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Some Domain Decomposition Methods for Discontinuous Coefficients

Marcus Sarkis

WPI

RICAM-Linz, October 31, 2011

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 1 / 38

Outline

Discretizations

◮ P1 conforming, RT0, P1 nonconforming, DG, Mortar ◮ Some important differences between them

Overlapping Schwarz Methods and/with Substructuring Methods

◮ COARSE SPACES ◮ Overlapping AS (additive Schwarz), ASHO (AS with harmonic

• verlap), RASHO (Restricted ASHO), OBBD (Overlapping BDD)

◮ Iterative Substructuring, BDDC

First we consider discontinuous coefficients however constant inside

• substructures. Then, we consider more general discontinuities

A general reference for DD: Andrea Toselli and Olof Widlund, “Domain Decomposition Methods - Algorithms and Theory”. Springer Series in Computational Mathematics, Vol. 34, 2005.

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 2 / 38

Geometrically Nonconforming Subdomain Partition

Ω = ∪N

i=1Ωi

Ωi disjoint shaped regular polygonal subdomains of diameter O(Hi) Thi(Ωi) shape regular triangulations In this talk: geometrically conforming partitions and matching meshes The Ωi are simplices (to drop later) with constant coefficient ρi Subdomains, faces, edges are open sets

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 3 / 38

Problem of Interest

Continuous PDE. Find u ∈ H1

0(Ω) such that

a(u, v) :=

N

• i=1
• Ωi

(ρi∇u, ∇v)dx =

• Ω

fvdx =: f (v) ∀v ∈ H1

0(Ω)

Conforming FEM. Vh(Ω) ⊂ H1

0(Ω) : continuous piecewise linear

functions in Ω. Find u ∈ Vh(Ω) such that a(u, v) = f (v) ∀v ∈ Vh(Ω) Nonconforming FEM. ˆ Vh(Ω): piecewise linear functions and continuous at the middle points of the edges of Th(Ω) and zero at the middle points of the edges of Th(∂Ω). Find u ∈ ˆ Vh(Ω) such that ˆ ah(u, v) :=

• τ∈Th(Ω)
• τ

(ρ∇u

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 4 / 38

Conforming FEM

P1 conforming: VH(Ω) ⊂ H1

0(Ω)

◮ Weighted L2− approximation and weighted H1− stability

u − IHu2

L2

ρ(Ω) H2|u|2

H1

ρ(Ω)

|IHu|2

H1

ρ(Ω) |u|2

H1

ρ(Ω)

◮ First inequality not always possible with constant independently of ρi

u2

L2

ρ(Ω) :=

• Ω

(ρu, u)dx |u|2

H1

ρ(Ω) :=

• Ω

(ρ∇u, ∇u)dx

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 5 / 38

Nonconforming FEM (both inequalities hold)

Step 1: take averages where the coefficient is constant: average on elements, boundary of elements, faces (edges) in 3D (2D) Step 2: obtain Poincar´ e-Friedrich type inequalities inside each Ωi Step 3: establish local H1− stability and L2− approximation in Ωi Step 4: establish global weighted H1− stab. and L2− appr. in Ω

◮ Crouzeix-Raviart P1 nonconforming:

VH(Ω) ⊂ H1

0(Ω)

◮ Broken weighted H1− stability and weighted L2− approximation

| IHu|

H1

ρ,H(Ω) |u|H1 ρ(Ω)

u − IHuL2

ρ(Ω) H|u|H1 ρ(Ω)

◮

IHu averages u on faces (edges) of Ωi |u|2

• H1

ρ,H(Ω) :=

N

• i=1
• Ωi

(ρi∇u, ∇u)dx

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 6 / 38

Nonconforming FEM (2D Local Analysis)

m1, m2, m3 middle points of edges E1, E2, E3 of Ωi ¯ um1, ¯ um2, ¯ um3 edge averages | IHu|2

H1(Ωi) |¯

um2 − ¯ um1|2 + |¯ um3 − ¯ um1|2 Note that |¯ um2 − ¯ um1|2 =

• E2(u − ¯

um1)ds

• E2 1ds
• 2

H−1u − ¯ um12

L2(E2)

H−2u − ¯ um12

L2(Ωi) + |u − ¯

um1|2

H1(Ωi)

By Friedrich’s inequality, the H1− stability in Ωi holds u − IHu has zero average on edges of Ωi so Friedrich holds on Ωi u − IHu2

