1. page 1 JASS 2004 Tobias Weinzierl Sophisticated construction ideas of ansatz- spaces How to construct Ritz-Galerkin ansatz-spaces for the Navier-Stokes equation that preserve the mass continuity. JASS 2004 – Tobias Weinzierl On some ideas of -Cornelia Blanke -Prof. Dr. Christoph Zenger -Dr. Miriam Mehl

2. page 2 JASS 2004 Tobias Weinzierl Principles of numerical simulations consistency convergence stability grid independence mass conservation laws of nature

3. page 3 JASS 2004 Tobias Weinzierl Importance of conservation laws mass / energy explosion incorrect formalization: X X h, τ t τ Some examples for X: energy mass momentum

4. page 4 JASS 2004 Tobias Weinzierl The simulated phenomenon Incompressible viscous fluid: three degrees of freedom: kinematic pressure p kinematic velocity u (vector field) constant density

5. page 5 JASS 2004 Tobias Weinzierl Mathematical stuff gradient divergence operator Laplace operator nonlinear term

6. page 6 JASS 2004 Tobias Weinzierl Navier-Stokes-equation velocity pressure momentum equation: nonlinear second order PDE konvection term is nonlinear diffusion is linear (Laplace operator) built in: energy and momentum conservation high Reynolds number: turbulent flow

7. page 7 JASS 2004 Tobias Weinzierl Agenda Navier-Stokes Problem Recapitulation of FEM ideas Grid types and evaluation of (bi-)linear ansatz functions Construction of a more sophisticated ansatz-space with respect to mass conservation Some further properties of our ansatz- functions What we work on now

8. page 8 JASS 2004 Tobias Weinzierl Things I will not talk about Handling nonlinearity Stability and consistence evaluations Handling sideconditions Hanging points Multilevel Time discretization

9. page 9 JASS 2004 Tobias Weinzierl The properties of FEM by Ciarlet – part I: The grid FEM approximates the solution function using linear combination of some basis functions (ansatz-functions). The solution function itself is approximated. dofs are no hanging points uniformity (Zlamal condition) geometric element

10. page 10 JASS 2004 Tobias Weinzierl The properties of FEM by Ciarlet – part II: The ansatz-functions Polyonomial Ansatz-functions Local support Affine Families Conformity:

11. page 11 JASS 2004 Tobias Weinzierl Grids for Navier-Stokes problems colocated grid fully staggered grid partially staggered grid u,p u u u u uu u u p p nonregular quadratic domain approximation finite element approximation: continous solutions Navier-Stokes-equations are ‚solved‘ use other proofs: „checkerboard instability“

12. page 12 JASS 2004 Tobias Weinzierl The discrete mass conservation control volume (Gauss) quadratic cells are control volumes should be valid for every fem solution set of linear equations (constraints on solution)

13. page 13 JASS 2004 Tobias Weinzierl Agenda Navier-Stokes Problem Recapitulation of FEM ideas Grid types and evaluation of (bi-)linear ansatz-functions Construction of a more sophisticated ansatz-space with respect to mass conservation Some further properties of our ansatz- functions What we work on now

14. page 14 JASS 2004 Tobias Weinzierl Bilinear ansatz-functions for velocity without proof linear on the axes of the coordinate system nonlinear on the support (see Maple sheet) discrete mass conservation doesn‘t imply pointwise mass conservation

15. page 15 JASS 2004 Tobias Weinzierl Lagrange ansatz-functions on triangles evaluate other grid types on a triangle the div is constant if two dofs are given, the side condition determines the value of the third ‚unknown‘

16. page 16 JASS 2004 Tobias Weinzierl Handling the side condition h h divide square into four triangles assumption: discrete conservation formula is valid new point isn‘t a real dof for its value is determined by other four points result is a piecewise linear function

17. page 17 JASS 2004 Tobias Weinzierl The sophisticated ansatz-function piecewise linear (see Maple sheet) if continuity is preserved on border, continuity is preserved in every inner point there are some other nice properties

18. page 18 JASS 2004 Tobias Weinzierl Agenda Navier-Stokes Problem Recapitulation of FEM ideas Grid types and evaluation of (bi-)linear ansatz-functions Construction of a more sophisticated ansatz-space with respect to mass conservation Some further properties of our ansatz- functions What we work on now

19. page 19 JASS 2004 Tobias Weinzierl Energy conservation energy of system modelled by Navier-Stokes-equations is constant iff diffusion isn‘t modelled if there‘s no friction, energy equality holds otherwise energy decreases roadmap: insert momentum equation apply continuity condition some technical work

20. page 20 JASS 2004 Tobias Weinzierl Discrete energy conservation convection diffusion mass matrix

21. page 21 JASS 2004 Tobias Weinzierl Navier-Stokes approximations No friction means D is zero and energy should be conserverd. Therefore: -C has to be antisymmetric -D has to be symmetric positiv definit Some calculations show: Constructed ansatz-space produces matrices with properties needed. For mass conservation is given pointwise adaptive grids and refinement processes are no problem for this ansatz-space.

22. page 22 JASS 2004 Tobias Weinzierl Agenda Navier-Stokes Problem Recapitulation of FEM ideas Grid types and evaluation of (bi-)linear ansatz-functions Construction of a more sophisticated ansatz-space with respect to mass conservation Some further properties of our ansatz- functions What we work on now

23. page 23 JASS 2004 Tobias Weinzierl What we work on now implement ideas within space- filling-curves framework use multigrid algorithms extract application domain independ techniques

24. page 24 JASS 2004 Tobias Weinzierl Lessons learned validate every approximation against laws of nature. This shows drawbacks / possible problems of solution use freedom of choosing ansatz- space properly try to use problem-specific ansatz- spaces: If dofs are coupled (here x 1 and x 2 ), perhaps the ansatz- functions have to be coupled, too ask application domain experts, too - not only mathematicians