A Mixed-Hybrid-Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems

Transcription

1 A Mixed-Hybrid-Discontinuous Galerkin Finite Element Metod for Convection-Diffusion Problems Herbert Egger Joacim Scöberl We propose and analyse a new finite element metod for convection diffusion problems based on te combination of a mixed metod for te elliptic and a discontinuous Galerkin metod for te yperbolic part of te problem. Te two metods are made compatible via ybridization and te combination of bot is appropriate for te solution of intermediate convection-diffusion problems. By construction, te discrete solutions obtained for te limiting subproblems coincide wit te ones obtained by te mixed metod for te elliptic and te discontinuous Galerkin metod for te limiting yperbolic problem, respectively. We present a new type of analysis tat explicitly takes into account te Lagrange-multipliers introduced by ybridization. Te use of adequate energy norms allows to treat te purely diffusive, te convection dominated, and te yperbolic regime in a unified manner. In numerical tests, we illustrated te efficiency of our approac and compare to results obtained wit oter metods for convection diffusion problems. 1 Introduction In tis paper we consider stationary convection-diffusion problems of te form div( ɛ u + βu) = f in Ω, u = g D on Ω D, ɛ u n + β νu = g N on Ω N, (1) were Ω is a bounded open domain in R d, d = 2, 3 wit boundary Ω = Ω D Ω N consisting of a Diriclet and a Neumann part, ɛ is a non-negative function and β : Ω R d is a d-dimensional vector field. Similar problems arise in many applications, e.g., in te modeling of contaminant transport, in electro-ydrodynamics or macroscopic models for semiconductor devices. A feature tat makes te numerical solution difficult is tat often convection plays te dominant role in (1). In te case of vanising diffusion, solutions of (1) will in general not be smoot, i.e., discontinuities are propagated along te caracteristic direction β; nonlinear problems may even lead to discontinuities or blow-up in finite time wen starting from smoot initial data. So appropriate numerical semes for te convection dominated regime ave to be able to deal wit almost discontinuous solutions in an accurate but stable manner. Anoter property tat is desireable to be reflected also on te discrete level is te conservation strucure inerent in te divergence form of (1). Due to te variety of applications, tere as been significant interest in te design and analysis of numerical scemes for convection dominated problems. Muc work as been devoted to devise Center for Computational Engineering Science, RWTH Aacen University, Germany 1

2 accurate and stable finite difference and finite volume metods for te solution of yperbolic systems by means of appropriate upwind tecniques including flux or slope limiters in te nonlinear case. A different approac to te stable solution of (almost) yperbolic problems is offered by discontinuous Galerkin metods, introduced originally for a linear yperbolic equation in neutron transport [31, 26, 24]. Starting from te 70 s, discontinuous Galerkin metods ave been investigated intensively and applied to te solution of various linear and nonlinear yperbolic and convection dominated elliptic problems wit great success, cf. [7, 6, 1], and [16] for an overview and furter references. Since in practical applications, convection respectively diffusion penomena may dominate, in different parts of te computational domain, several attempts ave been made to generalize discontinuous Galerkin metods also to elliptic problems [32, 29, 21], yielding numerical scemes very similar to interior penalty metods studied muc earlier [28, 5, 2]. For furter references on tis topic and a unified analysis of several discontinuous Galerkin metods for elliptic problems, we refer to [4]. For discontinuous Galerkin metods applied to convection diffusion problems we refer to [14, 8, 12], and to [11] for a multiscale version. Two disadvantages of discontinuous Galerkin metods applied for problems wit diffusion is tat, compared to a standard conforming discretization, te overall number of unknowns is increased substantially and tat te resulting linear systems are muc less sparse. Anoter very successful approac for te solution of convection dominated problems is te streamline diffusion metod [22, 25], were standard conforming finite element discretizations are stabilized by adding in a conforming way an appropriate amount of artificial diffusion in streamline direction. Tis metod is easy to implement and yield stable discretizations in many situations, but may lead to unpysically large layers near discontinuities and boundaries. For a comparison of ig order discontinuous Galerkin and streamline diffusion metods, we refer to [20]. For an appropriate treatment of boundary layers via Nitsce s metod, see [19]. In contrast to discontinuous Galerkin metods, te streamline diffusion metod does not yield conservative discretizations. Herre, we follow a different approac, namely te combination of upwind tecniques used in discontinuous Galerkin metods for yperbolic problems wit conservative discretizations of mixed metods for elliptic problems. In order to make tese two different metods compatible, we will utilize ybrid formulations for te mixed and te discontinuous Galerkin metods. It is wellknown [3, 10, 15] tat ybridization can be used for te efficient implementation of mixed finite elements for elliptic problems. Introducing te Lagrange-multipliers also in te discontinuous Galerkin metods allows us to couple bot metods naturally and yields a stable mixed ybrid discontinuous Galerkin metod wit te following properties: For β 0 te numerical solution coincides wit tat of a mixed metod, cf. [3, 10], and postprocessing tecniques can be used to increase te accuracy of te solution. For ɛ 0 te solution coincides wit tat obtained by a discontinuous Galerkin metod for yperbolic problems [26, 24]. Te intermediate convection-diffusion regime is treated automatically wit no need to coose stabilization parameters. For diffusion dominated regions, te stabilization can be omitted yielding a sceme tat was studied numerically in 1D in [18]. Our analysis in Section 4.2 also includes tis case. A particular advantage of our metod is tat it is formulated and can be implemented elementwise, i.e., it allows for static condensation on te element level yielding global systems for te Lagrange-multiplier only. In tis way we arrive at global systems wit less unknowns and muc sparser stencils tan oter discontinuous Galerkin metods, wile still utilizing te same upwind mecanisms. Oter extensions of mixed finite element metods to convection diffusion problems were considered in [13, 17]. Te outline of tis article is as follows: In Section 2, we review te ybrid formulation of te mixed metod for te Poisson equation, and ten introduce a ybrid version of te discontinuous 2

