Domain decomposition for non-stationary problems

1. Domain decomposition for non-stationary problems Yu. M. Laevsky (ICM&MG SB RAS) Novosibirsk, 2014

2. Content: • 1. Subdomains splitting schemes • 1.1. Methods with overlapping subdomains • 1.1.1. Method, based on the smooth partitioning of the unit • 1.1.2. Method with recalculating • 1.2. Methods without overlapping subdomains • 1.2.1. Like-co-component splitting method • 1.2.2. Discontinues solutions and penalty method • 2. Domain decompositionbasedon regularization • 2.1. Bordering methods • 2.2. Equivalent regularization • 2.3. Application of the fictitious space method • 3. Multilevel schemes and domain decomposition • 3.1. Dirichlet-Dirichlet decomposition • 3.2. Neumann-Neumann decomposition • 3.3. Example: propagation of laminar flame

3. Surveys: [1]. Yu.M. Laevsky,1993 (in Russian). [2]. T.F. Chanand T.P. Mathew, ActaNumerica, 1994. [3]. Yu.M. Laevsky, A.M. Matsokin,1999(in Russian). [4]. A.A.Samarskiy, P.N. Vabischevich, 2001(in Russian). [5]. Yu.M. Laevsky,Lecture Notes, 2003.

4. 1. Subdomains splitting schemes 1.1. Methods with overlapping of subdomains -regular overlapping - - -regular overlapping - -

5. 1. Subdomains splitting schemes 1.1. Methods with overlapping subdomains 1.1.1. Method based on smooth partitioning of the unit -smooth partitioning of the unit: in in Approximation by FEM gives:

6. 1. Subdomains splitting schemes 1.1. Methods with overlapping of subdomains 1.1.1. Method based on smooth partitioning of the unit Diagonalization of the matrix mass (the use of barycentric concentrating operators) and splitting give: Theorem -norm is the error in

7. 1. Subdomains splitting schemes 1.1. Methods with overlapping of subdomains 1.1.2. Method with recalculating unstable step

8. 1. Subdomains splitting schemes 1.1. Methods with overlapping subdomains 1.1.2. Method with recalculating Theorem -ellipticity is the constant of the error in -norm is

9. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.1. Like–co-component splitting method Approximation by FEM gives: Diagonalization of the matrix mass and splitting give:

10. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.1. Like–co-component splitting method Theorem The error in -norm is Example: The error in arbitrary “reasonable” norm is

11. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.2. Discontinues solutions and penalty method find Problem: in IBV: Red-black distribution on

12. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.2. Discontinues solutions and penalty method in in on Theorem:

13. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.2. Discontinues solutions and penalty method FE approximation: Red-black distribution of subdomains may use different meshes:

14. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.2. Discontinues solutions and penalty method Diagonalization of the matrix mass and splitting (according to red-black distribution of subdomains) give:

15. 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains 1.2.2. Discontinues solutions and penalty method Mathematical foundation Derivatives are uniformly bounded with respect to Theorem(penalty method) -norm is At the error in unconditional convergence

16. 2. Domain decomposition based on regularization 2.1. Bordering methods – implicit scheme Schur compliment

17. 2. Domain decomposition based on regularization 2.1. Bordering methods Explicit part of the scheme “works” in subspace.

18. 2. Domain decomposition based on regularization 2.1. Bordering methods Three-layer scheme – 2-d order of accuracy

19. 2. Domain decomposition based on regularization 2.1. Bordering methods Design of the operator is operator polynomial – theLantzos polynomial

20. 2. Domain decomposition based on regularization 2.1. Bordering methods Realization of the 2-d block of the scheme Iteration-like cycle: Theorem • schemes are stable. • Costs of “explicit part” is

21. 2. Domain decomposition based on regularization 2.2. Equivalent regularization • Standard spectral equivalence * • is in contrary with the requirement: • can be solved efficiently * • may be changed by two requirements:

22. 2. Domain decomposition based on regularization 2.2. Equivalent regularization Neumann-Dirichlet domain decomposition: Theorem -norm is the error in Fictitious domain method (space extension): Theorem -norm is the error in

23. 2. Domain decomposition based on regularization 2.3. Application of the fictitious space method Three-layer scheme Realization:inversion of the operator Stability:

24. 2. Domain decomposition based on regularization 2.3. Application of the fictitious space method Example:choosing by fictitious space method Mesh Neumann problem: Restriction operator: Extensionoperator:

25. 2. Domain decomposition based on regularization Fictitious space method (S.V. Nepomnyashchikh, 1991) Lemma.Let bethe Hilbert spaces with the inner products and be and , and let and selfadjoint positive definite bounded operators. linear and be linear operators such that Then let operator and for all and the inequalities are is identity valid are positive numbers. Then for any . where is the adjoint operator for

26. 3. Multilevel schemes and domain decomposition 3.1. Dirichlet-Dirichletdecomposition are symmetric, positive definite Localization of stability condition:

27. 3. Multilevel schemes and domain decomposition 3.1. Dirichlet-Dirichletdecomposition … *

28. 3. Multilevel schemes and domain decomposition 3.1. Dirichlet-Dirichletdecomposition Mathematical foundation *

29. 3. Multilevel schemes and domain decomposition 3.1. Dirichlet-Dirichletdecomposition Mathematical foundation Theorem (stability with respect to id) Theorem(stability with respect to rhs)

30. 3. Multilevel schemes and domain decomposition 3.2. Neumann-Neumann decomposition General framework

31. 3. Multilevel schemes and domain decomposition 3.2. Neumann-Neumann decomposition Domain decomposition

32. 3. Multilevel schemes and domain decomposition 3.3. Example: propagation of laminar flame – Arrhenius law For gas

33. 3. Multilevel schemes and domain decomposition 3.3. Example: propagation of laminar flame The problem is “similar”to hyperbolic problem: space and time “play the samerole”

34. Acknowledgements PolinaBanushkina Svetlana Litvinenko Alexander Zotkevich Sergey Gololobov