Discretization of Generalized Convection-Diffusion

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1 Discretization of Generalized Convection-Diffusion H. Heumann R. Hiptmair Seminar für Angewandte Mathematik ETH Zürich Colloque Numérique Suisse / Schweizer Numerik Kolloquium 8

2 Generalized Convection-Diffusion scalar convection diffusion: ε u + β gradu = f in Ω in Differential Forms: d dω + L β ω = f in Ω

3 Generalized Convection-Diffusion scalar convection diffusion: in Differential Forms: ε u + β gradu = f d dω + L β ω = f in Ω form in Ω exterior derivative Hodge operator Lie derivative Goal: convection diffusion for p forms ω p What is a Lie derivative? L β =?

4 Lie derivatives directional derivative u(x + tβ) u(x) (β gradu)(x) = lim t t M ϕ t (M ) β Lie derivative L β (transport of forms) with respect to flow ϕ t of velocity field β: < ω p,ϕ t (M p ) > < ω p,m p > L β ω p :=< L β ω p,m p >:= lim M t p t Cartan magic formula L β = i β d + di β with contraction i β (Bossavit) < i β ω p,m p >:= lim t < ω p,ext t (M p,β) > t ϕ t (M ) M β Ext t (M,β)

5 Generalized Convection-Diffusion scalar convection diffusion: u + β gradu = f in Ω in Differential Forms: d dω + L β ω = f in Ω generalized convection diffusion for p form ω p d dω p + L β ω p = f p or d dω p + i β dω p = f p or d dω p + di β ω p = f p example: magnetic convection curlcurla + β curla = F

6 Discrete Differential Forms differential forms ω p act on p dimensional manifolds M p! < ω p,m p >:= M p ω p discrete setting: prescribe ω p on finitely many M k p (vertices k = i, edges k = (e,e )...). interpolation of M p approximation < ω p,m p > = k a k(m p ) < ω p,m k p > M M limit procedure Whitney forms ω k p ω p (x) = ω h p(x) = k ωk p(x) < ω p,m k p >, p = : ω i Linear Finite Elements p = : ω e Edge Elements ω k p(x) := lim M p x a k(m p ) = back in FEM-setting, but conforming!

7 Lie derivative of discrete -forms Discrete version of β grad i β d CG =: L? G ei ω i =λ i := < gradλ i,e > Stokes = δ ie δ ie a i e Ti β ϕ t gradλ j a k β(aj) T j gradλ i a j a i gradλ k

8 Lie derivative of discrete -forms Discrete version of β grad i β d CG =: L? G ei ω i =λ i := < gradλ i,e > Stokes = δ ie δ ie a i C ie := < i β ω e,a i > = lim t < ω e,ext t(a i,β) > t upwind = β(a i ) ω e (a i ) Ti = G ei β(a i ) gradλ e/i Ti a k β(aj) e Ti β ϕ t gradλ j gradλ i T j a j a i gradλ k

9 Lie derivative of discrete -forms Discrete version of β grad i β d CG =: L? G ei ω i =λ i := < gradλ i,e > Stokes = δ ie δ ie C ie := < i β ω e,a i > = lim t < ω e,ext t(a i,β) > t upwind = β(a i ) ω e (a i ) Ti = G ei β(a i ) gradλ e/i Ti a i e Ti β ϕ t L ji := e = e C je G ei β(a j ) gradλ e/j Tj G ej G ei gradλ j a k β(aj) T j gradλ i a j = β(a j ) gradλ i Tj a i gradλ k L is M-matrix, inverse monoton!

10 FEM-Approach bilinear form and upwind quadrature (Tabata) b(u h,λ j ) := (β gradu h,λ j ) L u h Ph = β gradu h λ j T T β(a j ) gradu h Tj T supp(λ j ) T }{{} discr. Hodge P j a j T T j T b h (λ i,λ j ) = P j β(a j ) gradλ i Tj }{{} L ji T error analysis using Strang-Lemma und Bramble-Hilbert techniques. b h (u h,v h ) b(u h,v h ) Ch β, u h w h discrete Max. principle and L -stability since M-matrix

11 Numerical Experiments singular perturbed convection diffusion ε u + β gradu = f < ε or d ε dω + i β dω = f < ε instability in standard FEM upwind finite differences artificial viscosity Streamline Upwind Petrov Galerkin (SUPG/SDFEM)

12 Numerical Experiments: Convergence and Stability y β =, β = force data s.t. u ε (x,y) = xy y e x ε xe y ε + e x ε + y x y ε x convergence rate with ε = (left) and ε = (right) L error u (Up) L error u (FEM) L error u (SUPG) 8 L error u (Up) L error u (FEM) L error u (SUPG) 6 error 4 p =. error 4 p =. 5 p =. p =.9 6 h p =.49 p =.5 h

13 Numerical Experiments: Smoothing Γ (,) β = β =, f u on Γ Γ Γ4 profile line Γ u on Γ Γ 4 (,) Γ Solution for ε = 4 with upwind scheme and SUPG (mesh width=.7). upwind method SUPG profile profile line

14 Second Order Elements second order Lagrangian elements 6 local basis functions with dofs Σ 4 quadrature rules: Q(T) = (a i, T ω i ) i. O(h ). O(h ). O(h ) 4. O(h 4 ) bilinearform b h (u h,v h ) = T ω i (β gradu h ) Ti (a i )v h (a i ) T a i Q(T) = v i ω i (β gradu h ) Ti (a i ) T + T ω i (β gradu h ) T (a i )v h (a i ) a i Σ T:a i T T a }{{} i / Σ }{{} :=element boundary contribution :=element center contribution

15 Numerical Experiments: Convergence and Stability y β =, β = force data s.t. u ε (x,y) = xy y e x ε xe y ε + e x ε + y x y ε x convergence rate with ε = (left) and ε = (right). 8 error 4 6. p=.8. p=.7. p= p=.8 FEM p=.8 SUPG p=.99 h error 6 4. p=.65. p=.4. p= p=.49 FEM p=.8 SUPG p=.8 h

16 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ

17 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ..8 profile

18 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ.. profile

19 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ... x 6 profile

20 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ... x 6 profile

21 Numerical Experiments: Smoothing (,) β = cos(), β = sin(), f u on Γ, u on Γ ε = 4, h =.4 Γ (,) profile line Γ... x 6 profile SUPG Quadrature rule using vertices, midpoints and barycenter competes with SUPG!.5.5.5

22 Conclusions and Further Issues Lie derivative formalism reproduces upwind FEM! Can be extended to higher order! Choice of basis and quadrature? Proof of stability for nd+ order? Lie Derivative formalism brings Upwinding to discretization! β curla i β dω Stability of discretizations for + forms, e.g. magnetic convection? Boundary and gauge conditions for + forms?