3 Galerkin metod for te yperbolic subproblem. Te sceme for te intermediate convectiondiffusion regime ten results by a combination of te two metods for te limiting subproblems, and we sow consistency and conservation of all tree metods under consideration. Section 3 presents te main stability and boundedness estimates for te corresponding bilinear forms, and contains an a-priori error analysis in te energy norm wit empasis on te convection dominated regime. Details on super-convergence results and postprocessing for te diffusion dominated case are presented in Section 4. Results of numerical tests including a comparsion wit te streamline diffusion metod are presented in Section 5. 2 Hybrid mixed discontinuous Galerkin metods for convection diffusion problems Te aim of tis section is to formulate te problem under consideration in detail and to fix te relevant notation and some basic assumptions. By introducing te diffusive flux σ = ɛ u as a new variable, we rewrite (1) in mixed form σ + ɛ u = 0, div(σ + βu) = f in Ω u = g D on Ω D, ɛ u ν + βνu = g N on Ω N, wic will be te starting point for our considerations. Here and below, ν denotes te outward unit normal vector on te boundary of some domain. We refer to βu as te convective flux and call σ+βu te total flux. Existence and uniqueness of a solution to (2) follows under standard assumptions on te coefficients. For ease of presentation, let us make some simplifying assumptions. 2.1 Basic assumptions and notations We assume tat Ω is a polyedral domain and tat Ω D = Ω, i.e., Ω N =. Let T be a sape regular partition of Ω into simplices T and let E denote te set of facets E (element interfaces and element faces aligned to te boundary). We assume eac T and E is generated by an affine map Φ T or Φ E from a corresponding reference element ˆT respectively Ê. Wit T we denote te set of all element boundaries T (wit outward normal ν). Finally, by χ S we denote te caracteristic function of a set S Ω. Regarding te coefficients, we assume for simplicity tat g D = 0 and tat ɛ 0 is constant on elements T T. Furtermore, te vector field β is assumed to be piecewise constant wit continuous normal components across element interfaces, wic implies tat divβ = 0. Moreover, suc a vectorfield β induces a natural splitting of element boundaries into inflow and outflow parts, i.e., we define te outflow boundary T out := {x T : βν > 0} and T in = T \ T out. Te union of te element in-/outflow boundaries will be denoted by T in respectively T out and similarly, te symbols Ω in and Ω out are used for te in- and outflow regions of te boundary Ω. For our analysis we will utilize te broken Sobolev spaces H s (T ) := {u : u H s (T ), T T }, s 0, and for functions u H s+1 we define u [H s (T )] d to be te piecewise gradient. In a natural manner we define te inner products (u, v) T := uv dx and (u, v) T := (u, v) T, T T T wit te obvious modifications for vector valued functions. Te norm induced by te volume integrals (, ) T is denoted by u T := (u, u) T, and for piecewise constant α we define α (u, v) T := T (α u, v) T and α u T := α 2 (u, u) T. Norms and seminorms on te broken Sobolev spaces H s (T ) will be denoted by s,t and s,t. (2) 3

4 For te element interfaces we consider te function spaces L 2 (E ) := {µ : µ L 2 (E), E E }, and L 2 ( T ) := {v : v L 2 ( T ), T T }. Note tat functions in L 2 ( T ) are double valued on element interfaces and may be considered as traces of elementwise defined functions. Moreover, we can identify µ L 2 (E ) wit a function v L 2 ( T ) by duplicating te values at element interfaces, so in tis sense L 2 (E ) L 2 ( T ). For u, v L 2 ( T ) we denote integrals over element interfaces by λ, µ T := λµ ds and λ, µ T := λ, µ T, T T and te corresponding norms are denoted by u T := u, u T. Again, we write α u, v T wit te meaning T αu, v T. Let us now turn to te formulation of appropriate finite element spaces. We start from piecewise polynomials on te reference elements, and define te finite element spaces via appropriate mappings, cf. [9]. By P k ( ˆT ) respectively P k ( ˆF ), we denote te set of all polynomials of order k, and by RT k ( ˆT ) := P k ( ˆT ) x P k ( ˆT ) we denote te Raviart-Tomas (-Nedelec) element, cf. [30, 27] and [10]. Here te symbol is used to denote te union of two vector spaces. For our finite element metods we will utilize te following functions spaces: Σ := {τ [L 2 (Ω)] d : τ T = 1 Φ detφ T ˆτ Φ 1 T, ˆτ RT k( ˆT )}, T V := {v L 2 (Ω) : v T = ˆv Φ 1 T, ˆv P k(t )}, M := {µ L 2 (E ) : µ E = ˆµ Φ 1 E, µ = 0 on Ω, ˆµ Pk(Ê)}. For convenience we will sometimes use te notation W := Σ V M. Since we assumed tat our elements T are generated by affine maps Φ T, te finite element spaces could be defined equivalently as te appropriate polynomial spaces on te mapped triangles, cf. [10]. Tis would owever complicate a generalization to non affine elements. Let us now turn to te formulation of te finite element metods. We will start by recalling te ybrid mixed formulation for te elliptic subproblem (β 0) and ten introduce a ybrid version for te discontinuous Galerkin metod for te yperbolic subproblem (ɛ 0). Te sceme for te intermediate convection diffusion problem ten results by simply adding up te bilinear and linear forms of te limiting subproblems. 2.2 Diffusion For β 0 equation (2) reduces to te mixed form of te Diriclet problem σ = ɛ u, divσ = f, in Ω, u = 0 on Ω, (3) and te corresponding (dual) mixed variational problem reads 1 ɛ (σ, τ) T (u, divτ) T = 0 τ H(div, Ω) (divσ, v) T = (f, v) T v L 2 (Ω). Wile a conforming discretization of (3) allows to easily obtain conservation also on te discrete level, it also as some disadvantages: Te resulting linear system is a saddlepoint problem and involves considerably more degrees of freedom tan a standard (primal) H 1 conforming discretization of (3). Bot difficulties can be overcome by ybridization, cf. [3, 10, 15]. Let us sortly sketc te main ideas: Instead of requiring te discrete fluxes to be in H(div, Ω), one can use completely discontinuous piecewise polynomial ansatz functions, and ensure continuity of te normal fluxes 4

5 over element interfaces by adding appropriate constraints. Te corresponding discretized variational problem reads 1 ɛ (σ, τ ) T (u, divτ ) T + λ, τ ν T = 0, τ Σ (divσ, v ) T = (f, v) T, v V σ ν, µ T = 0, µ M. Note tat te coice of finite element spaces allows to eliminate te dual and primal variables on te element level, yielding a global (positive definite) system for te Lagrange multipliers only. Te global system as an optimal sparsity pattern and information on te Lagrange multipliers can furter be used to obtain better reconstructions by local postprocessing. We refer to [3, 10, 33] for furter discussion of tese issues, and come back to postprocessing later in Section 4. After integration by parts, we arrive at te following ybrid mixed finite element metod. Metod 1 (Diffusion) Find (σ, u, λ ) Σ V M suc tat B D (σ, u, λ ; τ, v, µ ) = F D (τ, v, µ ) (4) for all τ Σ, v V and µ M, were B D and F D are defined by and B D (σ, u, λ ; τ, v, µ ) := 1 ɛ (σ, τ ) T + ( u, τ ) T + λ u, τ ν T + (σ, v ) T + σ ν, µ v T, F D (τ, v, µ ) := (f, v ) T. (6) (5) We only mention tat te case ɛ = 0 on some elements T can be allowed in principle; for tese elements te term 1 ɛ (σ, τ ) T just as to be interpreted as σ T 0. Remark 1 Let Σ := [H 1 (T )] d, V := H 1 (T ) and M := {µ L 2 (E ) : µ = 0 on Ω}, and let W := Σ V M denote te continuous analogue to W. Te above bilinear form is ten defined for all (σ, u, λ; τ, v, µ ) W W W. Tis property will be used below to sow consistency of te metod and obtain Galerkin ortogonality. In fact, B D could be defined uniquely even for all (σ, u, λ) H(div; Ω) H 1 (T ) L 2 (E ) by just omitting te terms wit µ. Metod 1 is algebraically equivalent to te conforming RT k P k discretization of te dual mixed formulation of (3), and can be seen as a pure implementation trick. Below, we will analyze Metod 1 in a somewat non standard way, including te gradient of te primal variable and te Lagrange multipliers explicitly in te energy norm. Tis kind of analysis is quite close to tat of of discontinuous Galerkin metods for elliptic problems, and allows us to investigate te mixed metod togeter wit te discontinuous Galerkin metod for te yperbolic subproblem in a uniform framework. 2.3 Convection By setting ɛ 0 in (2), we arrive at te limiting yperbolic problem div(βu) = f in Ω, u = 0 on Ω in. (7) Multiplying (7) by a test function v H 1 (T ), and adding stabilization (see (36)), we obtain te discontinuous Galerkin metod for yperbolic problems [31, 26, 24] (div(βu), v) T + β ν (u + u), v T in = (f, v) T, were u + := u T + denotes te upwind value and T + is te upwind element, i.e., te element attaced to E were β ν T 0. To incorporate te boundary condition, we define u + = 0 on Ω in. After integration by parts, and noting tat u = u + on T out we obtain (u, β v) T β ν u +, v T in β νu, v T out = (f, v) T. 5

