Numerical performance of discontinuous and stabilized continuous Galerkin methods for convection diffusion problems

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1 Numerical performance of discontinuous and stabilized continuous Galerkin metods for convection diffusion problems Paola F. Antonietti MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, Milano, Italy Alfio Quarteroni MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, Milano, Italy CMCS, Ecole Polytecnique Federale de Lausanne (EPFL), Station 8, 5 Lausanne, Switzerland alfio.quarteroni@epfl.c We compare te performance of two classes of numerical metods for te approximation of linear steady state convection diffusion equations, namely, te discontinuous Galerkin (DG) metod and te continuous streamline upwind Petrov-Galerkin (SUPG) metod. We present a fair comparison of suc scemes considering bot diffusion dominated and convection dominated regimes, and present numerical results obtained on a series of test problems including smoot solutions, and test cases wit sarp internal and boundary layers. INTRODUCTION Convection diffusion equations occur in te matematical modeling of a wide range of penomena as semiconductor devices modeling, magnetostatics and electrostatic flows, eat and mass-transfer, and flows in porous media related to oil and groundwater applications. Te linear steady state convection diffusion equation wit constant coefficients posed on a domain Ω R d, d =,, is a boundary value problem of te form ε u + div(βu) = f in Ω, () u = on Ω, were f L (Ω) is a given real-valued function, ε > is te diffusion parameter and β R d, d =,, represents a constant (for simplicity) velocity field. In many pysical penomena te convection as muc greater magnitude tan te diffusion, i.e., β /ε. In suc cases, problem () is an example of a singularly perturbed problem (wit respect to ε): tat is, te solution in te case ε = (wit boundary conditions prescribed not on te wole boundary Ω but on Ω in = {x Ω : β n(x) < } wit n(x) denoting te unit outward normal vector to Ω at te point x Ω, see () below) is not equal, at all points, to te limit of te solutions as ε +. From te numerical viewpoint, te design of robust numerical scemes for te solution of () represents a callenging problem, and indeed tere as been an extensive development of discretizations metods for convection diffusion equations tat are robust in all te regimes, see (Roos et al. 996), for example (cf. also (Ferziger and Perić 999), (LeVeque 7), (LeVeque )). Among te numerical metods developed so far, we are interest ere in discontinuous and stabilized continuous Galerkin finite element metods wic restore two opposite paradigms witin te finite element approac, namely non-conforming (discontinuous) versus conforming (continuous) approximation spaces. Stabilized conforming finite element metods ave been extensively developed for yperbolic and parabolic conservation laws. Te original conforming stabilized metod for advection-diffusion equations, te Streamline-Upwind Petrov-Galerkin (SUPG), was developed by (Brooks and Huges 98) and ten analyzed by (Jonson et al. 98) (see also (Burman and Smit ) for te analysis of te SUPG metod for transient convection-diffusion equations). It provides an upwinding effect to standard finite element metods using te Petrov-Galerkin framework. It is well known tat te SUPG metod as te capability of improving numerical stability for convection dominated flows, wile satisfying a strong consistency property. Like many oter conforming stabil-

