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

1. Subdomains splitting schemes 1.1. Methods with overlapping subdomains Method, based on the smooth partitioning of the unit Method with recalculating 1.2. Methods without overlapping subdomains Like-co-component splitting method Discontinues solutions and penalty method 2. Domain decomposition based on 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 Content: 2

Surveys: [1]. Yu.M. Laevsky, 1993 (in Russian). [2]. T.F. Chan and T.P. Mathew, Acta Numerica, [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,

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

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

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

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

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

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

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

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

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

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

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

15 Mathematical foundation 1. Subdomains splitting schemes 1.2. Methods without overlapping subdomains Discontinues solutions and penalty method Derivatives are uniformly bounded with respect to Theorem (penalty method) the error in -norm is At 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-d order of accuracy 2. Domain decomposition based on regularization 2.1. Bordering methods Three-layer scheme

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

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

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

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

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

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

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

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

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

29 3. Multilevel schemes and domain decomposition 3.1. Dirichlet-Dirichlet decomposition 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 For gas – Arrhenius law

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 same role”

34 Acknowledgements Polina Banushkina Svetlana Litvinenko Alexander Zotkevich Sergey Gololobov

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