L2(Ωi) H2|u −

IHu|2

H1(Ωi) ≤ 2H2

|u|2

H1(Ωi) + |

IHu|2

H1(Ωi)

• Marcus Sarkis (WPI)

Domain Decomposition Methods RICAM-2011 7 / 38

Remark on CR elements

CR system is spectrally equivalent to the reduced hybridizable RT0 RT0 has mass conservation properties For discontinuous coefficients, RT0 gives better results for the velocity and pressure (after postprocessing) compared to conforming P1 In certain cases, discrete Poincar´ e-Friedrich-type inequalities with constants independently of coefficients are more difficult to obtain than for conforming elements

◮ Let V be a coarse vertex of TH(Ω). Let ΩV be the union of all Ωj

touching V (let us assume there are 6 Ωj)

◮ Coefficients (M, 1, 1, M, 1, 1) (anti-clockwise orientation). M very large ◮ Conforming P1: u − u(V )L2

ρ(ΩV ) H|u|H1 ρ(ΩV ) for u ∈ V H(ΩV )

◮ CR: broken weighted Poincar´

e does not hold in IH(ΩV )

Quasi-monotone coefficients: holds for IH(ΩV ) and H1(ΩV ) The preconditioning analysis for CR more complicated than conforming

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 8 / 38

CR: Boundary Element Averages

How to define the values at the subdomain edges? Let us take an edge E ij := ∂Ωi ∩ ∂Ωj Let ¯ ui, ¯ uj element averages on ∂Ωi and ∂Ωj Define the IHu by ¯ uij,β = ρβ

i

ρβ

i + ρβ j

¯ ui + ρβ

j

ρβ

i + ρβ j

¯ uj, β ∈ [1 2, ∞] Denote mij, mik, mil middle points of Eij, Eik, Eil Let ¯ umij, ¯ umik, ¯ umil be edge averages of u on Eij, Eik, Eil

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 9 / 38

CR: Element Averages (cont.)

Broken weighted H1− stability ρi|¯ uij,β −¯ uik,β|2 ≤ 3ρi

• |¯

uij,β − ¯ umij|2 + |¯ umij − ¯ umik|2 + |¯ umik − ¯ uik,β|2 ρi|¯ uij,β − ¯ umij|2 = ρi

• ρβ

i

ρβ

i + ρβ j

(¯ ui − ¯ umij) + ρβ

j

ρβ

i + ρβ j

(¯ uj − ¯ umij) 2 ρi

• ρβ

j

ρβ

i + ρβ j

2 ≤ ρj if β ≥ 1/2 Weighted L2− stability ρiu − IHu2

L2(Ωi) ρiH2|u −

IHu|2

H1(Ωi) + ρiH|u −

IHu|2

L2(Eij)

u − IHu = ρβ

i

ρβ

i + ρβ j

(u − ¯ ui) + ρβ

j

ρβ

i + ρβ j

(u − ¯ uj)

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 10 / 38

CR: Face Coarse Spaces

• VH and

VH ⊂ Vh Continuity at the middle points of the coarse edges ( VH) does not imply continuity at the middle points of the fine edges ( Vh) Edge coarse space (in 2D) and Face coarse space (in 3D) Step 1: On each subdomain edge Eij: make I E

H u equal to (

IHu)(mij) at every middle point of the fine triangulation on Eij Step 2: Discrete harmonic extension to define I E

H u inside Ωi

The resulting coarse space can be constructed easily for general Ωi

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 11 / 38

CR-Face Coarse Basis Functions

E ij := Ωi ∩ Ωj Face coarse basis functions θEij. On fine nodes, let:

◮ θEij : 1 on Eij ◮ θEij := 0 on all subdomain edges but Eij ◮ θEij : discrete harmonic extension inside Ωi and Ωj ◮ θEij := 0 elsewhere

Note that the support of θEij is Ωi ∪ Ωj What is I E

H ?

• I E

H u =

• Edges E∈TH(Ω)

¯ uEθE ˆ V E

H is the range of

I E

H

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 12 / 38

CR: NN Coarse Basis Functions

For each subdomain Ωi define θi =

• Eij⊂∂Ωi

ρβ

i

ρβ

i + ρβ j

θEij Support of θi is Ω

ext i

: union of all Ωj sharing an edge with Ωi

• I NN

H

given by

• I NN

H

=

N

• i=1

¯ uiθi

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 13 / 38

CR: Face Coarse Space (cont.)

