Adaptive mesh refinement

1. Adaptive mesh refinement • for discontinuous Galerkin method on quadrilateral non-conforming grids • Michal A. Kopera • PDE’s on the Sphere 2012

2. Motivation • Cut the number of elements down to a minimum necessary to sufficiently well resolve the problem • Tackle problems previously difficult or impossible to solve due to limited computational resources Source: NASA

3. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

4. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach level 0

5. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach level 0 level1

6. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach level 0 level1 level 2

7. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

8. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

9. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

10. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

11. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements ! !

12. Non-conforming quad-based DG • Non-conforming flux computation handled by the DG solver • Forest of quad-trees approach • Each parent element always replaced by four children • At most 2:1 size ratio of face-neighboring elements

13. How to compute flux? 1) Scatter data from the parent edge to children edges

14. How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case

15. How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case + 3) Gather fluxes from children edges to the parent edge

16. How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case 3) Gather fluxes from children edges to the parent edge 4) Apply fluxes like in a conforming case

17. How to move data through an interface? +

18. Let us define the space for both parent and child faces: with mappings Expanding variables yields

19. For each children face we require Substitution of expansions and reorganizing the terms yields

20. Let We require that + After splitting the integrals, plugging-in extensions, reorganizing and variable change we arrive at:

21. Refinement criterium

22. Refinement criterium What are the benefits and costs?

23. What are the benefits and costs?

24. What are the benefits and costs?

25. Analyzing mountain cases Multi-rate time-stepping CG AMR GPU 3D + MPI Multigrid ? Optimized data structures Shallow water Outlook

26. Shallow Water Equations 2D wave with 2D bathymetry Linear hydrostatic mountain