2 ized metods, te SUPG sceme contains an elementwise stabilization parameter τ tat as to be tuned in practice, and, except for simplified situations, te optimal value is not known. More precisely, for linear finite element discretizations on a uniform decomposition of a one-dimensional problem wit constant coefficients ε R +, β R, and constant data, it can be sown tat te coice τ = β ( cot(pe T ) Pe T ), () were Pe T is te mes Péclet number (cf. also (6), below), leads to a nodally exact approximate solution (Brooks and Huges 98), see also (Cristie et al. 976). In te multi-dimensional case, many different coices of te stabilization parameter τ ave been proposed see (Franca et al. ), and te references terein. It is also known tat te coice of te stabilization parameter τ also influences te beavior of te iterative solvers employed to solve te resulting linear system of equations. For example, GMRES converges more slowly for worse stabilization parameters, as pointed out in (Liesen and Strakos 5). On te oter and, discontinuous Galerkin (DG) metods are parameters free since tey do not rely on te addiction of ad-oc tuned streamline stabilization terms. Since teir introduction in 97s (Reed and Hill 97), (Douglas and Dupont 976), (Arnold 98), te discontinuous Galerkin metod as become a topic of extensive researc for te numerical solution of differential equations. DG metods employ piecewise polynomial spaces wic may be discontinuous across elements boundaries. Terefore, DG can accommodate non-matcing meses as well as variable polynomial approximation orders. Tese features make tem ideally suited for bot geometrical and functional adaptive discretizations. Moreover, DG metods are locally mass conservative and can capture possible discontinuities in te solution tanks to te discontinuous approximation spaces. For suc capabilities, DG metods ave been extensively developed for convection diffusion problems, cf. for example (Cockburn 999), (Baumann and Oden 999), (Houston et al. ), (Buffa et al. 6), (Antonietti and Houston 8), (Ayuso de Dios and Marini 9), and te references terein. In tis paper we aim at comparing DG and SUPG solutions, working, for te sake of simplicity, on a linear steady state convection diffusion equation wit constant coefficients. A detailed assessment of te performance of suc scemes (employing te lowest order elements) wit respect to accuracy is presented in a number of test problems featuring smoot solutions, and oter problems featuring sarp layers (eiter internal or boundary s). Te rest of te paper is organized as follows. In Section we introduce some notation and recall te variational formulation of problem (). In Section and Section we recall te DG and SUPG formulations and te corresponding error estimates, respectively. Section contains an extensive set of numerical experiments to compare te DG and SUPG solutions. Finally, in Section 6 we draw some conclusions. MODEL PROBLEM We consider te linear steady state convection diffusion equation wit constant coefficients () posed on Ω, a bounded convex open polygonal (resp. polyedral) domain in R (resp. R ). Te corresponding weak formulation reads as: Find u H(Ω) suc tat (ε u βu) v dx = fv dx v H(Ω). Ω Te existence of a unique solution of te above problem follows from te Lax-Milgram Teorem. For te analysis of more general convection diffusion equations see, e.g., (Quarteroni and Valli 99). For te design and analysis of DG metods in te case of a diffusion tensor wose entries are bounded, piecewise continuous real-valued functions defined on Ω and a piecewise continuous real-valued velocity field, we refer to (Ayuso de Dios and Marini 9). By n n(x) we denote te unit outward normal vector to Ω at te point x Ω, and define te sets Ω in and Ω out of inflow and outflow boundary, respectively, as Ω in = {x Γ : β n < }, Ω out = {x Γ : β n }. Ω () DG AND SUPG DISCRETISATIONS Te aim of tis section is to introduce te DG and SUPG discretisations. Let T be a conforming sape-regular quasiuniform partition of Ω into disjoint open triangles T were eac T T is te affine image of te reference open unit d-simplex in R d, d =,. Denoting by T te diameter of an element T T, we define te mes size = max T T T.. THE DG METHOD We start by introducing suitable trace operators, defined in te usual way (Arnold, Brezzi, Cockburn, and Marini ). Let F I and F B be te sets of all te interior and boundary (d )-dimensional (open) faces (if d =, face means edge ), respectively, we set F = F I F B. We also decompose F B = F Bout F B in Bout, were F = {F F B : F Ω out}

