derive an adjoint consistent discontinos Galerkin discretisation of te comressible Eler eqations. We demonstrate te effect of adjoint consistency on te accracy of te flow soltion, te smootness of te discrete adjoint soltion and te a osteriori error estimation wit resect to aerodynamical force coefficients on locally refined meses. Keywords: discontinos Galerkin discretisation; adjoint consistency; comressible Eler eqations; continos adjoint roblem; discrete adjoint roblem. Reference to tis aer sold be made as follows: Hartmann, R. (007) Error estimation and adjoint based refinement for an adjoint consistent DG discretisation of te comressible Eler eqations, Int. J. Comting Science and Matematics, Vol., Nos. /3/4, Biograical notes: Ralf Hartmann is Head of a Helmoltz jnior researc gro develoing discontinos Galerkin metods at te Institte of Aerodynamics and Flow Tecnology at DLR, Branscweig. He is one of te tree ators of te C++ finite element library deal.ii, see Bangert et al. (005, 006). He comleted is PD in te nmerical metods gro of Professor R. Rannacer, University of Heidelberg, in 00. Since ten e as worked on develoing discontinos Galerkin metods towards aerodynamical alications. Introdction Adjoint consistency, in addition to consistency, is te key reqirement for DG discretisations to be of otimal order in L as well as measred in terms of target fnctional. Frtermore, adjoint consistency is closely related to te smootness of discrete adjoint soltions. A tyical sitation is given by te SIPG and NIPG metods, i.e., te symmetric and te asymmetric interior enalty DG discretisation of Coyrigt 007 Inderscience Enterrises Ltd.

2 08 R. Hartmann ellitic PDEs. Wereas adjoint soltions based on te adjoint inconsistent NIPG metod are discontinos between element interfaces, were te jms in te adjoint soltions even ersist as te mes is refined, see Harriman et al. (003), te adjoint soltions based on te adjoint consistent SIPG metod are essentially continos, see also Harriman et al. (004) for an aroriate modification of a flx fnctional for ellitic PDEs. Frtermore, we refer to te toic of asymtotic adjoint consistency of stabilised finite element metods (Hoston et al., 000). Recently, L (005) and L and Darmofal (006) roosed a secific discretisation of te bondary flxes and te target fnctional to recover an adjoint consistent DG discretisation of te comressible Eler eqations. In tis aer, we rovide a general framework for analysing adjoint consistency of DG discretisations. Tis framework incldes te derivation of te continos adjoint roblems, inclding continos adjoint bondary conditions. Frtermore, it incldes te derivation of te discrete adjoint roblems, te derivation of rimal and adjoint residals and te discssion of te conditions nder wic residals vanis for te exact rimal and adjoint soltions, resectively. Wereas te standard DG discretisation of te comressible Eler eqations, e.g., Bassi and Rebay (997) and Hartmann and Hoston (00), is not adjoint consistent, we se te otlined framework to derive te adjoint consistent discretisation of te comressible Eler eqations originally stated in L (005) and L and Darmofal (006). Te roosed framework is generic and articlarly sefl for deriving adjoint consistent discretisations of more comlex non-linear roblems. We note tat tis framework is alied to te analysis and derivation of adjoint consistent DG discretisations of several linear and non-linear roblems, inclding te Poisson s eqation and te comressible Navier-Stokes eqations in Hartmann (006a). We also note tat te crrent blication is an extended version of Hartmann (006b). In tis aer we resent nmerical exeriments demonstrating te effect of adjoint consistency on te accracy of an inviscid comressible flow arond te NACA00 airfoil. Frtermore, we demonstrate te effect of adjoint consistency on te smootness of discrete adjoint soltions as well as on te a osteriori error estimation wit resect to aerodynamical force coefficients on adatively refined meses. Frter nmerical exeriments sow tat de to te smootness of te discrete adjoint soltion te effort of te standard error estimation aroac can be significantly redced. Given a flow soltion comted wit olynomial degree, te aroac in e.g., Hartmann and Hoston (00, 006b) comtes te discrete adjoint soltion wit olynomial degree +. Here, we sow tat for te adjoint consistent discretisation, sing a atc-wise interolation to olynomials of degree + of a discrete adjoint soltion comted wit te same degree as te flow soltion is sfficient for giving reasonable a osteriori error estimation reslts. Consistency and adjoint consistency On an bonded oen domain d wit bondary we consider following non-linear roblem N = 0 in, B = g on, ()

