domain decomposition for non-stationary problems
DESCRIPTION
Domain decomposition for non-stationary problems. Yu. M. Laevsky (ICM&MG SB RAS). Novosibirsk, 2014. Content:. 1. Subdomains splitting schemes 1.1. Methods with overlapping subdomains 1.1.1. Method , based on the smooth partitioning of the unit - PowerPoint PPT PresentationTRANSCRIPT
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Domain decomposition for non-stationary problemsYu. M. Laevsky(ICM&MG SB RAS) Novosibirsk, 2014
1. Subdomains splitting schemes1.1. Methods with overlapping subdomains 1.1.1. Method, based on the smooth partitioning of the unit 1.1.2. Method with recalculating1.2. 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 regularization2.1. Bordering methods 2.2. Equivalent regularization 2.3. Application of the fictitious space method
3. Multilevel schemes and domain decomposition3.1. Dirichlet-Dirichlet decomposition3.2. Neumann-Neumann decomposition3.3. Example: propagation of laminar flame Content:2Surveys:[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. 31. Subdomains splitting schemes4
-
-regular overlapping -1.1. Methods with overlapping of subdomains
-
-regular overlapping -
5
- smooth partitioning of the unit:
1. Subdomains splitting schemes1.1. Methods with overlapping subdomains
in
in
Approximation by FEM gives:
1.1.1. Method based on smooth partitioning of the unit
1.1.1. Method based on smooth partitioning of the unit6
the error in
1. Subdomains splitting schemes1.1. Methods with overlapping of subdomains Diagonalization of the matrix mass (the use of barycentric concentrating operators) and splitting give: is
Theorem-norm
71.1.2. Method with recalculating
1. Subdomains splitting schemes1.1. Methods with overlapping of subdomains unstable step81.1.2. Method with recalculating 1. Subdomains splitting schemes1.1. Methods with overlapping subdomains Theorem
the error in
is
is the constant of
-ellipticity-norm
91.2.1. Likeco-component splitting method
1. Subdomains splitting schemes1.2. Methods without overlapping subdomains
Approximation by FEM gives: Diagonalization of the matrix mass and splitting give:
10
1. Subdomains splitting schemes1.2. Methods without overlapping subdomains 1.2.1. Likeco-component splitting method Theorem
The error inis-norm
The error in arbitrary reasonable norm is
Example:111.2.2. Discontinues solutions and penalty method in
on
1. Subdomains splitting schemes1.2. Methods without overlapping subdomains Problem:IBV:find
Red-black distribution
121.2.2. Discontinues solutions and penalty method
in
1. Subdomains splitting schemes1.2. Methods without overlapping subdomains Theorem:
in
on
131.2.2. Discontinues solutions and penalty method 1. Subdomains splitting schemes1.2. Methods without overlapping subdomains FE approximation:
Red-black distribution of subdomains may use different meshes:14
1. Subdomains splitting schemes1.2. 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
Mathematical foundation
1. Subdomains splitting schemes1.2. 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 16
2. Domain decomposition based on regularization2.1. Bordering methods
implicit schemeSchur compliment17
2. Domain decomposition based on regularization2.1. Bordering methods
Explicit part of the scheme works in subspace.18
2-d order of accuracy2. Domain decomposition based on regularization2.1. Bordering methodsThree-layer scheme19
is operator polynomial
the Lantzos polynomial 2. Domain decomposition based on regularization2.1. Bordering methodsDesign of the operator
20Iteration-like cycle:
2. Domain decomposition based on regularization2.1. Bordering methodsschemes are stable. Costs of explicit part isTheorem
Realization of the 2-d block of the scheme21
2. Domain decomposition based on regularization2.2. Equivalent regularization Standard spectral equivalence
is in contrary with the requirement: can be solved efficiently **may be changed by two requirements: 22Neumann-Dirichlet domain decomposition:Fictitious domain method (space extension):
2. Domain decomposition based on regularization2.2. Equivalent regularization
the error inisTheorem-norm
Theorem
the error inis-norm
23
Realization: inversion of the operator
Stability:2. Domain decomposition based on regularization2.3. Application of the fictitious space method Three-layer scheme24Mesh Neumann problem: Example: choosing
by fictitious space method
Restriction operator:
Extension operator:
2. Domain decomposition based on regularization2.3. Application of the fictitious space method 25be the Hilbert spaces with the inner productsLemma. Let
and
and
, and let
and
be linear operators such that
operator and for allthe inequalities are
and
are positive numbers. Then for any
where is the adjoint operator for
. be andselfadjoint positive definite bounded operators. Fictitious space method (S.V. Nepomnyashchikh, 1991) linearThen let identity is valid2. Domain decomposition based on regularization
263. Multilevel schemes and domain decomposition3.1. Dirichlet-Dirichlet decomposition
are symmetric, positive definite
Localization of stability condition:
273. Multilevel schemes and domain decomposition3.1. Dirichlet-Dirichlet decomposition
*
283. Multilevel schemes and domain decomposition3.1. Dirichlet-Dirichlet decomposition*Mathematical foundation
293. Multilevel schemes and domain decomposition3.1. Dirichlet-Dirichlet decompositionMathematical foundationTheorem (stability with respect to id) Theorem (stability with respect to rhs)
303. Multilevel schemes and domain decomposition3.2. Neumann-Neumann decomposition
General framework
313. Multilevel schemes and domain decomposition3.2. Neumann-Neumann decomposition
Domain decomposition323. Multilevel schemes and domain decomposition3.3. Example: propagation of laminar flame
For gas
Arrhenius law
333. Multilevel schemes and domain decomposition3.3. Example: propagation of laminar flameThe problem is similar to hyperbolic problem:space and time play the same role
34AcknowledgementsPolina BanushkinaSvetlana LitvinenkoAlexander ZotkevichSergey Gololobov 35