6 In order to make te discontinuous Galerkin metod compatible wit te ybrid mixed metod formulated in te previous section, let us introduce te upwind value as a new variable λ := u +, and let us define te te symbol { λ, E T in {λ/u} := u, E T out, for all T T. Note tat λ = {λ/u} = u + on bot sides of E, so {λ/u} is just a new caracterization of te upwind value. After discretization we now arrive at te following ybrid version of te discontinuous Galerkin metod. Metod 2 (Convection) Find (u, λ ) V M suc tat for all (v, µ ) V M wit B C (u, λ ; v, µ ) = F C (v, µ ) (8) B C (u, λ ; v, µ ) := (u, β v ) T + β ν {λ /u }, µ v T, (9) and F C (v, µ ) := (f, v) T (10) By construction, Metod 2 is algebraically equivalent to te classical discontinuous Galerkin metod. Tis can easily be seen by testing wit µ = χ E wic yields tat λ = u + on te element interfaces. All terms of te bilinear form are again defined elementwise, wic allows us to use static condensation on te element level. Moreover, as in te case of pure diffusion, te bilinear form B C can be extended onto W W W, wic ten allows to derive consistency and use Galerkin ortogonality arguments. On facets E were β ν = 0, te Lagrange multiplier is not uniquely defined, and we set λ = 0 tere. 2.4 Convection-diffusion regime Let us now return to te original convection diffusion problem and consider te system σ + ɛ u = 0, div(σ + βu) = f in Ω, u = 0 on Ω. (11) Since we used te same spaces for te discretization of te elliptic and yperbolic subproblems, te two ybrid metods can be coupled in a very natural way by simply adding up teir bilinear and linear forms. Tis yields te following ybrid mixed discontinuous Galerkin metod for te intermediate convection diffusion regime. Metod 3 (Convection diffusion) Find (σ, u, λ ) (Σ, V, M ) suc tat B(σ, u, λ ; τ, v, µ ) = F(σ, u, λ ) (12) for all τ Σ, v V and µ M, were B and F are defined by B(σ, u, λ ; τ, v, µ ) := 1 ɛ (σ, τ ) T + ( u, τ ) T + λ u, τ ν T + (σ + βu, v ) T + σ ν + β ν {λ /u }, µ v T, (13) and F(τ, v ) := (f, v ) T. (14) By testing wit µ = χ E for E E we obtain tat σ ν E + βν E {λ /u } is continuous across element interfaces. Here, ν E denotes te unit normal vector on E wit fixed orientation. Tus λ and σ ν E + βν E {λ /u } ave unique values on te element interfaces and can be considered as discrete traces for u and te total flux σ + βu. 6

7 2.5 Consistency and conservation Before we turn to a detailed analysis of te finite element metods 1-3, let us summarize two important properties, wic follow almost directly from te corresponding properties of te mixed respectively te discontinuous Galerkin metod for limiting subproblems. For sake of completeness we sketc te proofs in te present framework. Proposition 1 (Consistency) Te metods 1-3 are consistent, i.e., let u denote te solution of te problem (3), (7), respectively (11) and define σ = ɛ u and λ = u. Ten te corresponding variational equations (4), (8) and (12) old, if σ, u, λ are replaced by σ, u and λ. Proof. Metod 1: Let u denote te solution of (3), and make te substitutions as mentioned in te proposition. Ten, we obtain by testing te bilinear form B D wit (τ, 0, 0) B D ( ɛ u, u, u; τ, 0, 0) = ( u, τ ) T + ( u, τ ) T u u, τ ν T \ Ω u, τ ν Ω = u, τ ν Ω = 0. Next we test wit (0, v, 0) and integrate by parts to recover B D ( ɛ u, u, u; 0, v, 0) = (div( ɛ u), v ) T = (f, v ) T, wic follows since u is te solution of (3). Finally, testing wit (0, 0, µ ) we obtain B D ( ɛ u, u, u; 0, 0, µ ) = ɛ u n, µ T = 0, wic olds since div(ɛ u) = f L 2 implies ɛ u H(div; Ω) and tus te normal flux ɛ u n is continuous across element interfaces. Note tat at tis point we formally require some extra regularity, e.g., u H 1 (Ω) H 3/2+ε (T ) or σ = ɛ u L s (Ω) for some s > 2, in order to ensure tat te moments ɛ u n, µ are well-defined for µ M, cf. [10]. As already mentioned in Remark 1 tis extra regularity assumption can be dropped by appropriately defining B D. Summarizing we ave sown tat Metod 1 is consistent. Next, consider Metod 2 and let u denote te solution of (7). Substituting u for u and λ in (8) - (10) and testing wit (v, 0) we obtain after integration by parts B C (u, u; v, 0) = (div(βu), v ) T β ν u, v Ω in = (f, v ) T. Now test wit (0, µ ), ten we ave B C (u, u; 0, µ ) = β ν u, µ T = 0, since u and µ are single valued and β ν appear two times wit different signs for eac element interface. Tus we ave proven consistency of Metod 2. Finally, Metod 3 is consistent as it is te sum of two consistent metods. Wile consistency is a key ingredient for te derivation of a-priori error estimates, conservation is a property of te discrete metods wic is desired for pysical reasons, since it inibits unpysical increase of mass or total carge. Tis is particularly important for time dependent problems. If a finite element sceme allows to test wit piecewise constant functions, conservation can be sown to old locally (for eac element) as well as globally as long as te discrete fluxes are single valued on element interfaces. Proposition 2 (Conservation) Te metods 1-3 are locally and globally conservative. 7

