About Discontinuous Galerkin Finite Elements

1. About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides, Strasbourg, France ackerer@imfs.u-strasbg.fr

2. OUTLINE 1. Introduction • 2. Solving advective dominant transport • 2.1.          Eulerian methods: Finite Volumes, Finite Elements • 3. Galerkin Discontinuous Finite Elements • 3.1.          1D discretization • 3.2.          General formulation • 3.3.          Numerical integration • 3.4. Slope limiter • 4. Numerical experiments • 4.1.          2D – 3D benchmarks • 4.2.          Comparisons with finite volumes 5. On going works

3.  t n+1 Finite volumes n n-1 xj-2 xj-1 xj xj+1 Finite elements j j+1  x j-1 j-2 Space/time discretization xj-2 xj-1 xj xj+1 xj-2 xj-1 xj xj+1 Discontinuous finite elements xj-1/2 xj+1/2 Introduction

4. Introduction Finite differences method (FD): Richardson (1922) was first to apply FD to weather forecasting. It required 3 months' worth of calculations to predict weather for next 24 hours. Basic ideas: 1. Use Taylor’s (1685-1731) series 2. Replace the derivatives

5. u xj-2 xj-1 xj xj+1 xj-1/2 xj+1/2 Introduction Finite Volumes methods FV have a very strong physical meaning

6. To avoid oscillation for this scheme To reduce numerical diffusion (R. Courant, K. Friedrichs & H. Lewy ,1924) Introduction Some key numbers (1D)

7. u FV xj-2 xj-1 xj xj+1 FE xj-2 xj-1 xj xj+1 with so that for any with i=1 to ne,which leads to ne equations with ne unknowns Introduction Galerkin Finite Elements method Basic ideas: 1. Approximate the unknown function by a sum of ‘simple’ functions 2. The numerical solution should be as close as possible to the exact solution over the domain

8. Basic ideas: 3. Choose which leads to 4. Standard Euler/implicit scheme for time discretization, for example written for i=1 to ne. Introduction The next steps are more or less easy mathematics ...

9. u FE DFE xj-1 xj xj+1 xj-2 xj-2 xj-1 xj xj+1 Yj(t) : degree of freedom (nodal conc., ….) 2. Defining on node/edge/face A inside of E and on edge/face A outside of E Discontinuous Finite Elements Galerkin Discontinuous Finite elements method Basic ideas: 1. Approximate the unknown function by a sum of ‘simple’ functions INSIDE an element E

10. Step 1: : the flux through A, positive if pointed outside : norm of A (length, surface). Step 2: Discontinuous Finite Elements Basic ideas: 3. Second order explicit Runge-Kutta scheme

11. xj-1 xj xj+1 xj-2 Discontinuous Finite Elements Basic ideas: 4. Oscillations avoided by slope limitation

12. Linear approximation Variational form DGFE : 1D discretization Hyperbolic 1D

13. E xi xi+1 xi+2 xi-1 Discretization Explicit formulation leads to a local system: DGFE : 1D discretization Galerkin formulation

14. E Step 1: xi xi+1 xi+2 xi-1 Step 2: DGFE : 1D discretization

15. E xi xi+1 xi+2 xi-1 DGFE : 1D discretization Slope limitation

16. Polynomial approximation Linear (2D): Bi-Linear (2D): Variational form : the flux through A, positive if pointed outside : norm of A (length, surface). DGFE : General formulation General formulation

17. Bilinear interpolation Linear interpolation DGFE : General formulation Standard interpolation functions

18. Step 1 : : the flux through A, positive if pointed outside : norm of A (length, surface). • Step 2 : : depending on the sign of DGFE : General formulation

19. DGFE : Numerical integration Numerical integration (1) Exact integration in reference element for E

20. DGFE : Numerical integration Numerical integration (2) Exact numerical integration with Simpson’s rule (pol. Ordre 2)

21. DGFE : Slope limiting

22. min(E)/max (E) : min/max value of over each element which has a common node with E. min(i)/max (i) : min/max of over each element containing i Optimization : Extrema : Constraints : then E DGFE : Slope limiting Step 3 : Multidimensional slope limiter (Bilinear function)

23. DGFE, CFL=1 FE, CFL=5 FE, CFL=1 DGFE : Numerical experiments 1D Benchmarks Flux discretisation Time discretization

24. Bilinear, CFL=0,6 Linear, CFL=0,6 Bilinear, CFL=0,1 Linear, CFL=0,6 DGFE : Numerical experiments 2D Benchmarks

25. Velocity field DGFE : Numerical experiments 3D Benchmarks

26. Bilin. DGFE Finite volume DGFE : Numerical experiments

27. D-GFE (CFL = 0.50) Finite volume (CFL = 0.50) EFD : 10000 cells, 30 000 unk. VF : 10000 cells, 10 000 unk., VF2: 40000 cells, 40 000 unk. DGFE : Numerical experiments Comparisons with Finite Volumes

28. BUT • Explicit scheme …… DGFE : Summary Summary • Discontinuous Galerkin: well known algorithms • Efficient in tracking fronts • Well adapted to change interpolation order from one element to the other

29. : the flux through A, positive if pointed outside : norm of A (length, surface). • Implicit upwind formulation • Time domain decomposition DGFE : On going work

30. t+3Dt/4 t+Dt/2 t+Dt/4 DGFE, CFL=1 DGFE : On going work • Time domain decomposition t+Dt t X

31. DGFE : On going work • Implicit upwind formulation

32. DGFE : On going work

33. DGFE : On going work

34. DGFE : On going work • Next to come ….