Domain decomposition for non-stationary problems Yu. M. Laevsky
(ICM&MG SB RAS) Novosibirsk, 2014
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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
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
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Surveys: [1]. Yu.M. Laevsky, 1993 (in Russian). [2]. T.F. Chan
and T.P. Mathew, Acta Numerica, 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. 3
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1. Subdomains splitting schemes 4 - -regular overlapping - 1.1.
Methods with overlapping of subdomains - -regular overlapping
-
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5 - smooth partitioning of the unit: 1. Subdomains splitting
schemes 1.1. Methods with overlapping subdomains in Approximation
by FEM gives: 1.1.1. Method based on smooth partitioning of the
unit
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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
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7 1.1.2. Method with recalculating 1. Subdomains splitting
schemes 1.1. Methods with overlapping of subdomains unstable
step
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8 1.1.2. Method with recalculating 1. Subdomains splitting
schemes 1.1. Methods with overlapping subdomains Theorem the error
in is is the constant of -ellipticity -norm
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9 1.2.1. Likeco-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:
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10 1. Subdomains splitting schemes 1.2. Methods without
overlapping subdomains 1.2.1. Likeco-component splitting method
Theorem The error in is-norm The error in arbitrary reasonable norm
is Example:
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11 1.2.2. Discontinues solutions and penalty method in on 1.
Subdomains splitting schemes 1.2. Methods without overlapping
subdomains Problem: IBV: find Red-black distribution
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12 1.2.2. Discontinues solutions and penalty method in 1.
Subdomains splitting schemes 1.2. Methods without overlapping
subdomains Theorem: in on
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13 1.2.2. 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:
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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:
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15 Mathematical foundation 1. Subdomains splitting schemes 1.2.
Methods without overlapping subdomains 1.2.2. Discontinues
solutions and penalty method Derivatives are uniformly bounded with
respect to Theorem (penalty method) the error in -norm is At
unconditional convergence
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16 2. Domain decomposition based on regularization 2.1.
Bordering methods implicit scheme Schur compliment
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17 2. Domain decomposition based on regularization 2.1.
Bordering methods Explicit part of the scheme works in
subspace.
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18 2-d order of accuracy 2. Domain decomposition based on
regularization 2.1. Bordering methods Three-layer scheme
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19 is operator polynomial the Lantzos polynomial 2. Domain
decomposition based on regularization 2.1. Bordering methods Design
of the operator
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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
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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:
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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
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23 Realization: inversion of the operator Stability: 2. Domain
decomposition based on regularization 2.3. Application of the
fictitious space method Three-layer scheme
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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
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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
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26 3. Multilevel schemes and domain decomposition 3.1.
Dirichlet-Dirichlet decomposition are symmetric, positive definite
Localization of stability condition:
28 3. Multilevel schemes and domain decomposition 3.1.
Dirichlet-Dirichlet decomposition * Mathematical foundation
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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)
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30 3. Multilevel schemes and domain decomposition 3.2.
Neumann-Neumann decomposition General framework
32 3. Multilevel schemes and domain decomposition 3.3. Example:
propagation of laminar flame For gas Arrhenius law
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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
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34 Acknowledgements Polina Banushkina Svetlana Litvinenko
Alexander Zotkevich Sergey Gololobov