8 Proof. Let us first sow te local conservation of Metod 1 by testing (4) wit (0, χ T, 0). Tis yields (f, 1) T = B D (u, λ, σ ; 0, χ T, 0) = σ ν, 1 T, tat is, te total flux over an element boundary equals te sum of internal sources, and ence te metod is locally conservative. By testing wit (0, 0, χ E ) for some E E, we obtain continuity of te normal fluxes σ ν across element interfaces, and so te sceme is also globally conservative. Now to Metod 2: Testing wit (χ T, 0) we get (f, 1) T = B C (u, λ ; χ T, 0) = β ν λ, 1 T in + β ν u, 1 T out, so te total flux over te element boundaries equals te sum of internal sources and fluxes over te boundary of te domain. Note tat β ν {λ /u } defines our unique flux on element interfaces. Now let E E suc tat E = T out 1 T in 2. By testing wit (0, χ E ), we obtain 0 = B C (u, λ ; 0, χ E ) = β ν {λ /u }, 1 T out 1 = β ν u, 1 T out 1 + β ν {λ /u }, 1 T in 2 + β ν λ, 1 T in 2, so te total outflow over a facet on one element levels te inflow over te same facet on te neigboring element. Finally, Metod 3 is conservative as it is te sum of two conservative metods. 3 A priori error analysis As already mentioned previously, our analysis of te ybrid metods under consideration is inspired by tat of discontinuous Galerkin metods [24, 4], in particular we will utilize similar mes dependent energy norms for proving stability and boundedness of te bilinear- and linear forms. We will sow stability of Metod 1 in te norm (τ, v, µ) D := ( 1 ɛ τ 2 T + ɛ v 2 T + ɛ λ u 2 T ) 1/2, (15) and stability of Metod 2 will be analysed wit respect to te norm ( ) 1/2 (u, λ) C := β β u 2 T + β ν λ u 2 T. (16) Here by β and β ν we understand appropriate bounds for β respectively β ν on single elements or facets. Note, tat for ɛ β (te crossover from diffusion dominated to convection dominated regime) all terms in (15) and (16) scale uniformly wit respect to ɛ, β and. For proving te boundedness of te bilinear forms we require sligtly different norms and (τ, v, µ) D, := ( (τ, v, µ) 2 D + ɛ τν 2 T ) 1/2, (17) (u, λ) C, := ( β u 2 T + β ν {λ/u} 2 T ) 1/2. (18) Tese norms scale again in te same manner wit respect to, ɛ and β as teir counterparts (15) and (16), and terefor it can be sown easily tat te additional terms do not disturb te approximation. 8

9 3.1 Pure diffusion - Metod 1 Below we will require te following prepraratory result. Lemma 1 Let v V and µ M be given. Ten tere exists a unique solution τ Σ defined elementwise by te variational problems ( τ, p) T = ( v, p) T, p [P k 1 (T )] d τν, q T = µ, q T, q P k ( T ). Moreover, tere exists a constant c I only depending on te sape of te elements suc tat olds. τ T c I ( v 2 T + µ 2 T ) 1/2 (19) Proof. Te existence of a unique solution τ follows wit standard arguments, and te norm estimate ten follows by te usual scaling argument and te equivalence of norms on finite dimensional spaces, cf. [10] for details. Since te estimate (19) uses an inverse inequality, te constant c I depends on te sapes of te elements. Lemma 1 now allows us to construct a suitable test function for establising te following stability estimate. Proposition 3 (Stability) Tere exists a positive constant c D independent of te messize suc tat te estimate B D (σ, u, λ ; τ, v, µ ) sup c D (σ, u, λ ) D, (20) (τ,v,µ ) 0 (τ, v, µ ) D olds for all (σ, u, λ ) Σ V M. Proof. Let us start wit testing te bilinear form (5) wit (σ, u, λ ), wic yields B D (σ, u, λ ; σ, v, µ ) = 1 ɛ σ 2 T. Now let τ be defined as in Lemma 1 wit µ replaced by ɛ (λ u ) and v replaced by ɛ u, so tat τ T c I ( ɛ2 λ u 2 T + ɛ 2 u 2 T ) 1/2 (21) olds wit constant c I independent of te messize. For γ > 0 we ten obtain B D (σ, u, λ ; γ τ, 0, 0) = γ 1 ɛ (σ, τ) T + γ( u, τ) T + γ λ u, τ T 1 2ɛ σ 2 T γ2 2ɛ τ 2 T + γ(ɛ u 2 T + ɛ λ u 2 T ) 1 2ɛ σ 2 T + (γ c Iγ 2 2 ) (ɛ u 2 T + ɛ λ u 2 T ), were we used (21) for te last estimate. Te assertion of te proposition now follows by coosing γ = 1/c I and combining te estimates for te two coices of test functions. We want to empasize at tis point tat ellipticity olds regardless of te value of c I in (21); only te ellipticity constant c D depends on c I and tus on te quality of te mes. Tis is certainly an advantage of te mixed discretization in comparison to, e.g., te symmetric interior penalty metod and also some oter discontinuous Galerkin metods for elliptic problems, were sufficiently muc penalization as to be added in order to ensure stability, see [4] for details. After using a Galerkin ortogonality in te analysis below, we will need boundedness of B D on te larger space W W W. 9

10 Proposition 4 (Boundedness) Tere exists a constant C D independent of suc tat te estimate B D (σ, u, λ; τ, v, µ ) C D (σ, u, λ) D, (τ, v, µ ) D (22) olds for all (σ, u, λ) W W and (τ, v, µ ) W. Proof. We consider only te term λ u, τ ν T in detail. Using te Caucy-Scwarz and a discrete trace inequality τ ν T c τ T, we obtain λ u, τ ν T c λ u T τ T. Te result ten follows by standard estimates for te remaining terms and summing up over all elements. Te above dicrete trace inequality cannot be used for te term involving σν, since σ W W. Terefore an additional term appears in te norm D,. 3.2 Pure convection - Metod 2 Since Metod 2 is equivalent to te discontinuous Galerkin metod for yperbolic problems, our analysis is carried out in a similar manner to tat presented in [24]. Proposition 5 (Stability) Tere exists a constant c C independent of te messize suc tat te estimate B C (u, λ ; v, µ ) sup c C (u, λ ) C (23) (v,µ ) (v, µ ) C olds for all (u, λ ) V M. Proof. We start by coosing test functions v = u and µ = λ. Since divβ = 0 we ave (u, β u ) T = 1 2 β νu, u T on eac element, and tus B C (u, λ ; u, λ ) = 1 2 β νu, u T + β ν {λ /u }, u T β ν {λ /u }, λ T = (1) + (2) + (3) = ( ). Recall tat λ equals 0 on Ω, and let us rearrange te terms (1)-(3) in te following way: (1) = 1 2 β νu, u T = 1 2 β ν u 2 T in (2) = β ν {λ /u }, u T = β ν u 2 T out (3) = β ν {λ /u }, λ T = β ν λ 2 T in 1 2 β ν u 2 T, out β ν λ, u T in, β ν λ, u T out. Now let T 1, T 2 denote two elements saring te facet E = T out 1 T in 2. Since λ is single valued on E by definition, we ave λ T out = λ 1 T in 2, tat means we can sift te terms only involving te Lagrange multiplier between neigboring elements. Summing up, we obtain ( ) = 1 2 β ν λ u 2 T. Let us now include a second term in te stability estimate by testing te bilinear form wit v = γ β β u for some γ 0, wic yields B C (u, λ ; v, 0) = γ β (u, β (β u )) T + γ β β ν{λ /u }, β u T = γ β β u 2 T + γ β β ν(λ u ), β u T in c γ( β β u 2 T β ν λ u 2 T ). 10