3 and F B in = {F F B : F Ω in}. Implicit in tese definitions is te assumption tat T respects te decomposition of Ωin te sense tat eac F F tat lies on Ωbelongs to te interior of exactly one of Ω out, Ω in. Let F F I be an interior face sared by te elements T + and T wit outward unit normal vectors n + and n, respectively. For piecewise smoot vector valued and scalar functions τ and z, respectively, let τ ± and z ± be te traces of τ and z on T ± taken witin te interior of T ±, respectively. We define te jump and te average across F by [[τ ]] = τ + n + + τ n, [[z]] = z + n + + z n, {{τ }} = (τ + + τ )/, {{z}} = (z + + z )/. On a boundary face F F B, we set, analogously, [[z]] = z n, {{τ }} = τ. We do not need eiter [[τ ]] or {{v}} on boundary faces, and leave tem undefined. On eac interior face F F I, we introduce an upwind average {{ }} β defined as βu + if β n + > {{βu}} β = βu if β n + < β {{u}} if β n + = On F F B, we set {{βu}} β = βu, if β n >, {{βu}} β = oterwise. For a given approximation order l, we introduce te DG finite element space as V DG V DG = {v L (Ω) : v T P l (T ) T T }, were P l (T ) is te set of polynomials of total degree l on T. We define te bilinear form A DG DG (, ) : V R as V DG A DG (u, v) = (ε u βu) v dx T T T F F + F F F ({{ε u}} [[v]] + [[u]] {{ε v}}) ds F ( ) σ F [[u]] [[v]] + {{βu}} β [[v]] ds wit te convention tat te application of te operator as to be intendend elementwise wenever te function to wic is applied is elementwise discontinuous. Here σ F is defined for all F F as σ F = α εl F F F, () wit α a positive real number at our disposal, and F te diameter of te face F F. Te DG approximation of problem () is defined as follows : find u DG V DG suc tat A DG (u DG, v ) = Ω fv dx v V DG. Remark.. For te approximation of te convection and diffusion terms we ave employed a DG metod wit upwinded fluxes (Brezzi et al. ) and te symmetric interior penalty (SIPG) metod (Arnold 98), respectively. Wit minor canges, any oter (possibly unsymmetric) DG approximation of te second order term could be considered. We close tis section observing tat, denoting by n F te normal to te face F F, it olds F F F {{βu}} β [[v]] ds = F F I F B out F + F F I {{βu}} [[v]] ds F β n F [[u]] [[v]] ds, for all u, v V DG. Terefore, te upwind value of βu, can be written as te sum of te usual (symmetric) average plus a jump penalty term. Suc equivalent representation as several advantages. For example, notice tat bot terms on te rigt and side are of te same kind as te ones already present in te treatment of te diffusive part of te operator. Tis can be favorably exploited in te implementation process.. THE SUPG METHOD Let te SUPG discrete space be defined as te subspace of V DG of continuous polynomials tat also satisfy te boundary condition, i.e., V SUPG = {v C (Ω) : v T P l (T ) T T, and v = on Ω},

4 For piecewise positive parameters τ T tat we will specify later, we define te bilinear form A SUPG (, ) : V SUPG V SUPG R as A SUPG (u, v) = (ε u βu) v dx Ω + τ T ( ε u + β u)β v dx, T T T and te functional F SUPG ( ) : V SUPG R F SUPG (v) = f (v + τ T β v) dx. T T T Ten, te SUPG formulation reads: find u SUPG suc tat V SUPG A SUPG (u SUPG, v ) = F SUPG (v ) v V SUPG. It is well known tat te coice of te stabilization parameters τ T, T T, is a delicate matter since it may influence te performance of te metod. For suc a reason, te problem of finding te optimal stabilization parameter as been a subject of an extensive researc in te last years, and many coices are available in te literature (cf. (Jon and Knobloc 7), (Jon and Knobloc 8), for example). Here, we consider a generalization of () to te multi-dimensional case and set τ T τ T = T β l ξ (Pe T ), (5) wit Pe T being te local Péclet number, i.e., Pe T = β T ε l, (6) and were β is te Euclidean norm of β. Te upwind function ξ( ) can be cosen, for instance, as ξ(θ) = cot(θ) /θ, θ >, (7) cf. Figure. Notice tat ξ(θ) for θ, and ξ(θ)/θ / for θ + (and te SUPG stabilization is not necessary for θ + ). Moreover, if convection strongly dominates diffusion, i.e., Pe T, we ave τ T T /( β l). As already pointed out, many oter definitions of τ T are possible. For example, in (5) and (6) T can be replaced by te element diameter in te direction of te convection β, or te upwind function ξ( ) can be approximated eiter by ξ(θ) max{, /θ} or by ξ(θ) min{, θ/}. We refer to (Jon and Knobloc 7), (Jon and Knobloc 8) (Brooks and Huges 98), for furter details. Figure : Stabilization function ξ(θ) = cot(θ) /θ. ERROR ESTIMATES In tis section we briefly recall te error estimates for te DG and SUPG formulations. To deal wit te convection dominated regime, we first introduce a norm wic controls te streamline derivative: ( ) / v β = τ / T β v,t v H (T ), T T (8) were τ is defined in (5), and were te piecewise broken Sobolev space H (T ) consists of functions v suc tat v T H (T ) for any T T. We endow te DG space wit te norm v DG = ( v + v β were β is defined in (8), and were ) / v V DG, (9) v = ε / v,t + α / F [[v]],f T T F F for all v V DG. Here α F is defined as wit σ F given in (). α F = σ F + β n F F, Provided te exact solution u is regular enoug and te stability parameter α in () is cosen sufficiently large, te following error estimates old for te DG solution u u DG ( l ε / + / β / u u DG DG ( ) () l ε / + / β /. ), θ

5 We refer to (Houston et al. ) and (Ayuso de Dios and Marini 9) for te proof (in a more general framework) and furter details. For te SUPG metod, we define te norm ( / v SUPG = ε / v L (Ω) β) + v () for all v V SUPG. Ten, te following estimate olds: ), () u u SUPG SUPG ( l ε / + / β / see (Quarteroni and Valli 99), for example. Note tat te two error estimates () and () feature te same rigt and side (apart from te idden multiplicative constants). Remark.. In te error estimates () and () te idden constants depend on te domain Ω, te sape regularity constant of te partition T, and te polynomial approximation degree l, but are independent of te mes size and te pysical parameters ε and β of te problem. 5 NUMERICAL RESULTS We compare te numerical performance of te DG and SUPG metods on four test cases. In te first example, we coose a smoot exact solution and compare te approximation properties of te two scemes. In te second and tird examples, te exact solutions exibit an internal and an exponential boundary layer, respectively. Finally, in te last example te (unknown) analytical solution features sarp internal and boundary layers. Trougout tis section 5. EXAMPLE In tis section, we aim at comparing te approximation properties of te DG and SUPG scemes on a smoot analytical solution. To tis aim, we cose β = (, ) T, and te source term f suc tat te solution of problem () is u(x, y) = sin(πx)(y y ), for any positive diffusion coefficient ε. Figure (loglog scale) sows te computed errors in te energy norms (9) and () versus /, for different coices of te diffusion coefficient ε, ε = k, k =,, 5, 9. Te expected convergence rates are clearly observed: O() and O( / ) convergence rates are attained in te diffusion and convection dominated regimes, respectively. We ave also compared te errors com- (a) ε = u u DG u u SUPG. (c) ε = 5 u u DG u u SUPG.5 (b) ε = u u DG u u SUPG. (d) ε = 9 u u DG u u SUPG.5 Figure : Example. Errors in te energy norms (9) and () for ε = k, k =,, 5, 9. (a) ε = (b) ε =.. u u L(Ω) - DG u u L(Ω) - SUPG u u L(Ω) - DG u u L(Ω) - SUPG Figure : Left: Initial triangulations wit mes size. Rigt: triangulation obtained after R = levels of refinement (mes size /).. 5 (c) ε = 5. 5 (d) ε = 9 we set Ω = (, ) (, ) and consider a sequence of uniform unstructured triangulations obtained by uniformly refining te initial mes sown in Figure (left). Denoting by te mes size of tis initial grid, after R =,,... refinements te corresponding mes size is given by = / R (cf. Figure (rigt), for R = ). Since te grids are quasi uniform, it olds / N, N being te number of elements of te partition. Finally, trougout te section, we restrict ourselves to piecewise linear elements, i.e., l =. u u L(Ω) - DG u u L(Ω) - SUPG. 5 u u L(Ω) - DG u u L(Ω) - SUPG. 5 Figure : Example. Errors in te L (Ω) norm for ε = k, k =,, 5, 9. puted in te L (Ω) and L (Ω) norms. Te results are 5

