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Discontinuous Galerkin methods and applications Scalar second-order PDEs Daniele A. Di Pietro A.A. 2015–2016 1 / 72 Setting I For f ∈ L2 (Ω) we consider the model problem −4u = f in Ω, on ∂Ω, u=0 The weak formulation reads with V := H01 (Ω), Z Find u ∈ V s.t. a(u, v) = f v for all v ∈ V , (Π) Ω where Z ∇u·∇v a(u, v) := Ω 2 / 72 Setting II The well-posedness of (Π) hinges on Poincaré’s inequality, ∀v ∈ H01 (Ω), kvkL2 (Ω) ≤ CΩ k∇vk[L2 (Ω)]d Indeed, a classical result is the coercivity of a, ∀v ∈ H01 (Ω), a(v, v) ≥ 1 kvk2H 1 (Ω) 1 + CΩ2 3 / 72 Setting III Lemma (Continuity of the potential and of the diffusive flux) Letting JvKF = {{v}}F = v for all F ∈ Fhb , it holds JuK = 0 J∇uK·nF = 0 ∀F ∈ Fh , ∀F ∈ Fhi . 4 / 72 Setting IV Assumption (Regularity of exact solution and space V∗ ) We assume that the exact solution u is s.t. u ∈ V∗ := V ∩ H 2 (Ω). We set V∗h := V∗ + Vh . This implies, in particular, that the traces of both u and ∇u·nF are square-integrable. 5 / 72 Symmetric Interior Penalty: Heuristic derivation I Vh := Pkd (Th ), k≥1 We derive a dG method for (Π) based on a bilinear form ah We proceed by enforcing consistency, symmetry, and coercivity For all (v, wh ) ∈ V∗h × Vh we set (0) ah (v, wh ) Z ∇h v·∇h wh = := Ω X Z T ∈Th ∇v·∇wh T 6 / 72 Consistency I Integrating by parts element-by-element we arrive at X Z X Z (0) ah (v, wh ) = − (4v)wh + (∇v·nT )wh T ∈Th T T ∈Th ∂T The second term on the RHS can be reformulated as follows: X Z T ∈Th (∇v·nT )wh = ∂T X Z i F ∈Fh F J(∇h v)wh K·nF + X Z b F ∈Fh (∇v·nF )wh , F since for all F ∈ Fhi with F = ∂T1 ∩ ∂T2 , nF = nT1 = −nT2 7 / 72 Consistency II Moreover, J(∇h v)wh K = {{∇h v}}Jwh K + J∇h vK{{wh }}, since letting ai = (∇v)|Ti , bi = wh |Ti , i ∈ {1, 2}, yields J(∇h v)wh K = a1 b1 − a2 b2 = 21 (a1 + a2 )(b1 − b2 ) + (a1 − a2 ) 12 (b1 + b2 ) = {{∇h v}}Jwh K + J∇h vK{{wh }}. As a result, and accounting also for boundary faces, X Z T ∈Th (∇v·nT )wh = ∂T X Z F ∈Fh F {{∇h v}}·nF Jwh K+ X Z i F ∈Fh F J∇h vK·nF {{wh }} 8 / 72 Consistency III In conclusion, (0) ah (v, wh ) = − X Z T ∈Th + T X Z F F ∈Fhi X Z (4v)wh + F ∈Fh F {{∇h v}}·nF Jwh K J∇h vK·nF {{wh }} To check consistency, set v = u. For all wh ∈ Vh , Z X Z (0) ah (u, wh ) = f wh + (∇u·nF )Jwh K Ω F ∈Fh F (0) Hence, we modify ah as follows: (1) ah (v, wh ) Z ∇h v·∇h wh − := Ω X Z F ∈Fh F {{∇h v}}·nF Jwh K 9 / 72 Symmetry I A desirable property is symmetry since It simplifies the solution of the linear system It is used to prove optimal L2 -error error estimates (1) We consider the following modification of ah : acs h (v, wh ) Z := ∇h v·∇h wh X Z − ({{∇h v}}·nF Jwh K + JvK{{∇h wh }}·nF ) Ω F ∈Fh F 10 / 72 Symmetry II Element-by-element integration by parts yields X Z X Z cs ah (v, wh ) = − (4v)wh + J∇h vK·nF {{wh }} T ∈Th − T X Z