11 For te last estimate we used Youngs inequality and a discrete trace inequality. Te result now follows by coosing γ = 1 4c and combining te estimates for te two different test functions. Note tat by inverse inequalities and due to our scaling of v wit / β, it follows tat (v, 0) C C (u, 0) C wit a constant C independent of te messize. Proposition 6 (Boundedness) Tere exists a constant C C independent of suc tat te estimate B C (u, λ; v, µ ) C C (u, λ) C, (v, µ ) C (24) olds for all u V V, λ M M, and (v, µ ) V M. Proof. Te assertion follows directly from te definition of te norms and te Caucy-Scwarz inequality. 3.3 Convection-diffusion - Metod 3 Due to te structure of Metod 3 as te combination of Metods 1 and 2, te stability and boundedness of te bilinear form (13) follows almost directly from te corresponding properties of te bilinear forms for te limiting subproblems. Te appropriate norms for te analysis of Metod 3 are given by (σ, u, λ ) = ( (σ, u, λ ) 2 D + (u, λ ) 2 C) 1/2 (25) and (σ, u, λ) = ( (σ, u, λ) 2 D, + (u, λ) 2 C, ) 1/2 (26) i.e., tey are just assembled from te norms used for te analysis of te elliptic and yperbolic subproblems. Note tat all terms in te norm scale appropriately, e.g., in te diffusion dominated case ( β ɛ) te terms comming from te convective part can be absorbed by te terms stemming from te stability of te diffusion part. Let us now state te properties of B in detail. Proposition 7 (Stability) Tere exists a positive constant c B not depending on te messize suc tat B(σ, u, λ ; τ, v, µ ) sup c B (σ, u, λ ) (27) (τ,v,µ ) (τ, v, µ ) olds for all (σ, u, λ ) Σ V M. Proof. We will sow te inf-sup stability by testing wit te functions used in te previous stability estimates, i.e., τ = σ + α τ, v = u + γ β β u, µ = λ. In view of Proposition 3 and 5, it only remains to estimate te additional term comming from te test function γ β β u inserted in te diffusion bilinear form, viz. B D (σ, u, λ ; 0, γ β β u, 0) = γ β (σ, (β u )) + γ β σ ν, β u = γ β (divσ, β u ) γ β divσ β u c γ ( 1 ɛ σ 2 + ɛ u 2) c γ (σ, u, λ ) 2 D. Tis term can be absorbed by te stability estimate for te diffusion problem as long as γ is cosen to be sufficiently small. Note tat γ does not depend on, ɛ or β, i.e., te stability constant c B does not depend on tese parameters. Te boundedness of te bilinear form follows directly by combining te two results for te limiting subproblems. 11

12 Corollary 1 (Boundedness) Tere exists a constant C B independent of te messize suc tat B(σ, u, λ; τ, v, µ ) C B (σ, u, λ) (τ, v, µ ) (28) olds for all (σ, u, λ) W W and (τ, v, µ ) W. As a last ingredient for deriving te a-priori error estimates, we ave to establis some approximation properties of our finite dimensional spaces wit respect to te norms under consideration. 3.4 Interpolation operators and approximation properties Let us start by introducing appropriate interpolation operators and ten recall some basic interpolation error estimates. For T T, E E, and functions u L 2 (T ), λ L 2 (E) we define te local L 2 projections Π T k u and ΠE k λ by respectively (u Π T k u, v ) T = 0, v P k (T ), (λ Π E k λ, µ ) E = 0, µ P k (E). Tese interpolation operators satisfy te following error estimates, cf [9]. Lemma 2 Let Π T k and ΠE k be defined as above. Ten te estimates u Π T k u T C s u s,t, 0 s k + 1, (u Π T k u) T C s u s+1,t, 0 s k, u Π T k u T + u Π E k u T C s+1/2 u s+1,t, 0 s k, old wit a constants C independent of. Te corresponding interpolation operators for functions on T respectively E are defined elementwise and are denoted by te same symbols. For te flux function σ we utilize te Raviart-Tomas interpolant defined by (σ Π RT k σ, p ) T = 0, p [P k 1 (T )] d, ((σ Π RT k σ)ν, µ ) E = 0, µ P k (E), E T. In order to make moments of σν be well-defined on single facets E one as to require some extra regularity, e.g., σ H(div, T ) L s (T ) for some s > 2 or σ H 1/2+ε (T ), cf. [10]. Under suc an assumption, te following interpolation error estimates old [10, 34]. Lemma 3 Let Π RT k be defined as above. Ten te estimates σ Π RT k σ T + 1/2 (σ Π RT k σ)ν T C s σ s,t, 1/2 < s k + 1, div(σ Π RT k σ) T C s divσ s,t, 1 s k + 1. old wit constant C independent of. Applying tese results elementwise, we immediately obtain te following interpolation error estimates for te mes dependent norms used above. Proposition 8 Let u H 1 (Ω) H 3/2+ε (T ), and set σ := ɛ u. Ten (σ Π RT k σ, u Π T k u, λ Π RT k u) D, C s ɛ u s+1,t, 1/2 < s k, (29) and for u H 1 (Ω) tere olds (u Π T k u, λ Π RT k u) C, C s+1/2 β u s+1,t, 0 s k, (30) wit constants C not depending on u or. Te same estimates old if te -norms are replaced by teir counterparts witout. 12

13 Remark 2 Te estimates of Proposition 8 old wit obvious modifications, if te smootness s or te polynomial degree k vary locally. We assume uniform polynomial degree and smootness only for ease of notation ere. Te interpolation error estimate (29) is suboptimal regarding te approximation capabilities of te flux interpolant. In fact by Lemma 3 one can obtain 1 ɛ σ Π RT k σ C s ɛ u s+1,t for 1/2 < s k + 1, so te best possible rate is k+1 instead of k as for D in (29). We will use tis fact in Section 4 to derive super convergence results for te primal variable u. 3.5 A priori error estimates Te error of te finite element approximation can be decomposed into an approximation error and a discrete error. Let (σ, u, λ ) denote te discrete solution of (12), and let u be te solution of (11) and define σ := ɛ u. Ten we ave (σ σ, u u, u λ ) (31) (σ Π RT k σ, u Π T k u, u Π E k u) + (Π RT k σ σ, Π T k u u, Π E k u λ ). Using stability and boundedness of te bilinear form, and applying Galerkin ortogonality, te second term can now also be estimated by te interpolation error. Proposition 9 Let (σ, u, λ ) W denote te solution of (12), and let u H 1 (Ω) H 3/2+ε (T ) be te solution of te convection diffusion problem (11). Ten tere exists a constant C independent of te messize suc tat te estimate (Π RT k ( ɛ u) σ, Π T k u u, Π E k u λ ) C s ( ɛ + 1/2 β ) u s+1,t olds for 1/2 < s k. Proof. Let us define σ = ɛ u, λ = u. By application of te stability estimate (20), Galerkin ortogonality and te boundedness (22) of te bilinear form, we obtain c B (Π RT k σ σ, Π T k u u, Π E k u λ ) sup B(Π RT k σ σ, Π T k u u, Π E k u λ ; τ, v, µ )/ (τ, v, µ ) (τ,v,µ ) 0 = sup B(Π RT k σ σ, Π T k u u, Π E k u u; τ, v, µ )/ (τ, v, µ ) (τ,v,µ ) 0 C B (Π RT k σ σ, Π T k u u, Π E k u u). Te assertion follows directly from (29). Te complete error estimate can now be derived by combining (31) and Proposition 8. Teorem 1 (Energy norm estimate) Let (σ, u, λ ) be te finite element solution of Metod 3, and let u H 1 (Ω) H 3/2+ε (T ) denote te solution of (11) and σ := ɛ u. Ten (σ σ, u u, u λ ) C s ( ɛ + 1/2 β ) u s+1,t olds for 1/2 < s k wit constant C independent of te messize. In te convection dominated case, te error estimate coincides wit te well known error estimates for te discontinuous Galerkin and te streamline diffusion metod for yperbolic problems, cf. [24, 25]. 13