Note that IHu and I E

H u have the same average on each coarse edge Eij

It is possible to show:

◮ Weighted L2− approximation

u − I E

H u2 L2

ρ(Ωi) H2|u|2

• H1

ρ,h(Ωi)

◮ Broken weighted H1− stability

| I E

H u|2

• H1

ρ,h(Ωi)

• 1 + log H

h

• |u|2
• H1

ρ,h(Ωi)

◮ Broken weighted H1− norm

|u|2

• H1

ρ,h(Ω) :=

N

• i=1
• τ∈Th(Ωi)
• τ

(ρi∇u, ∇u)dx

◮

I NN

H : the bounds are similar, however,

|u|2

• H1

ρ,h(Ωi)

replaced by |u|2

• H1

ρ,h(Ωext i

)

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 14 / 38

References (Two-level)

Two-level: ˆ Vh = ˆ V E(NN)

H

+ N

i=1 ˆ

V δ

h (Ωδ i )

• M. Sarkis. “Nonstandard coarse spaces and Schwarz methods for

elliptic problems with discontinuous coefficients using non-conforming elements”. Numer. Math., 77(3), 1997, pp. 383-406. An earlier version in: “Two-level Schwarz methods for P1 nonconforming finite elements and discontinuous coefficients. Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Number 3224, Part 2, pages 543-566, Hampton VA, 1993. NASA Important results on this paper:

◮ The β-weighting ◮ The face coarse spaces ◮ Iterative substructuring coarse space and overlapping Schwarz ◮ Tools to analyze preconditioners for nonconforming FEM: stable maps

between conforming and nonconforming spaces

◮ Order of (1 + H/δ) ∗ (1 + log H/h) bounds independent of coefficients Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 15 / 38

References (Multi-level)

Decomposition: ˆ VhL = ˆ V E(NN)

hL

+ ˆ VhL + L

l=0 Vhl (conforming spaces)

Bilinear forms: ˆ ah on ˆ V E(NN)

hL

, diag. of ˆ ah on ˆ VhL, a-BPX on L

l=0 Vhl

Order of (1 + log2 H/h) bounds independent of coefficients

• M. Sarkis, “Multilevel methods for P1 nonconforming finite elements

and discontinuous coefficients in three dimensions”. Decomposition Methods in Scientific and Engineering (Univesity Park, PA, 1993),

• Contemp. Math. , Vol. 180, Amer. Math. Soc., Providence, RI,

1994, pp. 119-124.

• M. Sarkis, “Schwarz preconditioners for elliptic problems with

discontinuous coefficients using conforming and non-conforming elements”. PhD Dissertation, Technical Report 671, Courant Institute, September 1994.

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 16 / 38

P1 Conforming FEM and Discrete Sobolev Inequalities

VH(Ω) ⊂ H1

0(Ω): P1 conforming FEM in the coarse triangulation

Vh(Ω) ⊂ VH(Ω): P1 conforming FEM in the fine triangulation Pointwise interpolation IH : Vh(Ω) → VH(Ω), defined by (IHu)(x) = u(x), x vertices of the Ωi Discrete Sobolev inequalities u2

L∞(Ωi) (1 + log H

h )u2

H1(Ωi)

in 2D u2

L∞(Ωi) H

h u2

H1(Ωi)

in 3D Same estimates hold for u − u(x)2

L∞(Ωi) . . . |u|2 H1(Ωi)

x ∈ Ωi

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 17 / 38

Pointwise Interpolation: L2 and H1 Poincar´ e Inequalities

L2− approximation for 2D. For u ∈ Vh(Ωi) u − IHu2

L2(Ωi) (1 + log H

h )H2|u|2

H1(Ωi),

∀u ∈ Vh(Ωi) H1− stability for 2D |IHu|2

H1(Ωi) (1 + log H

h )|u|2

H1(Ωi),

∀u ∈ Vh(Ωi) In 3D, replace (1 + log H

h ) by H h

Estimates done locally in Ωi, hence, holds for weighted norms in Ω H/h factor in 3D (not so good) Not trivial to extend to general Ωi and discontinuous coefficients

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 18 / 38

Discrete Sobolev Inequalities for Conforming FEM (3D)

Let uV be the value of u ∈ Vh(Ωi) at a vertex V of Ωi u − uV 2

L2(Ωi) (H

h )H2|u|2

H1(Ωi)

Let ¯ uE be the edge average of u ∈ Vh(Ωi) on an edge E of Ωi u − ¯ uE2

L2(Ωi) (1 + log H

h )H2|u|2

H1(Ωi)