3 Error estimation and adjoint based refinement for an adjoint consistent 09 were N is a non-linear differential (and Frécet-differentiable) oerator and B is a ossibly non-linear bondary oerator. Let J( ) be a non-linear target fnctional J ( ) = j ( )d x + j ( ) d s, wit Frécet derivative J [ ]( w ) = j [ ] w d x + j [ ] w d s, were j ( ) and j ( ) may be non-linear wit derivatives j and j. Here, denotes te Frécet derivative and te sqare bracket [ ] denotes te state were te Frécet derivative is evalated. Ten, te adjoint roblem to eqation () is given by ( N [ ])* z = j [ ] in, ( B [ ])* j z = [ ] on, (3) were (N'[])* and (B'[])* denote te adjoint oerators to N'[] and B'[], resectively. Let be sbdivided into sae-reglar meses = {} consisting of elements and let V be a discrete fnction sace on. Finally, let roblem () be discretised as follows: Find sc tat (, v) = 0 v V, (4) were is a semi-linear form. Ten, te discretisation (eqation (4)) is said to be consistent if te soltion to te rimal roblem () satisfies following eqation: ( v, ) = 0 v V, (5) were V is an aroriately cosen fnction sace. Te discretisation (eqation (4)) is said to be adjoint consistent if te exact soltion z to te adjoint roblem (3) satisfies following eqation: [ ]( w, z) = J [ ]( w) w V, (6) were '[] denotes te Frécet derivative of wit resect to its first argment. In oter words, a discretisation is adjoint consistent if te associated discrete adjoint roblem is a consistent discretisation of te continos adjoint roblem. Finally, we note, tat te definition in eqation (6) of adjoint consistency for non-linear roblems, see also L and Darmofal (006), generalises te definition of linear adjoint consistency, in tat for linear roblems and target fnctionals, it redces to te definition of linear adjoint consistency as given in e.g., Arnold et al. (00). For analysing consistency of a DG discretisation, we rewrite eqation (4) in te following rimal residal form: Find sc tat κ κ R ( ) vd x + r ( ) vd s + r( ) vds = 0 v, κ\ κ were R( ), r( ) and r ( ) denote te element, interior face and bondary residals, resectively. According to eqation (5), te discretisation is consistent if te exact soltion to eqation () satisfies () (7)

4 0 R. Hartmann κ κ R ( ) vd x + r ( ) vd s + r( ) vds = 0 v, κ\ κ (8) wic olds if satisfies R ( ) = 0 in κ, κ, r ( ) = 0 on κ \, κ, (9) r ( ) = 0 on. To analyse adjoint consistency, we rewrite te discrete adjoint roblem: Find z sc tat [ ]( w, z ) = J [ ]( w) w V, (0) in adjoint residal form: Find z sc tat κ κ w R*( z )d x + w r*( z )d s + w r* ( z )ds = 0 w, κ\ κ () were R*(z ), r*(z ) and r*( z ) denote te element, interior face and bondary adjoint residals, resectively. According to eqation (6), te discretisation (eqation (4)) is adjoint consistent if te exact soltion z to eqation (3) satisfies κ κ w R*( z)d x + w r*( z)d s + w r* ( z)ds = 0 w, wic olds if z satisfies κ\ κ R*( z) = 0 in κ, κ, r*( z) = 0 on κ \, κ, (3) r* ( z) = 0 on. Given a target fnctional of te form (eqation ()), we see tat R*(z ) deends on j ( ), and r*( z ) deends on j ( ). For obtaining adjoint consistent discretisations, it is, in some cases, see e.g., te following section, necessary to modify te target fnctional as follows J ( ) = J( i( )), (4) were i( ) is a vector-valed fnction and will be secified in Section 3. A modification of a target fnctional is called consistent if J ( ) = J( ) olds for te exact soltion to te rimal roblem (). Tereby, te modification in eqation (4) is consistent if te exact soltion satisfies i() =. Altog te tre vale of te target fnctional is ncanged, te comted vale J( ) of te target fnctional is modified, and more imortantly, J [ ] differs from J'[ ]. Tis modification can be sed to recover an adjoint consistent discretisation. ()