14 Corollary 2 Let ɛ β on eac element, and let te conditions of Teorem 1 old. Ten te estimate (σ σ, u u, u λ ) C s+1/2 β u s+1,t olds for 1/2 < s k wit constant C independent of te parameters ɛ, β and. Tis estimate olds in particular for te limiting yperbolic problem (ɛ 0) in wic case σ = σ 0 and (τ, v, µ) = (v, µ) C, and so Metod 3 collapses wit Metod 2, i.e., te discontinuous Galerkin metod for yperbolic problems. In analogy to standard error estimates for mixed metods for te Poisson problem, we obtain te following convergence result in te diffusion dominated regime. Corollary 3 Let ɛ β and let te conditions of Teorem 1 old. Ten te estimate (σ σ, u u, λ λ ) C s ɛ u s+1,t olds for 1/2 < s k wit constant C independent of ɛ, β and. Moreover, we ave D. Clearly, tis estimate olds also for Metod 1 in te case of pure diffusion. Let us remark once again tat all terms in te a-priori error estimates are defined locally, so te smootness index s and te polynomial degree k can vary locally, allowing for p-adaptivity. 4 Super convergence and postprocessing for diffusion dominated problems Te best possible rate for 1 ɛ σ σ guaranteed by Teorem 1 and Corollary 3 is k, wic is one order suboptimal regarding te interpolation error estimate of Lemma 3. It is well-known owever tat in te purely elliptic case, te optimal rate k+1 can be obtained by a refined analysis, and we will derive corresponding results below. Since we consider te case of dominating diffusion in tis section, we assume for ease of notation tat ɛ 1 in te sequel. 4.1 Refined analysis for pure diffusion Altoug te estimate (29) is optimal concerning te approximation error wit respect to te norm D, we can obtain better error estimates for σ = u, i.e., we will sow tat σ σ depends only on te interpolation error σ Π RT k σ, and tus optimal convergence for σ can be expected. We refer to [3, 10, 33] for corresponding results in te mixed framework. Proposition 10 Let (σ, u, λ ) denote te solution of (4), and let u, σ := u be te solution of problem (3). Ten (σ σ, u Π T k u, λ Π E k u) D C s u s+1,ω (32) olds for 1/2 < s k + 1 wit constant C independent of. Proof. Let us first consider te following term. B D (Π RT k σ σ, Π T k u u, Π E k u u; τ, v, λ ) = (Π RT k σ σ, τ ) T (Π T k u u, divτ ) T + Π E k u u, τ ν T + (div(π RT k = (Π RT k σ σ, τ ) T, σ σ), v ) T + (Π RT k σ σ)ν, µ T were te last equality follows from te definition of te interpolants. Ten in te same manner as in te proof of Proposition 9 we obtain c D (Π RT k σ σ, Π T k u u, Π E k u λ ) D Π RT k σ σ T, 14

15 and te statement follows by application of te triangle inequality and te interpolation error estimate (29). Note tat for te modified error (32), te best possible rate now is k+1, wic is optimal in view of te interpolation error estimates. As we sow next, te estimates for (Π T k u u ) and (Π E k u λ ) can even be improved if we assume tat te domain Ω is convex, cf. [33] for similar results in te mixed framework. Proposition 11 Let Ω be convex and u H 1 (Ω) H 3/2+ε (T ) be te solution of (3). Moreover, let u denote te discrete solution obtained by Metod 1. Ten te estimate Π T k u u 0 C s+1 { u s+2,t, k = 0 u s+1,t, k > 0 (33) olds for 1/2 < s k + 1 (resp. 0 s 1 for k = 0). If in addition f is piecewise constant, ten olds also for k = 0. Π T 0 u u 0 C s+1 u s+1,t (34) Proof. Let φ H 1 0 (Ω) denote te solution of te Poisson equation φ = Π T k u u wit omogeneous Diriclet conditions, and let z := φ. Due to convexity of Ω we ave φ 2,Ω c Π T k u u 0 and φ Π T k φ c min(k+1,2) Π T k u u 0. Using te definition of φ and z we obtain Π T k u u 2 0 = (Π T k u u, divz) = (Π T k u u, div(π RT k z)) = (σ σ, Π RT k z) = (σ σ, Π RT k z φ) (div(σ σ ), φ Π T k φ) σ σ 0 Π RT k z φ 0 + div(σ σ ) 0 φ Π T k φ 0. Te first estimate now follows by Lemma 3. If f is piecewise polynomial of order k ten div(σ σ ) 0, so te last term in te above estimate vanises and we conclude te second assertion. 4.2 Te diffusion dominated case Let us sow now tat similar results still old in te presence of convection as long as diffusion is sufficiently dominating. In tis case, we can discretize te convective term witout upwind stabilization, and we terefore consider te following bilinear form instead of (9) B NU C (u, λ ; v, µ ) := (u, β v ) T + β ν λ, µ v T. (35) Suc a discretization for te convective part was investigated numerically but not analysed previously in [18] for a 1D problem. Tere, te autors conjectured tat tis discretization already introduces some stabilization, wic is not te case as is clear from our analysis. Consistency and conservation: Substituting te continuous solution u for u and λ in (35) we obtain after integration by parts tat (div(βu), v ) T + β ν u, µ T = (div(βu), v ) T = ( f, v ) T, so te bilinear form BC NU is consistent. Te sceme is also conservative, since te flux β ν λ in (35) is single valued on element interfaces. Moreover, we ave B C (u, λ ; v, µ ) = B NU C (u, λ ; v, µ ) + β ν λ u, µ v T out, (36) wic clarifies wat kind of upwind was used for te discontinuous Galerkin stabilization in (9). 15