F ∈Fh F F ∈Fhi F JvK{{∇h wh }}·nF This shows that acs h retains consistency since J∇h uKF ·nF = 0 ∀F ∈ Fhi , JuKF = 0 ∀F ∈ Fh 11 / 72 Coercivity I For all vh ∈ Vh it holds acs h (vh , vh ) = k∇h vh k2[L2 (Ω)]d X Z −2 F ∈Fh F {{∇h vh }}·nF Jvh K The difficulty is that the second term has no a priori sign We further modify acs h as follows: For all (v, wh ) ∈ V∗h × Vh , cs asip h (v, wh ) := ah (v, wh ) + sh (v, wh ), with stabilization bilinear form sh (v, wh ) := X F ∈Fh η hF Z F JvKJwh K 12 / 72 Coercivity II We aim at asserting coercivity in the norm ∀v ∈ V∗h , 12 |||v|||sip := k∇h vk2[L2 (Ω)]d + |v|2J , with jump seminorm |v|J := (η −1 1 2 sh (v, v)) = X F ∈Fh 1 kJvKk2L2 (F ) hF ! 12 The following discrete Poincaré’s inequality holds with σ2 > 0 independent of h: ∀vh ∈ Vh , kvh kL2 (Ω) ≤ σ2 |||vh |||sip 13 / 72 Coercivity III The choice for sh is justified by the following result. Lemma (Bound on consistency and symmetry terms) For all (v, wh ) ∈ V∗h × Vh , 1 2 X Z X X 2 ≤ h k∇v| ·n k { {∇ v} }·n Jw K |wh |J . 2 F T F F h h L (F ) F ∈Fh F T ∈Th F ∈FT Moreover, if v = vh ∈ Vh , X Z 1 {{∇h vh }}·nF Jwh K ≤ Ctr N∂2 k∇h vh k[L2 (Ω)]d |vh |J . F F ∈Fh 14 / 72 Coercivity IV For all Fhi 3 F = ∂T1 ∩ ∂T2 , the Cauchy–Schwarz inequality yields Z F Z {{∇h v}}·nF Jwh K = F ≤ 1 (a1 + a2 )Jwh K 2 1 2 1 −1 2 2 hF 2 kJwh KkL2 (F ) hF (ka1 kL2 (F ) + ka2 kL2 (F ) ) 2 where ai := (∇v)|Ti ·nF , i ∈ {1, 2} For all F ∈ Fhb with F = ∂T ∩ ∂Ω, Z 1 −1 {{∇h v}}·nF Jwh K ≤ hF2 k∇v|T ·nF kL2 (F ) × hF 2 kJwh KkL2 (F ) F Summing over Fh , using the Cauchy–Schwarz inequality, and regrouping element-wise yields the first inequality 15 / 72 Coercivity V To prove the second inequality, let v = vh ∈ Vh Using hF ≤ hT together with the discrete trace inequality yields X X X X hF k∇vh |T ·nF k2L2 (F ) ≤ hT k∇vh |T ·nF k2L2 (F ) T ∈Th F ∈FT T ∈Th F ∈FT 2 ≤ Ctr X X k∇vh k2[L2 (T )]d T ∈Th F ∈FT ≤ 2 Ctr N∂ k∇h vh k2[L2 (Ω)]d Plugging this result into the first inequality yields the second 16 / 72 Coercivity VI Lemma (Discrete coercivity) 2 2 For all η > η := Ctr N∂ it holds with Cη := (η − Ctr N∂ )(1 + η)−1 , ∀vh ∈ Vh , 2 asip h (vh , vh ) ≥ Cη |||vh |||sip . 17 / 72 Coercivity VII asip h (v, wh ) = Z ∇h v·∇h wh − Ω + X Z F ∈Fh X F ∈Fh η hF F {{∇h v}}·nF Jwh K + JvK{{∇h wh }}·nF Z F JvKJwh K, Using the bound on consistency and symmetry terms, 1/2 2 2 asip h (vh , vh ) ≥ k∇h vh k[L2 (Ω)]d − 2Ctr N∂ k∇h vh k[L2 (Ω)]d |vh |J + η|vh |J For all β ∈ R+ , η > β 2 , x, y ∈ R, it holds x2 − 2βxy + ηy 2 ≥ η − β2 2 (x + y 2 ) 1+η 1/2 Let β = Ctr N∂ , x = k∇h vh k[L2 (Ω)]d , y = |vh |J to conclude 18 / 72 Basic energy error estimate I By construction, asip h enjoys consistency and coercivity It only remains to prove boundedness To this end, we recall the estimate: For all (v, wh ) ∈ V∗h × Vh , 1 2 X Z X X 2 hF k∇v|T ·nF kL2 (F ) |wh |J {{∇h v}}·nF Jwh K ≤ F ∈Fh F T ∈Th F ∈FT This suggests to consider the augmented norm ! 