5 Error estimation and adjoint based refinement for an adjoint consistent 3 Te adjoint consistency analysis In tis section we erform te adjoint consistency analysis for te DG discretisation of te comressible Eler eqations. To tis end, we consider te two-dimensional stationary comressible Eler eqations = ( ) 0 in, (5) sbject to varios bondary conditions, e.g., sli-wall bondary conditions at solid wall bondaries w =, were a vanising normal velocity B = v n = vn + vn= 0 on W (6) is imosed. In two sace-dimensions, te vector of conservative variables and te convective flxes () = (f (), f ()) are defined by = (, v, v, E), ρv ρv ρv + ρvv f = f = ρvv ρv+ ρhv ρhv ( ) and ( ), were, v = (v, v ), and E denote te density, velocity vector, ressre and secific total energy, resectively. Additionally, H is te total entaly given by H = E + ( /ρ) = e + (/)v + ( /ρ), were e is te secific static internal energy, and te ressre is determined by te eqation of state of an ideal gas = ( )e, were = c /c v is te ratio of secific eat caacities at constant ressre, c, and constant volme, c v ; for dry air, =.4. Te most imortant target qantities in inviscid comressible flows are te ressre indced drag and lift coefficients, c d and c l, defined by J ( ) = j ( )d s = d s, C n (7) W were j() = (/C )n on w and j() 0 elsewere. Here, C M l c l l = (/) γ = (/) γ( v / ) = (/) ρ v, were M denotes te Mac nmber, c te sond seed defined by c = /, l denotes a reference lengt, and is given by d = (cos(), sin()) or l = ( sin(), cos()) for te drag and lift coefficient, resectively. Sbscrits, indicate free-stream qantities. In order to derive te continos adjoint roblem, we mltily te left and side of eqation (5) by z, integrate by arts and linearise abot to obtain ( ( [ ]( w)), z) = ( [ ]( w), z) + ( n [ ]( w), z), were [] denotes te Frécet derivative of wit resect to. Tereby, te variational formlation of te continos adjoint roblem is given by: Find z sc tat ( w,( [ ]) z) + ( w,( n [ ]) z) = J [ ]( w),

6 R. Hartmann for all w, and te continos adjoint soltion z satisfies following roblem in strong form ( [ ]) z = 0 in, ( n [ ]) z = j [ ] on. (8) Using () n = (0, n, n, 0) on W, and te definition of j( ) in eqation (7) we obtain [ ](0, n, n,0) z = [ ] n on, C wic redces to te bondary conditions of te adjoint comressible Eler eqations, 3 W ( B [ ])* z = n z + n z = n on. (9) C We note tat, in general, it is nclear weter te rimal roblems (5) and (6), and te adjoint roblems (8) and (9), are well-osed. Terefore, in te sbseqent analysis we mst assme tat tese roblems are well-osed. Frtermore, we note tat te analysis assmes sfficient reglarity of te rimal and adjoint soltions. Before introdcing te DG discretisation of eqation (5) we define te finite element sace V of discontinos iecewise vector-valed olynomial fnctions of degree >0 by 4 4 = { [ L ( )] : κ σκ [ ( κ)] if κ = c, and 4 v ˆ ˆ ˆ κ σκ κ κ = s κ V v v W ˆ ˆ ˆ [ ( )] if ; }, α were : san{ ˆ = x :0 αi i, =,}, san{ ˆ : 0 }, ˆ = x α α c is te nit sqare and ŝ is te nit triangle. On interior edges e = ' between two adjacent ± elements and ', by v κ (v ± for sort) we denote te traces of v taken from witin te interior of and ', resectively. Frtermore, we define te jm of by = +. Ten, te DG discretisation of eqation (5) is given by: Find sc tat (, v) ( ): vd x+ (,, n ) v ds κ \ κ (, ( ), n ) v ds = 0, for all v V, were and may be any Liscitz continos, consistent and conservative nmerical flx fnctions, see e.g., Hartmann and Hoston (00), aroximating te normal flx n ( ). takes into accont te ossible discontinities + of at element interfaces. On te bondary, may deend on te interior trace + and a consistent bondary fnction ( ). We note tat may be different from. In fact, we will see below tat, deending on te secific coice of te DG discretisation (eqation (0)) is eiter adjoint consistent or not. Using integration by arts we obtain eqation (7) were te rimal residals are given by (0)

