1

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