12 |||v|||sip,∗ := |||v|||2sip + X hT k∇v|T ·nT k2L2 (∂T ) T ∈Th 19 / 72 Basic energy error estimate II Lemma (Boundedness) There is Cbnd , independent of h, s.t. ∀(v, wh ) ∈ V∗h × Vh , asip h (v, wh ) ≤ Cbnd |||v|||sip,∗ |||wh |||sip . 20 / 72 Basic energy error estimate III asip h (v, wh ) = Z X Z ∇h v·∇h wh − Ω + F ∈Fh X F ∈Fh η hF F {{∇h v}}·nF Jwh K− X Z F ∈Fh F JvK{{∇h wh }}·nF Z F JvKJwh K := T1 + T2 + T3 + T4 . The Cauchy–Schwarz inequality yields |T1 | ≤ k∇h vk[L2 (Ω)]d k∇h wh k[L2 (Ω)]d , |T4 | ≤ η|v|J |wh |J Moreover, the choice of |||·|||sip,∗ is s.t. |T2 | ≤ |||v|||sip,∗ |wh |J ≤ |||v|||sip,∗ |||wh |||sip Finally, 1 1 |T3 | ≤ Ctr N∂2 |v|J k∇h wh k[L2 (Ω)]d ≤ Ctr N∂2 |||v|||sip |||wh |||sip 21 / 72 Basic energy error estimate IV Find uh ∈ Vh s.t. asip h (uh , vh ) = Z f vh for all vh ∈ Vh Ω Theorem (Energy error estimate) Assume u ∈ V∗ and η > η. Then, there is C, independent of h, s.t. |||u − uh |||sip ≤ C inf |||u − vh |||sip,∗ . vh ∈Vh 22 / 72 Basic energy error estimate V Corollary (Convergence rate in |||·|||sip -norm) Additionally assume u ∈ H k+1 (Ω). Then, it holds |||u − uh |||sip ≤ Cu hk , with Cu = CkukH k+1 (Ω) and C independent of h. Remark (Lowest-order dG methods) It is clear from the above estimate that convergence requires k ≥ 1, i.e., we cannot take k = 0 23 / 72 L2 -norm error estimate I Using the broken Poincaré inequality of [Brenner, 2004] one can infer ku − uh kL2 (Ω) ≤ σ20 Cu hk This estimate is suboptimal by one power in h An optimal estimate can be recovered exploiting symmetry Further regularity for the problem needs to be assumed 24 / 72 L2 -norm error estimate II Definition (Elliptic regularity) Elliptic regularity holds for problem (Π) if there is Cell , only depending on Ω, s.t., for all ψ ∈ L2 (Ω), the solution to the problem: Z Find ζ ∈ H01 (Ω) s.t. a(ζ, v) = ψv for all v ∈ H01 (Ω), Ω is in V∗ and satisfies kζkH 2 (Ω) ≤ Cell kψkL2 (Ω) . 25 / 72 L2 -norm error estimate III Theorem (L2 -norm error estimate) Let u ∈ V∗ solve (Π) and assume elliptic regularity. Then, there is C, independent of h, s.t. ku − uh kL2 (Ω) ≤ Ch|||u − uh |||sip,∗ . Corollary (Convergence rate in k·kL2 (Ω) -norm) Additionally assume u ∈ H k+1 (Ω). Then, it holds ku − uh kL2 (Ω) ≤ Cu hk+1 . with Cu = CkukH k+1 (Ω) and C independent of h. 26 / 72 L2 -norm error estimate IV We consider the auxiliary problem Find ζ ∈ H01 (Ω) s.t. a(ζ, v) = Z (u − uh )v for all v ∈ H01 (Ω) Ω 27 / 72 L2 -norm error estimate V Since ζ ∈ V∗ , consistency of asip h implies Z sip ah (ζ, u − uh ) = (−4ζ)(u − uh ) Ω Exploiting the symmetry of asip h and since −4ζ = u − uh , we obtain 2 asip h (u − uh , ζ) = ku − uh kL2 (Ω) Using again consistency and the fact that k ≥ 1 =⇒ πh1 ζ ∈ Vh , 1 asip