7 Error estimation and adjoint based refinement for an adjoint consistent 3 R ( ) = ( ) in κκ,, r ( ) = n ( ) (,, n ) on κ \, κ, r ( ) = n ( ) (, ( ), n ) on. Given te consistency of te nmerical flx, (w, w, n) = n (w), and te consistency of te bondary fnction, i.e., () = for te exact soltion to eqation (5), we find tat satisfies R ( ) = 0 in κκ,, r ( ) = 0 on κ \, κ, r ( ) = 0 on. We conclde tat eqation (0) is a consistent discretisation of eqation (5). Given te target fnctional J( ) defined in eqation (7) wit Frécet derivative, J []( ), te discrete adjoint roblem is given by eqation (0), were [ ]( w, z ) ( [ ] w): z dx ( (,, n ) w + (,, n )w ) z ds κ \ κ ( (, ( ), n ) + (, ( ), n ) ( )) w z d s. + Here v ( v +, v +, n ) and v ( v +, v, n ) denote te derivatives of te flx fnction (,, ) wit resect to its first and second argments, resectively. As te nmerical flx is conservative, (v, w, n) = (w, v, n), we conclde ( vwn,, ) = ( vwn,, ) w = ( vw,, n) = + ( wv,, n). w Using tis, te discrete adjoint roblem (0) is rewritten as follows: Find z tat for all roblem ( [ ] w): z d x+ (,, n ) w z ds κ \ κ s J () sc + ( (, ( ), n ) + (, ( ), n ) ( )) w z d = [ ]( w), w V. We see tat te discrete adjoint soltion z mst satisfy following ( [ ]) z = 0 in κκ,, () sbject to inter-element conditions + + ( + (,, n )) z = 0 on κ\, κ, (3) and bondary conditions

8 4 R. Hartmann ( + + ( )) z = j [ ] on, (4) in a weak sense, were : = (, ( ), n ) and : = (, ( ), n ). Comaring te discrete adjoint bondary condition (4) and te continos adjoint bondary condition in eqation (8), we see, tat not all coices of give rise to an adjoint consistent discretisation. In fact, we reqire to ave te following roerties: In order to incororate bondary conditions in te rimal discretisation (0), mst deend on ( + ), ence 0. Frtermore, we reqire + = 0, as oterwise te left and side of eqation (4) involves two smmands wic is in contrast to eqation (8). Finally, we recall tat is consistent, ( vvn,, ) = n ( v), and conclde tat is given by (, ( ), n) = n ( ( )). Emloying a modified target fnctional J ( ) = J( i( )), eqation (4) yields ( n ( [ ( )]) ( )) z = j [ i( )] i ( ). (5) We find te modification i( ) = ( ) wic is consistent as i() = () = olds for te exact soltion. Tereby, eqation (5) redces to + + ( n [ ( )]) z = j [ ( )], (6) wic reresents a discretisation of te continos adjoint bondary condition in eqation (8). In order to obtain a discretisation of te adjoint bondary condition at solid wall bondaries (eqation (9)), we reqire B ( + ) = 0 on W. Tis condition is satisfied by n nn 0 ( ) = on, W (7) 0 nn n wic originates from by sbtracting te normal velocity comonent of, i.e., v = (v, v ) is relaced by v = v (v n)n wic ensres tat te normal velocity comonent vanises, v n = 0. In smmary, let be given by eqation (7) and and J be defined by (, ( ), n) = n ( ), J ( ) = J ( ), (8) were ( ): = ( ( )), J ( ): = J( ( )) and j ( ): = j( ( )), ten te discrete adjoint roblem (0) is given by: Find z sc tat ( [ ]( w)): z d x+ (,, n ) w z ds κ \ κ ( n [ ]) w z d s = J [ ]( w), for all w V. Hence, we ave te adjoint residal form (eqation ()) were te adjoint residals are given by (9)