6 reported in Figure (log-log scale) and in Figure 5 (log-log scale), respectively. For all te cases, a quadratic convergence rate is clearly observed, for any value of te diffusion coefficient ε. We ave run te same set of experiments on a sequence of structured triangular grids, te results (not reported ere, for te sake of brevity) are completely analogous. From te (a) ε = u u L (Ω) - DG u u L (Ω) - SUPG 5.5 (c) ε = 5 u u L (Ω) - DG u u L (Ω) - SUPG 5.5 (b) ε = u u L (Ω) - DG u u L (Ω) - SUPG. 5 (d) ε = 9 u u L (Ω) - DG u u L (Ω) - SUPG 5.5 Figure 5: Example. Errors in te L (Ω) norm for ε = k, k =,, 5, 9. Concerning te error in average (cf. Figure ) te DG metod seems to be sligtly more accurate tan te SUPG; wereas if we measure te error in te L (Ω) norm te SUPG metod seems to be sligtly better tan te DG metod (cf. Figure 5). Neverteless, te difference between te DG and SUPG solutions are negligible, as igligted by te results reported in Figure 6 were we ave plotted, for te same coices of ε, te pointwise absolute value of te difference between te DG and SUPG numerical solutions, on a grid wit mes size = /. 5. EXAMPLE In te second example, we set again β = (, ) and let ε vary, owever now te exact solution reads ( ) (x /) + (y /) /6 u(x, y) = atan, ε (te source term f and te non-omogeneous boundary conditions being set accordingly). We notice tat, as ε +, te solution exibits a sarp internal layer, cf. Figure 7 (left) for ε = and Figure 7 (rigt) for ε = 5. (a) ε = (b) ε =...5 x x Figure 7: Example. Exact solutions for ε = (left) and ε = 5 (rigt)... (c) ε = 5.5 x (d) ε = 9.5 x Figure 6: Example. Absolute value of te pointwise difference between te DG and SUPG approximate solutions for ε = k, k =,, 5, 9. Grids wit mes sizes = /. results reported in Figures,, 5, we can infer tat in te case of a smoot analytical solution, te DG and SUPG enjoy almost te same accuracy, for any value of te diffusion coefficient ε. Indeed, te error curves reported in Figures sow tat te errors computed in te energy norms (9) and () are almost identical. 8 6 Figure 8 compare te DG and SUPG discrete solutions, respectively, obtained on an unstructured triangular grid of mes size = /, for ε = k, k =,,,, 5. We can observe tat te discrete solutions approximate quite well te corresponding analytical solution. Neverteless, in te convection dominated regime bot metods exibit spurious oscillations near te internal layer. We next compare te numerical solutions provided by te two metods. In Figure 9 te pointwise absolute error of te DG and SUPG approximate solutions is reported on a grid wit mes size = /, for ε = k, k =,,,, 5. Notice tat, in tis case, te approximate solutions provided by te two metods are almost identical in te diffusion dominated regime (cf. Figure 9, top), wereas tey differ substantially near te internal layer in te convection dominated regime (cf. Figure 9, bottom). Indeed, as ε becomes small, tey bot exibit oscillations near te internal layer, but of different kind. 6

7 (a) ε = (a) ε =, DG (b) ε =, SUPG.... x x (b) ε = (c) ε =, DG (d) ε =, SUPG (c) ε = (e) ε =, DG (f) ε =, SUPG (d) ε = (g) ε =, DG () ε =, SUPG (e) ε = 5 (i) ε = 5, DG (j) ε = 5, SUPG Figure 8: Example. DG (left) and SUPG (rigt) approximate solutions on an grid of mes size = / for ε = k, k =,,,, 5. Figure 9: Example. Pointwise absolute error of te DG and SUPG approximate solutions for ε = k, k =,,,, 5. Mes size = /. 7