Let ¯ uF be the face average of u ∈ Vh(Ωi) on a face F of Ωi u − ¯ uF2

L2(Ωi) H2|u|2 H1(Ωi)

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 19 / 38

Coarse Basis Functions (3D)

Similar construction as we did for the nonconforming case I B

H u =

• V

uV θV +

• E

¯ uEθE +

• F

¯ uFθF

◮ θV := 1 at vertex V and zero at remaining interface nodes ◮ θE := 1 at nodes on edge E and zero at remaining interface nodes ◮ θF := 1 at nodes on face F and zero at remaining interface nodes ◮ θV , θE and θF : discrete harmonic inside the subdomains

H1−norm of the basis functions |θV |2

H1(Ωi) h

|θE|2

H1(Ωi) H

|θF|2

H1(Ωi) (1 + log H

h )H

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 20 / 38

Stability and Approximation

The θV , θE and θF form a partition of unity on Ωi I B

H u =

• V

(¯ u − uV )θV +

• E

(¯ u − ¯ uE)θE +

• F

(¯ u − ¯ uF)θF Let us bound for instance |(¯ u − ¯ uF)θF|H1(Ωi), the others are similar |(¯ u − ¯ uF)θF|2

H1(Ωi) |¯

u − ¯ uF|2(1 + log H h )H |¯ u−¯ uF|2 |u−¯ u|2+|u−¯ uF|2 1 H3

• u − ¯

u2

L2(Ωi) + u − ¯

uF2

L2(Ωi)

• H1− stability |I B

H u|2 H1(Ωi) (1 + log H h )|u|2 H1(Ωi)

L2− approximation u − I B

H u|2 L2(Ωi) H2|u − I B H u|2 H1(Ωi)

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Subspace Decomposition

Global space V . Example: V = Vh(Ω) ⊂ H1

0(Ω)

◮ Global bilinear form a(u, v) for u, v ∈ V . Example:

a(u, v) =

• Ω

(ρ∇u, ∇v)dx

Local spaces Vi, i = 1 : N. Example: Vi = Vh(Ω) ∩ H1

0(Ωδ i )

◮ Local bilinear forms ai(ui, vi) for ui, vi ∈ Vi. Example:

ai(ui, vi) =

• Ωδ

i

(ρ∇ui, ∇vi)dx

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 22 / 38

Subspace Decomposition (cont)

Coarse space V0. Example: V0 = V B

h

• r V H

h

• r V NN

h

◮ Coarse bilinear form a0(u0, v0) for u0, v0 ∈ V0. Example:

a0(u0, v0) = a(u0, v0)

Extension operators RT

i

: Vi → V . Examples:

◮ Local spaces: u = RT

i ui is the zero extension H1 0(Ωδ i ) to H1 0(Ω)

◮ Coarse space: u = RT

0 u0 is the “identity” since V0 ⊂ V

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 23 / 38

Abstract Theory of Schwarz Methods

Additive Schwarz Tas = (N

i=0 RT i A−1 i

Ri)A = (B)A Lower bound: C −2

0 a(u, u) ≤ a(Tasu, u)

◮ Lions’ Lemma: Find a C0 > 0 such that, for any u ∈ V , there exists a

decomposition u = N

i=0 RT i ui such that N

• i=0

ai(ui, ui) ≤ C 2

0 a(u, u)

Upper bound: a(Tasu, u) ≤ ωρ(E)a(u, u)

◮ Inexact solvers: Find an ω where for any i = 0 : N and ui ∈ Vi

a(RT

i ui, RT i ui) ≤ ωai(ui, ui)