9 Error estimation and adjoint based refinement for an adjoint consistent 5 R*( z) = ( [ ]) z in κκ,, + + r*( z) = ( + (,, n )) z on κ \, κ, + r* ( z ) = j [ ] ( n [ ]) z on. In articlar, te discretisation (eqation (0)) togeter wit eqation (8) is adjoint consistent as te exact soltion z to te continos adjoint roblem (8) satisfies R*( z) = 0 in κκ,, r*( z) = 0 on κ \, κ, (30) r* ( z) = 0 on. We note, tat te standard DG discretisations of te comressible Eler eqations, see e.g., Bassi and Rebay (997) and Hartmann and Hoston (00, 006a), among several oters, take te same nmerical flx fnction on te bondary as in te interior of te + + domain, and simly relace in (,, n) by te bondary fnction ( ) + + reslting in (, ( ), n). Frtermore, te definition of in e.g., Bassi and Rebay (997) and Hartmann and Hoston (00) based on v = v (v n)n ensres a vanising average normal velocity, vn = (/ )( v+ v ) n= 0. However, v n = 0 + and B ( ) = 0, as reqired in eqation (6), is not satisfied. Tereby, te DG discretisation based on te standard coice of and is not adjoint consistent. In fact, already te nmerical exeriments in Hartmann and Hoston (00) indicated large gradients i.e., an irreglar adjoint soltion near solid wall bondaries. Recently, in L (005) and L and Darmofal (006), it as been demonstrated for an inviscid comressible flow over a bm, tat a discretisation based on eqation (8) is adjoint consistent and gives rise to a smoot discrete adjoint soltion. 4 Nmerical exeriments In tis section, we will demonstrate te effect on te smootness of te discrete adjoint soltion wen emloying te adjoint consistent discretisation based on eqation (8) in comarison to te standard or classical aroac of sing + + (, ( ), n), and J ( ), (3) i.e., of sing te same nmerical flx on te bondary as in te interior of te domain and evalating an nmodified fnctional. Frtermore, we sow te effect of te smootness of te adjoint soltion on te a osteriori error estimation, see Hartmann and Hoston (00). To tis end, we revisit te M = 0.5, = 0 inviscid flow arond te NACA00 airfoil test case considered in Hartmann and Hoston (00). In Figre we comare te flow soltions for te standard and te adjoint consistent DG discretisations and see no visible difference. However, wen comaring te adjoint soltions, see Figre, we notice tat te discrete adjoint soltion to te standard DG discretisation is irreglar near and stream of te airfoil. In contrast to tat, te adjoint soltion to te adjoint consistent discretisation is entirely smoot.

10 6 R. Hartmann Figre M = 0.5; α = 0 inviscid flow arond te NACA00 airfoil: Mac isolines of te (rimal) flow soltion to (left) te standard and (rigt) te adjoint consistent DG discretisation Figre M = 0.5; α = 0 inviscid flow arond te NACA00 airfoil: z isolines of te discrete adjoint soltion z to (left) te standard and (rigt) te adjoint consistent DG discretisation In Tables and we collect te data of a goal-oriented (adjoint-based) adative refinement algoritm (Hartmann and Hoston, 00) tailored to te accrate comtation of te drag coefficient c d for te standard and te adjoint consistent DG discretisation, resectively. Here, we sow te nmber of elements and degrees of freedom, te tre error J() J( ) based on te tre vale J() = 0, te estimated error based on te aroximate error reresentation η = (, z ) =η (3) κ κ wit z, and te vale η = ηκ κ after alying te trianglar ineqality, togeter wit te corresonding effectivity indices = / J() J( ) and θ = η/ J( ) J( ). Wereas bot istories of adatively refined meses are almost

11 Error estimation and adjoint based refinement for an adjoint consistent 7 identical, we see tat on all corresonding meses te adjoint consistent discretisation is, by a factor of abot.3.4, more accrate tan te standard DG discretisation. Frtermore, we see tat in bot cases te error estimation is qite accrate, reresented by te fact tat is close to one. Te error estimation for te adjoint consistent discretisation is imroved on coarser grids bt sligtly degraded on finer meses as comared to te standard DG discretisation. Finally, and in Table differ significantly, indicating tat te standard DG discretisation cases an extensive error cancellation, wic is also seen in te irreglar discrete adjoint soltion. In contrast to tat te discrete adjoint soltion to te adjoint consistent DG discretisation is smoot, no cancellation effects occr and and in Table coincide. Table Error estimation for te standard DG discretisation wit z # el. # DoFs J() J( η = ηκ ) κ θ η = ηκ κ e e e e 03.53e e e e e e 04.63e e e e e θ Table Error estimation for te adjoint consistent DG discretisation wit z # el. # DoFs J() J( ) η = η κ κ θ η = ηκ κ θ e e e e e e e 04.39e e e e e e e e We note tat, de to N(, v ) = 0 for all v, see eqation (0), te aroximate error reresentation (3) vanises and is ence, rendered seless if evalated based on a discrete adjoint soltion z aroximated in te same discrete fnction sace as te flow soltion. One aroac, sed in e.g., Hartmann and Hoston (00, 006b) and in te comtations of Tables and, is to evalate te error reresentation based on + z, i.e., on a discrete adjoint soltion comted wit an increased olynomial degree. As te adjoint roblem reqires te soltion of one linear roblem in comarison to several linear roblems associated wit an imlicit soltion metod for te flow roblem, tis is a viable aroac. However, in order to redce te additional effort associated wit te soltion of te adjoint roblem, in te following we test an alternative aroac originally roosed in Becker and Rannacer (996) of comting te discrete adjoint soltion z wit te same olynomial degree as te flow soltion and sing a atc-wise interolation to V. Here, sing a atc we denote te aggregation of e.g., for qadrilateral elements. Tyically, in a ierarcically refined mes, a moter element is slit into for cild elements wic togeter form a atc. On tis atc te