8 Finally, for ε = k, k =,,,, Figure (loglog scale) and Figure (log-log scale) sow te computed errors in te energy norms (9) and () and in te L (Ω) norm, respectively. Te results reported in Figure and Figure ave been obtained on unstructured triangular grids as te ones sown in Figure ; analogous results (not reported ere) ave been obtained on structured triangular grids and te same convergence beavior as been observed. Te error in te energy norms goes to zero at te predicted rate, namely linearly in te diffusion dominated case and at a rate of / in te convection dominated regime. Neverteless, notice tat suc a rate seems to be acieved only asymptotically for ε = by bot te DG and SUPG metods. Concerning te mean error (a) ε = u u DG u u SUPG (b) ε = u u DG u u SUPG a quadratic convergence rate, and te DG sceme seems to be sligtly more accurate tan te SUPG metod. If we measure te errors in te L (Ω) norm (cf. Figure ) we observe again a quadratic convergence rate but in tis case te SUPG metod seems to be sligtly more accurate. (a) ε =. 5 u u L (Ω) - DG u u L (Ω) - SUPG (c) ε = u u L (Ω) - DG u u L (Ω) - SUPG (b) ε =. 5 u u L (Ω) - DG u u L (Ω) - SUPG (d) ε = u u L (Ω) - DG u u L (Ω) - SUPG. (c) ε =.5 (d) ε =.. Figure : Example. Errors in te L (Ω) norm for ε = k, k =,,,. u u DG u u DG u u SUPG u u SUPG.5.5 Figure : Example. Errors in te energy norms for ε = k, k =,,, (a) ε = u u L (Ω) - DG u u L (Ω) - SUPG. (c) ε = u u L (Ω) - DG u u L (Ω) - SUPG 5 5. (b) ε = u u L (Ω) - DG u u L (Ω) - SUPG. (d) ε = u u L (Ω) - DG u u L (Ω) - SUPG Figure : Example. Errors in te L (Ω) norm for ε = k, k =,,,. (cf. Figure ), we observe tat bot metods acieve 5. EXAMPLE In tis example, we consider a test case wit an exponential boundary layer taken from (Houston et al. ) and (Ayuso de Dios and Marini 9), wic is te multi-dimensional version of te one-dimensional test proposed by (Melenk and Scwab 999). We set β = (, ), and vary te diffusion coefficient ε. Te rigt and side f and te non-omogeneous boundary condition are cosen in suc a way tat te exact solution is given by: u(x, y) = x + y xy + e /ε e ( x)( y)/ε e /ε. Figure sows te discrete solutions computed wit te DG and SUPG metods for ε = k, k =,,, 9 on a grid wit mes size = /. We can observe tat, for ε =, te two solutions are almost identical, wereas in te intermediate regime ε = te DG metod exibits some spurious oscillations near te boundary layer, wile te SUPG solution seems to be definitely more accurate. Finally, we observe tat in te convection dominated regime, te DG solution is totally free of spurious oscillations but te boundary layer is not captured. Suc a beavior, already observed in (Ayuso de Dios and Marini 9), is due to te fact tat te boundary conditions are imposed weakly. On te oter and, for ε = 9 te SUPG metod results in oversoot near te point 8

9 (, ). Te above considerations are also confirmed by te results reported in Figure were te absolute value of te pointwise error is sown on grid wit mes size = /. (a) ε = x.5.5 x (a) ε = (b) ε =.9 x (b) ε = (c) ε = (c) ε = (d) ε = (d) ε = Figure : Example. DG (left) and SUPG (rigt) discrete solutions for ε = k k =,,, 9. Mes size = / Figure : Example. Pointwise absolute error of te DG and SUPG approximate solutions for ε = k, k =,,, 9. Mes size = /. Finally, Figure 5 sows te computed errors in te L (Ω) and L (Ω) norms for ε = k, k =,, 9. In te diffusion dominated regime and for bot te metods te errors go to zero quadratically as te mes is refined, wereas for ε = 9 te DG metod is quadratically convergent in bot te L (Ω) and L (Ω) norm wile te SUPG metod convergences at a rate of order O( / ) and O() in te L (Ω) and L (Ω) norms, respectively. 9

10 (a) ε = (b) ε = 9 (a) ε = u u L (Ω) - DG u u L (Ω) - SUPG u u L (Ω) - DG u u L (Ω) - SUPG (c) ε = 5. (d) ε = (b) ε =.5 u u L (Ω) - DG u u L (Ω) - SUPG u u L (Ω) - DG u u L (Ω) - SUPG Figure 5: Example. Errors in te L (Ω) (top) and L (Ω) (bottom) norms for ε = k, k =, (c) ε = EXAMPLE In te last example, always wit Ω = (, ), we consider te following convection diffusion equation: ε u + β u = in Ω, u = g on Ω. Te Diriclet boundary conditions are set as follows: if y =, g = if x = and y /, oterwise, te velocity β is cosen as β = (cos(π/), sin(π/)), and we let ε vary. Notice tat in tis case, a close expression for te exact solution is not available. We know owever tat, as ε tends to zero, te exact solution exibits a strong internal and boundary layer. We compare te discrete solutions computed by te DG and SUPG scemes for ε = k, k =,, 9. Te results on te unstructured triangular mes wit mes size = /8 are sown in Figure 6. Te following