h (u − uh , πh ζ) = 0 28 / 72 L2 -norm error estimate VI Elliptic regularity yields kζkH 2 (Ω) ≤ Cell ku − uh kL2 (Ω) Hence, 1 ku − uh k2L2 (Ω) = asip h (u − uh , ζ − πh ζ) . |||u − uh |||sip,∗ |||ζ − πh1 ζ|||sip,∗ (Boundedness) . |||u − uh |||sip,∗ hkζkH 2 (Th ) (Approximation) . |||u − uh |||sip,∗ hku − uh kL2 (Ω) (Ell. reg.) 29 / 72 Liftings I Liftings map jumps onto vector-valued functions defined on elements Liftings play a key role in several developments Flux and mixed formulations Computable lower bound for η Convergence to minimal regularity solutions The theoretical developments will eventually allow us to analyze dG methods for nonlinear problems such as the Navier–Stokes equations 30 / 72 Liftings II For an integer l ≥ 0, we define the (local) lifting operator rlF : L2 (F ) −→ [Pld (Th )]d , as follows: For all ϕ ∈ L2 (F ), Z rlF (ϕ)·τh Ω Z {{τh }}·nF ϕ = ∀τh ∈ [Pld (Th )]d F We observe that supp(rlF ) = [ T T ∈TF 31 / 72 Liftings III Lemma (Bound on local lifting) Let F ∈ Fh and let l ≥ 0. For all ϕ ∈ L2 (F ), it holds −1 k rlF (ϕ)k[L2 (Ω)]d ≤ Ctr hF 2 kϕkL2 (F ) . 32 / 72 Liftings IV Using the Cauchy–Schwarz and discrete trace inequalities, k rlF (ϕ)k2[L2 (Ω)]d = Z rlF (ϕ)· rlF (ϕ) = Z Ω ≤ {{rlF (ϕ)}}·nF ϕ F Z 1 hF |ϕ|2 1 2 Z 1 2 × hF |{{rlF (ϕ)}}|2 F F 1 ≤ −1 hF 2 kϕkL2 (F ) × Ctr card(TF )−1 X Z T ∈TF 2 | rlF (ϕ)|2 T The result follows since card(TF )−1 ≤ 1 and X Z T ∈TF ! 12 | rlF (ϕ)|2 = k rlF (ϕ)kL2 (Ω)d T 33 / 72 Liftings V For all l ≥ 0 and v ∈ H 1 (Th ), we define the (global) lifting Rlh (JvK) := X rlF (JvK) ∈ [Pld (Th )]d F ∈Fh Rlh (JvK) maps the jumps of v into a global, vector-valued volumic contribution which is homogeneous to a gradient 34 / 72 Liftings VI Lemma (Bound on global lifting) Let l ≥ 0. For all v ∈ H 1 (Th ), it holds k Rlh (JvK)k[L2 (Ω)]d 1 2 ≤ N∂ ! 21 X k rlF (JvK)k2[L2 (Ω)]d , F ∈Fh so that 1 k Rlh (JvK)k[L2 (Ω)]d ≤ Ctr N∂2 |v|J . 35 / 72 Liftings VII Accounting for the support of local liftings, it holds X (Rlh (JvK))|T = (rlF (JvK))|T F ∈FT Using the Cauchy–Schwarz inequality we obtain k Rlh (JvK)k2[L2 (Ω)]d 2 X Z X l = rF (JvK) T ∈Th T F ∈FT X X Z ≤ card(FT ) | rlF (Jvh K)|2 T ∈Th F ∈FT ≤ max card(FT ) T ∈Th X T X Z T ∈Th F ∈FT T | rlF (Jvh K)|2 X k rlF (JvK)k2[L2 (Ω)]d = max card(FT ) T ∈Th {z } F ∈Fh | N∂ The second bound is a direct consequence of the stability for rlF 36 / 72 Discrete gradients I For l ≥ 0, we define the discrete gradient operator Glh : H 1 (Th ) −→ [L2 (Ω)]d , as follows: For all v ∈ H 1 (Th ), Glh (v) := ∇h v − Rlh (JvK) The discrete gradient accounts for inter-element and boundary jumps 37 / 72 Discrete gradients II Lemma (Bound on discrete gradient) Let l ≥ 0. For all v ∈ H 1 (Th ), it holds 1 2 kGlh (v)k[L2 (Ω)]d ≤ (1 + Ctr N∂ ) 2 |||v|||sip . Proof. Let v ∈ H 1 (Th ). The triangle inequality and the bound on Rlh (JvK) yield kGlh (v)k[L2 (Ω)]d ≤ k∇h vk[L2 (Ω)]d + k Rlh (JvK)k[L2 (Ω)]d 1 ≤ k∇h vk[L2 (Ω)]d + Ctr N∂2 |v|J . 