12 8 R. Hartmann discrete fnction reresented by a discontinos iecewise olynomial of degree, in fact by for olynomials of degree, is ten interolated to one olynomial of degree + on tat atc. In Tables 3 and 4 we collect te data of te goal-oriented adative refinement algoritm analogos to Tables and bt now based on te atc-wise interolation of z to V instead of z. First, we note tat te istories of te local mes refinement in Tables 3 and 4 are qite similar to te istories in Tables and. From tis we see tat te qality of te adjoint based indicators and reslting adative mes refinement is not significantly decreased wen evalated wit z as comared to te more costly aroac of comting z. However, for te standard DG discretisation in Table 3 te accracy of te error estimation is significantly redced reresented by te fact, tat te effectivity indices θ vary between 0.5 and 0.7. In contrast to tat te indices θ for te adjoint consistent DG discretisation in Table 4 are, excet for te coarsest mes, constant (close to /3), sowing tat te error estimation as te same beavior nder mes refinement as te tre error. Tis difference is attribted to te fact tat te atc-wise interolated aroximation of te adjoint soltion is significantly more accrate for te adjoint consistent discretisation were te discrete adjoint soltion is smoot tan for te standard discretisation wit irreglar discrete adjoint soltions. Table 3 Error estimation for te standard DG discretisation wit interolated to V z atc-wise # el. # DoFs J() J( η = ηκ ) κ θ η = ηκ κ e 03.44e e e e e e e e e e e e e e θ Table 4 Error estimation for te adjoint consistent DG discretisation wit interolated to V z atc-wise # el. # DoFs J() J( η = ηκ ) κ θ η = ηκ κ e e e e e e e 04.5e e e e e e 05.86e e θ

13 Error estimation and adjoint based refinement for an adjoint consistent 9 5 Conclding remarks In tis aer, we ave rovided a general framework for analysing te adjoint consistency roerty of DG discretisations, introdced consistent modifications of target fnctional, and derived an adjoint consistent DG discretisation for te comressible Eler eqations originally roosed in L (005) and L and Darmofal (006). Tis inclded te analysis of te continos adjoint eqations and te adjoint bondary conditions of te comressible Eler eqations as well as te discrete adjoint eqations and te discretisation of bondary conditions and target fnctionals. Nmerical exeriments ave demonstrated tat te discrete flow soltions are visibly indistingisable for te adjoint consistent comared to a standard (i.e., adjoint inconsistent) discretisation, bt tere is a clear difference in te discrete adjoint soltions: Wile te discrete adjoint soltion to te standard discretisation is irreglar near and stream of te airfoil, te discrete adjoint soltion to te adjoint consistent discretisation is entirely smoot. In fact, te discrete adjoint soltion for an adjoint consistent discretisation is a consistent discretisation of te continos adjoint soltion and ts inerits te smootness roerties of te continos adjoint soltion. Frter nmerical exeriments ave demonstrated te effect of adjoint consistency on te a osteriori error estimation on locally refined meses. Wile large error cancellation occrs for te standard discretisation de to te irreglar and oscillating discrete adjoint soltion near te airfoil, tere is virtally no error cancellation for te adjoint consistent discretisation wic again indicates an entirely smoot discrete adjoint soltion. Frtermore, drag coefficients comted based on te adjoint consistent discretisation were by a factor of abot two more accrate tan based on te standard discretisation. Finally, we ave tested an alternative aroac of a osteriori error estimation based on a atc-wise interolated discrete adjoint soltion comted wit te same olynomial degree as te flow soltion. Wereas tis aroac yields a oor error estimation for te standard DG discretisation it rerodces te beavior of te exact error of te adjoint consistent discretisation. Ts, in addition to accracy imrovements of te flow soltion already mentioned, adjoint consistent discretisations ave te otential to significantly decrease te additional amont of work associated wit solving discrete adjoint roblems reqired for error estimation and adjoint based refinement. Ftre researc will be dedicated to te adjoint consistency analysis of DG discretisations of more comlex non-linear roblems, see e.g., Hartmann (006a) for te adjoint consistency analysis of te interior enalty DG discretisation of te comressible Navier-Stokes eqations (Hartmann and Hoston, 006a). Acknowledgements All comtations ave been erformed wit te DG flow solver PADGE wic is based on te C++ finite element library deal.ii, (Bangert et al., 005, 006). Tis work as been sorted by te President s Initiative and Networking Fnd of te Helmoltz Association of German Researc Centres.