conclusions can be drawn: i) in te diffusion dominated regime, te DG and SUPG metods are substantially equivalent; ii) in te intermediate regime, te DG metod exibits spurious oscillations, wereas te SUPG metod seems to be more stable and only an oversooting effect is observed near te point (, ); iii) in te convection dominated regime, DG metod exibits some spurious oscillations on te internal layer wereas no oscillations along boundary layer are present, owever unfortunately te Figure 6: Example. DG (left) and SUPG (rigt) discrete solutions for ε = k, k =,, 9. Mes size = / boundary layer is completely missed. On te oter and SUPG metod exibits again an oversooting penomenon near te point (, ) and also some spurious oscillations along te internal layer. Te above considerations are also confirmed by te results reported in Figure 7 were te absolute value of te pointwise difference between te DG and SUPG approximate solutions for ε = k, k =,, 9, are reported. Results sown in Figure 7 (left) ave been obtained on a grid wit mes size = /8; te analogous ones obtained on a finer triangulation wit mes size = /6 are reported in Figure 7 (rigt). 6 CONCLUSIONS In tis paper we ave recalled te formulation and principal properties of DG and SUPG approximations of linear steady state convection diffusion equations. Te aim was to carry out a careful comparison of teir performance (as of stability, accuracy and numerical robustness) in tree different solution regimes: diffusion dominated, intermediate, and convection dominated. Our comparative analysis is made for piecewise linear elements on a number of significant test problems. As expected, DG and SUPG metods are substantially equivalent in te diffusion dominated regime. In te intermediate regime, SUPG.5

11 looks more robust (even if oversooting may occur in te boundary layer). In te convection dominated regime, SUPG still exibits oversooting wereas DG is oscillation free, at least wen te exact solution features sarp boundary layers. However, it is fair to notice tat te DG metod is not able to capture te boundary layers (indeed, te latter are completely missed because of te weak enforcement of boundary conditions). An in dept comparison between discontinuous and stabilized continuous Galerkin metods in te case of iger order polynomial approximations and transient convection diffusion problems is currently under investigation (Antonietti and Quarteroni ). (a) ε = ACKNOWLEDGMENTS P.F.A. as been partially supported by Italian MIUR PRIN 8 Analisi e sviluppo di metodi numerici avanzati per EDP. A.Q. acknowledges te financial support of Italian MIUR PRIN 9 Modelli numerici per il calcolo scientifico e applicazioni avanzate (b) ε = References Antonietti, P. F. and P. Houston (8). A preprocessing moving mes metod for discontinuous Galerkin approximations of advectiondiffusion-reaction problems. Int. J. Numer. Anal. Model. 5(), Antonietti, P. F. and A. Quarteroni (). In preparation. Arnold, D. N. (98). An interior penalty finite element metod wit discontinuous elements. SIAM J. Numer. Anal. 9(), Arnold, D. N., F. Brezzi, B. Cockburn, and L. D. Marini (). Unified analysis of discontinuous Galerkin metods for elliptic problems. SIAM J. Numer. Anal. 9(5), (electronic). Ayuso de Dios, B. and L. D. Marini (9). Discontinuous Galerkin metods for advectiondiffusion-reaction problems. SIAM J. Numer. Anal. 7(), 9. Baumann, C. E. and J. T. Oden (999). A discontinuous p finite element metod for convection-diffusion problems. Comput. Metods Appl. Mec. Engrg. 75(-),. Brezzi, F., L. D. Marini, and E. Su li (). Discontinuous Galerkin metods for first-order yperbolic problems. Mat. Models Metods Appl. Sci. (), Brooks, A. N. and T. J. R. Huges (98). Streamline upwind/petrov-galerkin formulations for convection dominated flows wit particular empasis on te incompressible Navier-Stokes (c) ε = Figure 7: Example. Pointwise absolute difference between te DG and SUPG approximate solutions for ε = k, k =,, 9. Mes size = /8 (left) and = /6 (rigt).

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