1. 23/5/20051 ICCS congres, Atlanta, USA May 23, 2005 The Deflation Accelerated Schwarz Method for CFD C. Vuik Delft University of Technology c.vuik@ewi.tudelft.nl http://ta.twi.tudelft.nl/users/vuik/ J. Verkaik, B.D. Paarhuis, A. Twerda TNO Science and Industry

2. 23/5/20052 Contents Problem description Schwarz domain decomposition Deflation GCR Krylov subspace acceleration Numerical experiments Conclusions

3. 23/5/20053 Problem description CFD package TNO Science and Industry, The Netherlands simulation of glass melting furnaces incompressible Navier-Stokes equations, energy equation sophisticated physical models related to glass melting GTM-X:

4. 23/5/20054 Problem description Incompressible Navier-Stokes equations: Discretisation: Finite Volume Method on “colocated” grid

5. 23/5/20055 Problem description SIMPLE method: pressure- correction system ( )

6. 23/5/20056 Schwarz domain decomposition Minimal overlap: Additive Schwarz:

7. 23/5/20057 inaccurate solution to subdomain problems: 1 iteration SIP, SPTDMA or CG method complex geometries parallel computing local grid refinement at subdomain level solving different equations for different subdomains Schwarz domain decomposition GTM-X:

8. 23/5/20058 Deflation: basic idea Solution: “remove” smallest eigenvalues that slow down the Schwarz method Problem: convergence Schwarz method deteriorates for increasing number of subdomains

9. 23/5/20059 Deflation: deflation vectors +

10. 23/5/200510 Property deflation method: systems with have to be solved by a direct method Deflation: Neumann problem singular Problem: pressure-correction matrix is singular: has eigenvector for eigenvalue 0 Solution: adjust non-singular 

11. 23/5/200511 for general matrices (also singular) approximates in Krylov space such that is minimal Gram-Schmidt orthonormalisation for search directions optimisation of work and memory usage of Gram-Schmidt: restarting and truncating Additive Schwarz: Property: slow convergence Krylov acceleration GCR Krylov acceleration GCR Krylov method: Objective: efficient solution to

12. 23/5/200512 Numerical experiments

13. 23/5/200513 Numerical experiments Buoyancy-driven cavity flow

14. 23/5/200514 Numerical experiments Buoyancy-driven cavity flow: inner iterations

15. 23/5/200515 Numerical experiments Buoyancy-driven cavity flow: outer iterations without deflation

16. 23/5/200516 Buoyancy-driven cavity flow: outer iterations with deflation Numerical experiments

17. 23/5/200517 Buoyancy-driven cavity flow: outer iterations varying inner iterations Numerical experiments

18. 23/5/200518 Numerical experiments Glass tank model

19. 23/5/200519 Numerical experiments Glass tank model: inner iterations

20. 23/5/200520 Numerical experiments Glass tank model: outer iterations without deflation

21. 23/5/200521 Numerical experiments Glass tank model: outer iterations with deflation

22. 23/5/200522 Glass tank model: outer iterations varying inner iterations Numerical experiments

23. 23/5/200523 Heat conductivity flow Numerical experiments Q=0 Wm -2 h=30 Wm -2 K -1 T=303K T=1773K K = 1.0 Wm -1 K -1 K = 0.01 Wm -1 K -1 K = 100 Wm -1 K -1

24. 23/5/200524 Heat conductivity flow: inner iterations Numerical experiments

25. 23/5/200525 using linear deflation vectors seems most efficient a large jump in the initial residual norm can be observed higher convergence rates are obtained and wall-clock time can be gained implementation in existing software packages can be done with relatively low effort deflation can be applied to a wide range of problems Conclusions