The CG1-DG2 method for conservation laws

Transcription

1 for conservation laws Melanie Bittl 1, Dmitri Kuzmin 1, Roland Becker 2 MoST 2014, Germany 1 Dortmund University of Technology, Germany, 2 University of Pau, France

2 CG1-DG2 Method - Motivation hp-adaptivity for transport problems: 1 non-smooth solution: continuous linear elements + FCT (flux corrected transport) only h-adaptivity 2 smooth solution: continuous higher order elements without stabilization 2 / 18

3 CG1-DG2 Method - Motivation hp-adaptivity for transport problems: 1 non-smooth solution: continuous linear elements + FCT (flux corrected transport) only h-adaptivity 2 smooth solution: continuous higher order elements without stabilization Consider the stationary problem u + (vu) = f in Ω = (0, 1) (0, 1) v(x, y) = (1, 1). 2 / 18

4 CG1-DG2 Method - Motivation Figure : Error for continuous Galerkin with quadratic elements (CG2) Figure : Error for discontinuous Galerkin with quadratic elements (DG2) 3 / 18

5 Idea A new method which fulfills 1 Stability 2 Optimal convergence rate 3 Lower computational cost than the discontinuous Galerkin (DG) method 4 / 18

6 Idea A new method which fulfills 1 Stability 2 Optimal convergence rate 3 Lower computational cost than the discontinuous Galerkin (DG) method : Enriching the space of continuous linear finite elements with discontinuous quadratic basis functions: CG1-DG2 functions are continuous at the vertices but may have jumps across edges. 4 / 18

7 CG1-DG2 space for triangles On each triangle K of a shape-regular triangulation K h of Ω we have: 1 linear basis function ϕ i K, i = 1, 2, 3 2 quadratic basis function ψk i := ϕi mod 3 Kϕi+1 K, i = 1, 2, 3 The space of discontinuous quadratic basis functions is defined D h := {v h L 2 (Ω) : v h K D K } with D K := {ψ i K : 1 i 3}. The CG1-DG2 space is defined by: V 1,2 h := V 1 h D h. where V 1 h is the space of linear finite elements. Figure : blue: continuous linear basis functions, white: discontinuous quadratic basis functions 5 / 18

8 Some analysis for transport equations Model problem in 2d: σu + div(vu) = f in Ω, u = g on Ω We will assume that 1 σ div v σ 0 > 0, 2 sup x Ω σ(x) + div v(x) =: σ 1 <. 3 σ L (Ω) and v 1, v 2 W 1, (Ω), v = (v 1, v 2 ) T. 6 / 18

9 Weak Formulation On each element K K h : Multiplying with test function ϕ and integration by parts on each element K. Summation over all elements K: (vu) ϕ + σuϕ dx + v n û[ϕ] ds + v n uϕ ds = fϕ dx v n gϕ ds. K h K h S int h S,+ h }{{} a(u,ϕ) S, h }{{} where û is the upwind value, v n = v n, S int interior sides, S boundary sides. f(ϕ) 7 / 18

10 Stability 1 Lemma There exists a mesh-independent constant γ > 0 such that a(u h, v h ) sup v h V 1,2 h \{0} v h γ u h. u h DG := σ 0 u h v n u 2 2 h + 1 v n [u h ] Ω 2 2, S h u h v := δ K v u h 2 K, δ K = h K, v K K h,k u h := u h DG + u h 2 v. 1 R. Becker, M. Bittl, and D. Kuzmin. Analysis of a combined CG1-DG2 method for the transport equation. In: SINUM, submitted (september 2013). 8 / 18

11 A priori estimate 2 Theorem Let h := max K Kh h K. Let 0 k 2 and u H k+1 (Ω). Then we have the a priori error estimate u u h C h k+1/2 u H k+1 (Ω) 2 R. Becker, M. Bittl, and D. Kuzmin. Analysis of a combined CG1-DG2 method for the transport equation. In: SINUM, submitted (september 2013). 9 / 18

12 Numerical examples: Stationary convection problem u + (vu) = f in Ω = (0, 1) (0, 1) v(x, y) = (1, 1). Figure : exact solution 10 / 18

13 Numerical examples: Stationary convection problem u + (vu) = f in Ω = (0, 1) (0, 1) v(x, y) = (1, 1). CG2 CG1-DG2 DG2 h u u h u u h u u h 1/ e e e-02 1/ e e e-03 1/ e e e-04 1/ e e e-05 EOC Figure : exact solution 10 / 18

14 MoST 2014, Germany Numerical examples: Stationary convection problem (a) Error for CG2 (b) Error for CG1-DG2 (c) Error for DG2 11 / 18

15 Stationary convection problem: DOFs h CG1 CG2 CG1-DG2 DG1 DG2 1/ / / / Table : DOFs on triangular meshes for the unit square 12 / 18

16 Solid body rotation u t + (vu) = 0 in Ω = (0, 1) (0, 1) v(x, y) = (0.5 y, x 0.5) 13 / 18

17 Solid body rotation (a) CG2 (b) CG1-DG2 (c) DG2 14 / 18

18 MoST 2014, Germany Solid body rotation - error (a) Error for CG2 (b) Error for CG1-DG2 (c) Error for DG2 15 / 18

19 Solid body rotation - hp-adaptivity CG1+FCT (non-smooth) and CG1-DG2 (smooth) 16 / 18

20 Summary & Outlook We introduced the CG1-DG2 method: 1 Stability 2 A priori error estimate 3 Numerical studies In the future: Euler equations Inviscid 2-fluid model 17 / 18

21 Thank you for your attention! 18 / 18