16 Stability: Testing te bilinear form B NU C wit v = u and µ = λ, we obtain BC NU (u, λ ; u, λ ) = (u, β u ) T β ν λ, λ u T = 1 2 β νu, u T β ν λ, λ u T = 1 2 β ν u λ 2 T in 1 2 β ν u λ 2 T out Note tat by adding stabilization te stabilization term β ν u λ 2 T, te last term becomes out strictly positive, i.e., B C (u, λ ; u, λ ) = BC NU (u, λ ; u, λ ) + β ν λ u 2 T out = 1 2 β ν λ u 2 T, and we recover te first part of te stability estimate of Proposition 5. Following te approac for te convection dominated case, we now consider te following metod for for te diffusion dominated regime, cf. also [18]: Metod 4 (no upwind) Find (σ, u, λ ) W suc tat olds for all (τ, v, µ ) W were B NU := B D + B NU C. B NU (σ, u, λ ; τ, v, µ ) = F(v, µ ), (37) For te proof of stability of te bilinear form B NU, we require tat te convection is sufficiently small. A sufficent condition is given by β ν λ u 2 T c D (σ, u, λ ) 2 D, (σ, u, λ ) W. (38) Remark 3 Recall tat te stability constant c D and tus te validity of condition (38) depends only on te constant of an inverse inequality and tus on te sape of te elements. Moreover, since bot norms are defined elementwise, it is possible to decide for eac element separately if stabilization sould be added or not. Clearly, (38) can be sown to old if β c T ɛ is valid on eac element wit constant c T only depending on te sape of te individual elements. Using (38) as te caracterization of dominating diffusion, we can now prove te following stability result. Proposition 12 Let (38) be valid. Ten te estimate B NU (σ, u, λ ; τ, v, µ ) sup c D (τ,v,µ ) 0 (τ, v, µ ) D 2 (σ, u, λ ) 2 D (39) olds for all (σ, u, λ ) W wit c D denoting te stability constant of Proposition 3. Since te convective terms can be absorbed by te diffusion terms, te boundedness result of Corollary 1 applies wit ( ) replaced by D,( ). Using te stability estimate (39), te following a-priori error estimate is obtained in a similar manner as Proposition 10 for te purely elliptic case. Proposition 13 Let condition (38) be valid and (σ, u, λ ) denote te solution of Metod 4. Moreover, let u H 1 (Ω) H 3/2+ε (T ) denote te solution of problem (11), and set σ := u. Ten (σ σ, u Π T k u, λ Π E k u) D C s u s+1,ω olds for all 1/2 < s k + 1 wit constant C independent of. 16

17 Proof. In view of Proposition 10 we only ave to ensure tat te convective term does not disturb te estimate. Following te proof of Proposition 10, i.e., testing wit te same test functions as tere, we obtain te additional term B NU C (Π T k u u, Π E k u u; v, µ ) = (Π T k u u, β v ) T + β ν (Π E k u u), µ v T = 0, since β v P k (T ) on eac element, and β ν (µ v ) P k (E) for eac facet. Te result now follows along te lines of te proof of Proposition 10. Proposition 13 allows us to derive a superconvergence estimate for Π T k u u T like in te purely elliptic case. Proposition 14 Let Ω be convex and u be te solution of (11) wit β satisfying (38). Moreover, let u denote te discrete solution of Metod 4. Ten Π T k u u T C s+1 { u s+2,t k = 0, u s+1,t k > 0, olds for 1/2 < s k + 1 respecively 0 s 1 in case k = 0. Proof. By means of Proposition 13, te result follows in te same way as Proposition 11. Due to te lack of a condition div(σ σ ) 0, wic is valid in te purely elliptic case, we can not obtain (34) ere. So in te lowest order case, superconvergence olds only under some additional smootness of te solution u. 4.3 Postprocessing Te super convergence results of te previous section can now be utilized to construct better approximations ũ P k+1 (T ) by local postprocessing. Here, we follow an approac proposed by Stenberg [33] for te mixed discretization of te Poisson equation (3), and construct our postprocessed solution from te approximations of te primal and te dual variable. Alternative approaces based on te Lagrange multipliers can be found in [3, 10]. Let us define ũ P k+1 (T ) elementwise by te variational problems ( u, v) T = (σ, v) T, v P k+1 (T ) : (v, 1) T = 0 (u, 1) E = (u, 1) T. Ten te following order optimal error estimate olds. Proposition 15 Let Ω be convex and u denote te solution of (11) wit (38) being valid. Moreover, let (σ, u, λ ) be te solution of Metod (4) and u be defined as above. Ten and (u u) T C s u s+1,t u u T C s+1 { u s+2,t, k = 0 u s+1,t, k > 0, for all 1/2 < s k + 1 wit constant C independent of te messize. For k = 0, te second estimate olds for 0 s 1. Proof. Let ũ H 1 (Ω) P k+1 (T ) denote te finite element solution of te standard H 1 conforming finite element metod applied to te solution of (3). Ten (u ũ ) T C s u s+1,t for 0 s k + 1. Moreover, u ũ C s+1 u s+1,t for 0 s k + 1, since we assumed convexity of Ω and f L 2. Now define ṽ := (I Π T 0 )(ũ u ). Ten ṽ 2 T = ( (I Π T 0 )(ũ u ), ṽ ) T = ( (ũ u ), ṽ ) T = ( (ũ u), ṽ ) T + ( u + σ, ṽ ) T ṽ T ( (u ũ ) T + σ + u ) T. 17

18 Summing up over all elements and using te estimates for (u ũ ) and Proposition 13 yields (u u ) T (u ũ ) T + (ũ u ) T = (u ũ ) T + ṽ T C s u s+1,t, wic already is te first part of te result. In order to establis te L 2 estimate, note tat by Π T 0 ṽ = 0 we obtain ṽ T C ṽ T via an inverse inequality. Hence u u T u ũ T + ũ u T u ũ T + ṽ T + Π T 0 (ũ u ) T = u ũ T + ṽ T + Π T 0 (ũ u) T + Π T 0 (u u ) T. Summing up over all elements, and using tat Π T 0 (ũ u) T ũ u T C s+1 u s+1,t and Proposition 14, we conclude te L 2 estimate. Remark 4 In te purely elliptic case (β 0) wit f piecewise constant, we can obtain te optimal estimate u u s+1 u s+1,t also for te case k = 0 by using te estimate (34) instead of Proposition Implementation and numerical tests Let us now illustrate te teoretical results derived in te previous section by some numerical tests. As a model problem, let us consider ɛ u + β u = f in Ω := (0, 1) 2 u = g on Ω, (40) were ɛ and β are constant on te wole domain. Since for te limiting yperbolic problem our metod is equivalent to te discontinuous Galerkin metod, we will compare our results mainly to tose obtained by te streamline diffusion metod [22, 25, 23]. For a detailed comparison of p-versions of te streamline diffusion metod wit discontinuous Galerkin metods for first order yperbolic problems we refer to [20]. Te variational form of te streamline diffusion metod is formally derived by using v + αβ v as a test function in te variational formulation of (40). Assuming g = 0 for simplicity, tis yields Metod 5 (streamline diffusion) Find u H 1 0 (Ω) H 2 (T ) suc tat ɛ( u, v) T + (β u, v) T + α[ ɛ( u, β v) T + (β u, β v) T ] = (f, v) T + α(f, β v). In order to obtain stability of te metod, te stabilization parameter as to be coses appropriately, depending on te sape of te elements in te mes. Typically, te stabilization parameter is in te order of / β, were is te local mes size. For iger order metods, also te polynomial degree influences te coice of α, cf. [20]. For our numerical tests we us α = max{ β 2ɛ, 0}/ β 2. In tis way, stabilization is turned off in te diffusion dominant regime. In a similar manner, we add edge stabilization (36) to our mixed ybrid discontinuous Galerkin metod wit a factor α = max{ β ν 2ɛ, 0}/( β ν ). 18