38 / 72 Reformulation of asip h I Let l ∈ {k − 1, k} and set Vh := Pkd (Th ) with k ≥ 1 It holds for all vh , wh ∈ Vh , acs h (vh , wh ) = Z Z ∇h vh ·∇h wh − Ω Ω ∇h vh · Rlh (Jwh K)− Z Ω ∇h wh · Rlh (Jvh K) Indeed, ∇h vh ∈ [Pld (Th )]d with l ≥ k − 1, Z Z ∀F ∈ Fh , {{∇h vh }}·nF Jwh K = ∇h vh · rlF (Jwh K) F Ω Using the definition of discrete gradients, acs h (vh , wh ) = Z Ω Glh (vh )·Glh (wh ) − Z Ω Rlh (Jvh K)· Rlh (Jwh K) 39 / 72 Reformulation of asip h II Plugging the above expression into asip h , asip h (vh , wh ) Z = Ω Glh (vh )·Glh (wh ) + ŝsip h (vh , wh ), with ŝsip h (vh , wh ) := X F ∈Fh η hF Z F Z Jvh KJwh K − Ω Rlh (Jvh K)· Rlh (Jwh K) Dropping the negative term in ŝsip h leads to the Local Discontinuous Galerkin (LDG) method of [Cockburn and Shu, 1998] This method has the drawback of having a significantly larger stencil 40 / 72 Stencil Z ∇h vh ·∇h wh Ω Z ∇h vh · Rlh (Jwh K)+∇h wh · Rlh (Jvh K) , Ω X η Z Jvh KJwh K hF F F ∈F h Z Ω Rlh (Juh K)· Rlh (Jvh K), Z Ω Glh (vh )·Glh (wh ) Figure: Stencil of the different terms 41 / 72 Numerical fluxes I Discrete gradients can be used to identify discrete fluxes and prove local conservation properties Such properties are important when the flux is used as an advective velocity in transport problems, as is the case in porous media flow Unlike other FE methods, dG methods often possess local conservation properties on the computational mesh (as opposed to vertex-centered or face-centered macro-elements) 42 / 72 Numerical fluxes II Let T ∈ Th , ξ ∈ Pkd (T ). Element-by-element IBP yields Z Z Z Z f ξ = − (4u)ξ = ∇u·∇ξ − (∇u·nT )ξ T T T ∂T Hence, letting ΦF (u) := −∇u·nF and T,F = nT ·nF , Z Z Z X ∇u·∇ξ + T,F ΦF (u)ξ = fξ T F ∈FT F T Our goal is to identify a similar relation for the discrete solution uh 43 / 72 Numerical fluxes III Using vh = ξχT as test function we obtain Z T f ξ = asip h (uh , ξχT ) = Z ∇uh ·∇ξ − T − X Z F ∈FT X Z F ∈FT F F {{(∇ξ)χT }}·nF Juh K {{∇h uh }}·nF JξχT K + X Z F ∈FT F η Juh KJξχT K hF Let l ∈ {k − 1, k}. For all T ∈ Th and all ξ ∈ Pkd (T ), Z Z Z X l Gh (uh )·∇ξ+ T,F φF (uh )ξ = f ξ, T with F ∈FT F T η φF (uh ) := − {{∇h uh }}·nF + Juh K hF | {z } | {z } consistency penalty 44 / 72 Numerical fluxes IV Taking ξ ≡ 1 we infer X F ∈FT Z T,F Z φF (uh ) = F f T 45 / 72 SIP as a mixed dG method I Liftings and discrete gradients appear in mixed dG methods The mixed form has been made popular by the unified analysis of [Arnold et al., 2002] The focus is on the SIP and LDG methods 46 / 72 SIP as a mixed dG method II One mixed formulation of the homogeneous Poisson problem consists in finding (σ, u) ∈ X := [L2 (Ω)]d × H01 (Ω) s.t. 2 d m(σ, τ ) + b(τ, Z u) = 0 ∀τ ∈ [L (Ω)] , −b(σ, v) = fv ∀v ∈ H01 (Ω) (Πm ) Ω where, for all σ, τ ∈ [L2 (Ω)]d and all v ∈ H01 (Ω), Z Z m(σ, τ ) := σ·τ, b(τ, v) := τ ·∇v Ω Ω Clearly, (σ, u) ∈ X solves (Πm ) iff σ = −∇u and u solves (Π) 47 / 72 SIP as a mixed dG method III Let k ≥ 1, l ∈ {k − 1, k} and set Σh := [Pld (Th )]d , Uh := Pkd (Th ), Xh := Σh × Uh The discrete problem consists in finding (σh , uh ) ∈ Xh s.t. m(σh , τh ) + bh (τh , uh ) = 0 Z −bh (σh , vh ) + ŝsip f vh h (uh , vh ) = ∀τh ∈ Σh , ∀vh ∈ Uh , (Πm,h ) Ω with discrete bilinear form Z bh (τh , vh ) := τh ·Glh (vh ) Ω 48 / 72 SIP as a mixed dG method IV Proposition (Elimination of discrete diffusive flux) The pair (σh , uh ) ∈ Xh solves (Πm,h ) iff σh = −Glh (uh ), and uh ∈ Uh is s.t. Z Z sip l l asip (u , v ) = G (u )·G (v ) + ŝ (u , v ) = f vh h h h h h h h h h h Ω ∀vh ∈ Uh . Ω 49 / 72 SIP as a mixed dG method V The first equation in (Πm,h ) yields Z (σh + Glh (uh ))·τh = 0 ∀τh ∈ Σh Ω Since l ≥ k − 1, Glh (uh ) = ∇h uh − Rlh (Juh K) ∈ Σh , hence σh = −Glh (uh ) To conclude, plug this relation into the second equation of (Πm,h ) The converse is straightforward 50 / 72 Numerical fluxes and variations of SIP I Let, for the sake of simplicity, l = k, so that Σh = [Pkd (Th )]d , Uh = Pkd (Th ) Let T ∈ Th , ζ ∈ [Pkd (T )]d , and ξ ∈ Pkd (T ) Local IBP on T ∈ Th yields for the exact solution Z Z Z X σ·ζ − u∇·ζ + T,F uF (ζ·nF ) = 0, T T − σ·∇ξ + T F F ∈FT Z X Z (σF ·nF )ξ = T,F F ∈FT Z F f ξ, T with uF and σF single-valued on F 51 / 72 Numerical fluxes and variations of SIP II At the discrete level, uF and σF are replaced by ûF and σ̂F , with ûF scalar-valued and σ̂F is vector-valued We obtain, for all T ∈ Th , all ζ ∈ [Pkd (T )]d , and all ξ ∈ Pkd (T ), Z Z σh ·ζ − T uh ∇·ζ + T σh ·∇ξ + T Z T,F X F ∈FT ûF (ζ·nF ) = 0, F F ∈FT Z − X Z Z (σ̂F ·nF )ξ = T,F F fξ T 52 / 72 Numerical fluxes and variations of SIP III For the mixed formulation of SIP, the numerical fluxes are ( {{uh }} ∀F ∈ Fhi , ûF = 0 ∀F ∈ Fhb , σ̂F = −{{∇h uh }} + ηh−1 F Juh KnF ∀F ∈ Fh A variant of the SIP method consists letting σ̂F = {{σh }} + ηh−1 F Juh KnF As a result, we obtain the LDG method 53 / 72 Numerical fluxes and variations of SIP IV σh can still be locally eliminated since ûF only depends on uh The corresponding primal bilinear form is Z aldg (u , v ) = ∇h uh ·∇h vh h h h Ω X Z − ({{∇h uh }}·nF Jvh K + {{∇h vh }}·nF Juh K) F ∈Fh Z + Ω Z = Ω F Rkh (Juh K)· Rkh (Jvh K) Gkh (uh )·Gkh (vh ) + + X F ∈Fh X F ∈Fh η hF η hF Z F Juh KJvh K Z F Juh KJvh K Coercivity holds for all η > 0 at the price of an enlarged stencil 54 / 72 The BR2 method I Following [Bassi et al., 1997], we consider cs br2 abr2 h (vh , wh ) := ah (vh , wh ) + sh (vh , wh ) with, for η > 0 and l ∈ {k − 1, k}, sbr2 h (vh , wh ) := X F ∈Fh Z η Ω rlF (Jvh K)· rlF (Jwh K) Coercivity holds for all η > N∂ with the same stencil as SIP 55 / 72 The BR2 method II ! 