14 0 R. Hartmann References Arnold, D., Brezzi, F., Cockbrn, B. and Marini, D. (00) Unified analysis of discontinos Galerkin metods for ellitic roblems, SIAM J. Nmer. Anal., Vol. 39, No. 5, Bangert, W., Hartmann, R. and Kanscat, G. (005) deal.ii, Differential Eqations Analysis Library, Tecnical Reference, 5. ed., Setember, First ed. 999, tt:// Bangert, W., Hartmann, R. and Kanscat, G. (006) deal.ii, A general rose object oriented finite element library, ACM Transactions on Matematical Software, Vol. 33, No. 4, To aear, Available as Tecnical Reort ISC06-0-MATH, Texas A&M University, tt:// Bassi, F. and Rebay, S. (997) Hig-order accrate discontinos finite element soltion of te d Eler eqations, J. Com. Pys., Vol. 38, Becker, R. and Rannacer, R. (996) A feed-back aroac to error control in finite element metods: basic analysis and examles, East West J. Nmer. Mat., Vol. 4, Harriman, K., Hoston, P., Senior, B. and Süli, E. (003) Version discontinos Galerkin metods wit interior enalty for artial differential eqations wit nonnegative caracteristic form, Recent Advances in Scientific Comting and Partial Differential Eqations, AMS, Vol. 330 of Contemorary Matematics, Harriman, K., Gavagan, D. and Süli, E. (004) Te Imortance of Adjoint Consistency in te Aroximation of Linear Fnctionals sing te Discontinos Galerkin Finite Element Metod, Oxford University Comting Laboratory, Tecnical Reort, ft://ft.comlab. ox.ac.k/b/docments/tecreorts/na-04-8.s.gz. Hartmann, R. (006a) Adjoint consistency analysis of discontinos Galerkin discretizations, SIAM J. Nmer. Anal., In review. Hartmann, R. (006b) Derivation of an adjoint consistent discontinos Galerkin discretization of te comressible Eler eqations, in Lbe, G. and Rain, G. (Eds.): Proceedings of te BAIL 006 Conference, ISBN: , See also tt:// ni-goettingen.de/bail. Hartmann, R. and Hoston, P. (00) Adative discontinos Galerkin finite element metods for te comressible Eler eqations, J. Com. Pys., Vol. 83, Hartmann, R. and Hoston, P. (006a) Symmetric interior enalty DG metods for te comressible Navier Stokes eqations I: metod formlation, Int. J. Nm. Anal. Model., Vol. 3, No.,. 0. Hartmann, R. and Hoston, P. (006b) Symmetric interior enalty DG metods for te comressible Navier Stokes eqations II: goal oriented a osteriori error estimation, Int. J. Nm. Anal. Model., Vol. 3, No.,.4 6. Hoston, P., Rannacer, R. and Süli, E. (000) A osteriori error analysis for stabilised finite element aroximations of transort roblems, Comt. Met. Al. Mec. Engrg., Vol. 90, Nos., L, J. (005) An a osteriori Error Control Framework for Adative Precision Otimization sing Discontinos Galerkin Finite Element Metod, PD Tesis, MIT, tt://raael.mit.ed/ LPD_Tesis.df. L, J. and Darmofal, D.L. (006) Dal-consistency analysis and error estimation for discontinos Galerkin discretization: alication to first-order conservation laws, IMA Jornal of Nmerical Analysis, Sbmitted.