19 5.1 Numerical tests Wit a first example, we want to illustrate tat our metod is capable of dealing wit boundary layers near outflow boundaries very well, and in contrast to te streamline diffusion metod, we do not obtain large layers. In a second example, we ten sow tat even witout any kind of sock stabilization, discontinuities can be treated rater well even for almost yperbolic problems. As a general remark we would like to empasize tat we always compare our metod using polynomials of order k wit te streamline diffusion metod using polynomials of degree k + 1. Tus, formally te approximation properties of our finite element spaces is one order less. However, as our numerical results indicate, tis affects te results only in te diffusion dominated case, were according to our teory we can increase te approximations by local postprocessing. Tis is illustrated in Example 3. Example 1: In te first test we set g = 0 and f = β 1 [y + (e β2y/ɛ 1)/(1 e 1/ɛ )] + β 2 [1 + (e β1x/ɛ 1)/(1 e β1/ɛ )]. For ɛ > 0, te exact solution to (40) is ten given by u(x, y) = [x + (e β1x/ɛ 1)/(1 e β1/ɛ )] [y + (e β2y/ɛ 1)/(1 e β2/ɛ )], i.e., te solution as boundary layers at te top and rigt outflow boundaries. For a numerical study, we set ɛ = 0.01 and β = (2, 1), and ten solve te problem numerically for various messizes and polynomial degrees k. Table 1 displays te errors of te numerical solutions obtained wit Metod 3 and te streamline diffusion metod. streamline diffusion mixed ybrid DG ne k=1 k=2 k=3 k=0 k=1 k= Table 1: L 2 errors obtained for Example 1 on uniform meses wit messize and ne elements using order k polynomials. Te exact solution is almost bilinear away from te boundary layers. Terefore one cannot expect to gain muc from furter increasing te polynomial degree. Note owever, tat in te mixed-ybrid-dg(0) metod, te solution u is approximated only by piecwise constant functions. Neverteless, te error is smaller tan te one obtained by any of te streamline-diffusion(k) metods. Tis indicates tat most of te error in te streamline diffusion metod actually stems from adding te stabilization term. Tis also explains wy te streamline-diffusion(k) metods do not give better results for increased polynomial degree, wile our mixed-ybrid-dg metod does. Since in our example te location of boundary layers is determined a-priori, one sould of course use local mes refinement towards te outflow boundaries. Te exact solution using adaptive grids are plotted in Figure 1. In Table 2 we summarise te numerical results obtained on adaptive meses. In order to obtain an error of less tan we can use te streamline-diffusion(2) metod wit 898 elements, te mixed-ybrid-dg(0) metod wit 320 elements or te mixed-ybrid-dg(1) metod wit only 8 elements. Similar results are also obtained in te second Example below. If we continue to locally refine te mes towards te boundary layer, te problem becomes diffusion dominant (at least in te boundary layers), and te streamline-diffusion(k+1) metod gives better results ten te mixed-ybrid-dg(k) metod. In te current example, tis appens at about 5000 elements. In te diffusion dominant region, we can owever improve te solution obtained by our ybrid metod by local postprocessing, cf. Example 3 below. 19

20 Figure 1: Example 1: exact solution and locally adapted mes wit 878 elements. streamline diffusion mixed ybrid DG ne k=1 k=2 k=3 k=0 k=1 k= Table 2: L 2 error of numerical solutions obtained wit te streamlinediffusion(k) and te mixed-ybrid-dg(k) metod for adaptively refined mes wit ne elements and minimal messize. Example 2: For a second test, we set f = 0, β = (2, 1) as before, and ɛ = 10 6, so we are dealing wit an (almost) yperbolic problem. Additionally, we introduce a discontinuity in te boundary conditions, i.e., we set u(0, y) = H(y 0.5) on te left inflow boundary (H( ) denotes te Heavyside function), and we set u = 0 on te remaining part of te boundary. Te exact solution for ɛ = 0 (te boundary conditions at te outflow boundaries ave to be omitted in tis case) is given by { 1 y > 0.5(1 + x) u(x, y) = 0 else. Below we use te solution of te purely yperbolic problem also for calculation of te numerical errors of te finite element solutions. Again, we solve on uniform meses (not aligned to te discontinuity) and compare te solutions obtained wit Metod 3 and te streamline upwind metod for different polynomial degrees. streamline diffusion mixed ybrid DG ne k=1 k=2 k=3 k=0 k=1 k= Table 3: L 2 errors of streamline-diffusion(k) and mixed-ybrid-dg(k) metod for uniformly refined meses wit messize and ne elements for polynomial degree k. Since te exact solution is piecewise constant, increasing te polynomial degree can only yield improvements of te approximations in te boundary and internal layers. As in te previous 20

21 example, we can igly improve te performance by using locally adapted meses. Te corresponding numerical results are listed in Table 4. In Figure 2 we plot solutions obtained wit te streamline diffusion mixed ybrid DG ne k=1 k=2 k=3 k=0 k=1 k= Table 4: L 2 errors of streamline-diffusion(k) and mixed-ybrid-dg(k) metod for adaptively refined meses wit ne elements and minimal messize for polynomial degree k. streamline-diffusion(3) and mixed-ybrid-dg(2) metod on an appropriate adaptive mes. Bot Figure 2: Streamline-diffusion(3) and mixed-ybrid-dg(2) solution obtained on appropriate adapted meses wit 878 respectively 798 elements. solutions resolve te discontinuity at te interior layer rater stably and oscillations are located only witin te elements toucing te discontinuity. Moreover, altoug te mes is not aligned wit te flow direction β, te numerical diffusion is very small and te jump of te solution stays witin one element trougout te domain. Note also tat te mixed ybrid DG metod deals wit te outflow boundary in a more natural way tan te streamline diffusion metod, so no mes refinement is needed at te boundary layers. Let us now turn to a diffusion dominated problem an illustrate te increase in accuracy obtained by local post-processing discussed in Section 4.3. Example 3: Set β = (2, 1), ɛ = 0.1 and f = 1 in (40), and let u = 0 at te boundary. As an approximation for te true solution we take te conforming finite element solution wit polynomial degree 8. We solve problem (40) wit Metod 3 and 4 respectively, and compare te numerical results wit tose obtained by te streamline diffusion metod. As outlined above, we decrease te stabilization parameter wit te messize suc tat no stabilization is added for β < 2ɛ. Te results obtained for various mes sizes and polynomial degrees are summarized in Table 5. Since te problems become diffusion dominant for a messize of 0.1, te streamlinediffusion(k+1) metods give better results tan te ybrid(k) metods in te last two lines of Table 5. For improving te approximation for te mixed ybrid DG metod in tat case, we now apply local postprocessing as discussed in Section 4. In Table 6 we list te results obtained wit te 21