12 |||vh |||br2 := k∇h vh k2[L2 (Ω)]d + X F ∈Fh k rlF (Jvh K)k2[L2 (Ω)]d Lemma (Uniform norm equivalence) Let k ≥ 1 and let l ≥ 0. it holds, for all vh ∈ Vh , 1 1 2 2 (1 + 2Cr−2 )− 2 kvh kdG ≤ |||vh |||br2 ≤ max(1, Ctr ) kvh kdG , with Cr stability constant for |·|J with respect to the |||·|||br2 norm. 56 / 72 The BR2 method III Lemma (Discrete coercivity) Assume η > N∂ . Then, it holds ∀vh ∈ Vh , 2 abr2 h (vh , vh ) ≥ Csta kvh kdG , with Csta := Cη (1 + 2Cr−2 )−1 and Cη = (η − N∂ )(1 + η)−1 . 57 / 72 The BR2 method IV 1 k Rlh (JvK)k[L2 (Ω)]d ≤ Ctr N∂2 |v|J It holds, for all vh ∈ Vh , abr2 h (vh , vh ) = Z |∇h vh |2 −2 Ω Z Ω ∇h vh · Rlh (Jvh K)+η X F ∈Fh k rlF (Jvh K)k2[L2 (Ω)]d Hence, using the bound for k Rlh (Jvh K)k[L2 (Ω)]d , 2 abr2 h (vh , vh ) ≥ k∇h vh k[L2 (Ω)]d + η X F ∈Fh k rlF (Jvh K)k2[L2 (Ω)]d 1 1 2 − 2N∂ k∇h vh k[L2 (Ω)]d 2 X F ∈Fh k rlF (Jvh K)k2[L2 (Ω)]d Proceeding as for the SIP bilinear form yields the assertion 58 / 72 The BR2 method V Let l ∈ {k − 1, k}. For all T ∈ Th and all ξ ∈ Pkd (T ), it holds Z Z Z X Glh (uh )·∇ξ + T,F φF (uh )ξ = f ξ, T F ∈FT F T where, this time, φF (uh ) := −{{∇h uh }}·nF + η{{rlF (Juh K)}}·nF The corresponding mixed formulation is obtained for ( ûF = {{uh }} ∀F ∈ Fhi , 0 ∀F ∈ Fhb , σ̂F = −{{∇h uh }} + η{{rlF (Juh K)}}·nF ∀F ∈ Fh 59 / 72 Setting I Let κ ∈ L∞ (Ω) be uniformly bounded from below in Ω For f ∈ L2 (Ω), we are interested in the problem −∇·(κ∇u) = f in Ω, on ∂Ω u=0 The weak form of this problem is Z Find u ∈ V s.t. a(u, v) = f v for all v ∈ V (Πκ ) Ω with energy space V = H01 (Ω) and bilinear form Z a(u, v) := κ∇u·∇v Ω 60 / 72 Setting II Assumption (Partition of Ω) There is a partition PΩ := {Ωi }1≤i≤NΩ of Ω s.t. Each Ωi , 1 ≤ i ≤ NΩ , is a polyhedron; The restriction of κ to each Ωi , 1 ≤ i ≤ NΩ , is constant. 61 / 72 Setting III Figure: Exact solutions corresponding to different heterogeneity ratios 62 / 72 The SWIP method I Assumption (Mesh compatibility) We suppose that the admissible mesh sequence TH is s.t., for each h ∈ H, each T ∈ Th is a subset of only one set Ωi of the partition PΩ . In this situation, the meshes are said to be compatible with the partition PΩ . 63 / 72 The SWIP method II Definition (Weighted averages) To any interface F ∈ Fhi with F = ∂T1 ∩ ∂T2 , we assign two nonnegative real numbers ωT1 ,F and ωT2 ,F s.t. ωT1 ,F + ωT2 ,F = 1. For a scalar-valued function v smooth enough, we define its weighted average on F s.t., for a.e. x ∈ F , {{v}}ω,F (x) := ωT1 ,F v|T1 (x) + ωT2 ,F v|T2 (x). For all F ∈ Fhb with F = ∂T ∩ ∂Ω, we set {{v}}ω,F (x) := v|T (x). 