Sophisticated construction ideas of ansatz-spaces

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

2. consistency mass conservation convergence stability laws of nature grid independence Principles of numerical simulations

3. mass / energy explosion incorrect formalization: Importance of conservation laws τ • Some examples for X: • energy • mass • momentum X X h,τ t

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

5. Mathematical stuff gradient Laplace operator divergence operator nonlinear term

6. velocity pressure Navier-Stokes-equation • 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. 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 Agenda

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

9. 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. The properties of FEM by Ciarlet – part II: The ansatz-functions Conformity: • Polyonomial Ansatz-functions • Local support • Affine Families

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

12. 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. 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 Agenda

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

15. 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. 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 Handling the side condition h h

17. The sophisticated ansatz-function • piecewise linear (see Maple sheet) • if continuity is preservedon border, continuity ispreserved in every innerpoint • there are some other niceproperties

18. 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 Agenda

19. 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. convection mass matrix diffusion Discrete energy conservation

21. 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. Navier-Stokes approximations

22. 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 Agenda

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

24. 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 x1 and x2), perhaps the ansatz-functions have to be coupled, too • ask application domain experts, too - not only mathematicians