64 / 72 The SWIP method III We consider the SWIP bilinear form: For all (vh , yh ) ∈ Vh × Vh , aswip (vh , yh ) := h Z κ∇h vh ·∇h yh + Ω − X F ∈Fh X Z F ∈Fh F η γκ,F hF Z F Jvh KJyh K ({{κ∇h vh }}ω ·nF Jyh K + Jvh K{{κ∇h yh }}ω ·nF ) where ( γκ,F := 2κ1 κ2 κ1 +κ2 γκ,F := κ|T ∀F ∈ Fhi s.t. F = ∂T1 ∩ ∂T2 , ∀F ∈ Fhb s.t. F = ∂T ∩ ∂Ω We observe that γκ,F ≤ 2 min(κ1 , κ2 ) 65 / 72 The SWIP method IV Assumption (Regularity of exact solution and space V∗ ) We assume that the exact solution u to (Πκ ) is s.t. u ∈ V∗ with V∗ := V ∩ H 2 (PΩ ). We set V∗h := V∗ + Vh . 66 / 72 The SWIP method V 1 |||v|||2swip := kκ 2 ∇h vk2[L2 (Ω)]d + |v|2J,κ , |v|2J,κ := X γκ,F kJvKk2L2 (F ) hF F ∈Fh Theorem (|||·|||swip -norm error estimate and convergence rate) There is C, independent of h and κ, s.t. |||u − uh |||swip ≤ C inf |||u − vh |||swip,∗ . vh ∈Vh Moreover, if u ∈ H k+1 (PΩ ), 1 |||u − uh |||swip ≤ Cu kκkL2 ∞ (Ω) hk , with Cu = CkukH k+1 (PΩ ) and C independent of h and κ. 67 / 72 A weighted method for diffusion-advection-reaction I ∇·(−κ∇u + βu) + µ̃u = f u=0 in Ω, on ∂Ω We assume κ and β as for the heterogeneous diffusion and pure advection problems The reaction coefficient µ̃ ∈ L∞ (Ω) is s.t. 1 1 Λ := µ̃ + ∇·β = µ − ∇·β ≥ µ0 a.e. in Ω. 2 2 68 / 72 A weighted method for diffusion-advection-reaction II swip adar (v, wh ) + aupw h (v, wh ) = ah h (v, wh ), aswip (v, wh ) h Z := κ∇h v·∇h wh + Ω F ∈Fh X Z − F ∈Fh aupw h (v, wh ) F Z = X γκ,F η hF Z F JvKJwh K ({{κ∇h v}}ω ·nF Jwh K + JvK{{κ∇h wh }}ω ·nF ) Z [µ̃vwh + ∇h ·(βv)wh ] + (β·n) vwh ∂Ω X Z X Z − (β·nF )JvK{{wh }} + γβ,F JvKJwh K Ω F ∈Fhi F F ∈Fhi F 69 / 72 A weighted method for diffusion-advection-reaction III |||v|||2da[ := |||v|||2swip + |v|2β + τc−1 kvk2L2 (Ω) X |||v|||2da] := |||v|||2da[ + βc−1 hT kβ·∇vk2L2 (T ) T ∈Th Theorem (Error estimate) There is C, independent of h and the data κ, β and µ̃, s.t. −1 |||u − uh |||da] ≤ C max(1, τc−1 µ−1 0 , Cη ) inf |||u − vh |||da],∗ . vh ∈Vh 70 / 72 A weighted method for diffusion-advection-reaction IV If u ∈ H k+1 (Ω), the estimate yields 1 1 1 − 21 −1 2 2 2 |||u − uh |||da[ ≤ Cu0 max(1, τc−1 µ−1 0 , Cη )(κ + βc h + τc h)hk , with κ := kκkL∞ (Ω) Cu0 = C 0 kukH k+1 (Ω) , and C 0 6= C 0 (κ, β, µ̃) For high Peclet numbers hβc /κ (dominant advection), − 21 |u − uh |β + τc 1 ku − uh kL2 (Ω) ≤ Chk+ 2 For low Peclet numbers (dominant diffusion), |||u − uh |||swip ≤ Chk These results extend to the case of vanishing diffusion 71 / 72 References Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D. (2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779. Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., and Savini, M. (1997). A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In Decuypere, R. and Dibelius, G., editors, Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, pages 99–109. Brenner, S. C. (2004). Korn’s inequalities for piecewise H 1 vector fields. Math. Comp., 73(247):1067–1087 (electronic). Cockburn, B. and Shu, C.-W. (1998). The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal., 35:2440–2463. 72 / 72