◮ Strengthened Cauchy-Schwarz: Find a upper bound for the spectral

radius of E = {ǫij}i,j=0:N a(RT

i ui, RT j uj) ≤ ǫija(RT i ui, RT i ui)1/2a(RT j uj, RT j uj)1/2

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 24 / 38

Decomposition

Partition of Unity φi(x) ∈ Vh(Ω) for the overlapping subdomains Ωδ

i

supp(φi) ⊂ Ω

δ i

0 ≤ φi(x) ≤ 1, x ∈ Ω

δ i N

• i=1

φi(x) = 1, x ∈ Ω ∇φi∞ ≤ C/δ, 1 ≤ i ≤ N Decomposition of u = u0 + N

i=1 ui

◮ u0 = I B

h u. Let w = u − u0

◮ ui = Ih(φiw) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 25 / 38

Lower bound estimation

We need to estimate a(u0, u0) +

N

• i=1

ai(ui, ui) It follows from H1

ρ(Ω)− stability

a(u0, u0) = |I B

h u|2 H1

ρ(Ω) (1 + log H

h )|u|2

H1

ρ(Ω) = (1 + log H

h )a(u, u) Easy to see: |∇Ih(φiw)|2 |w∇φi|2 + |φi∇w|2

• Ωδ

i

(ρ∇Ih(φiw), ∇Ih(φiw))dx 1 δ2

• Ωδ

i

ρw2 +

• Ωδ

i

ρ|∇w|2dx (1 + log H h )(H δ )2|u|2

H1

ρ(Ωext i

) + (1 + log H

h )|u|2

H1

ρ(Ωext i

)

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 26 / 38

Small Overlapping Technique

To improve the factor (H/δ)2 to H/δ Note that the support of ∇φi is a δ-layer near ∂Ωi Divide the support of ∇φi in pieces: the layer inside Ωi, and the layers Ωδ

i ∩ Ωj. Note that in each of these layers, ρ is constant

We can estimate the L2− norm in each layer by the L2− and H1−seminorms on the corresponding subdomain the layer belongs 1D: Estimate w2

L2(0,δ) in terms of w2 L2(0,H) and |w|2 H1(0,H)

Use Fundamental Theorem of Calculus and Cauchy-Schwarz: w2(x) w2(y) + H|w|2

H1(0,H)

∀x, y ∈ (0, H) Integrate x in (0, δ) and y in (0, H) Hw2

L2(0,δ) δw2 L2(0,H) + H2δ|w|2 H1(0,H)

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 27 / 38

References

Two-Level: C. R. Dohrmann, A. Klawonn, and O. B. Widlund. “Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions”, March 2007, SIAM J. Numer. Anal.,

• Vol. 46(4), 2008, 2153–2168.

Multi-level: M. Dryja, M. Sarkis and O. Widlund. “Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions”. Numer. Math., 72(3), 1996, pp. 313-348

• M. Dryja and O. B. Widlund. ”Domain Decomposition Algorithms

with Small Overlap”, SIAM J. Sci. Stat. Comput., Vol. 15, No. 3, May 1994, 604-620.

• M. Dryja, B. F. Smith, and O. B. Widlund. “Schwarz Analysis of

Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions”, SIAM J. Numer. Anal., Vol. 31, No. 6, December 1994, 1662-1694.

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 28 / 38

Primal Iterative Substructuring Methods

Eliminate all the interior variables inside of the Ωi The idea: replace the local overlapping Dirichlet solvers by nonoverlapping Neumann solvers Local spaces Vi := Vh(Ωi) with vertex, edge, face, or/and the whole boundary average constraint Local bilinear forms ai(ui, vi) =

• Ωi

(ρi∇ui, ∇vi)dx Extension operator RT

i

: ui → u

◮ Nodes x ∈ ∂Ωi define u(x) = ui(x)(ρβ

i )/( j ρβ j )

◮ Nodes x ∈ ∂Ωj\∂Ωi define u(x) = 0 ◮ Extend discrete harmonically inside the subdomains

Note that this algorithm requires Neumann matrices, so, not algebraic

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 29 / 38

References

• M. Dryja and O. B. Widlund. “Schwarz Methods of

Neumann-Neumann Type for Three- Dimensional Elliptic Finite Element Problems”, Comm. Pure Appl. Math., Vol. 48, No. 2, February 1995, 121-155.

• M. Dryja, B. F. Smith, and O. B. Widlund. “Schwarz Analysis of

Iterative Substructuring Algorithms for Elliptic Problems in Three Dimensions”, SIAM J. Numer. Anal., Vol. 31, No. 6, December 1994, 1662-1694.

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BDDC

The idea is to use the same local bilinear forms to define the coarse basis functions The coarse basis functions are discontinuous across interfaces The local and global solvers are solved in parallel and the weights are used in the sum to make the iterative solutions continuous Reference: J. Mandel and C. R. Dohrmann, “ Convergence of a Balancing Domain Decomposition by Constraints and Energy Minimization” , Numer. Lin. Alg. Appl. 10(2003) 639-659. See also [TW] book

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Partition of Unity Coarse Spaces

Choose φi as coarse basis functions ¯ ui : Average of u in Ωδ

i and define

I PU

H u = N

• i=1

¯ uiφi For constant coefficients: H/δ bound. [S3] Obs: For φi based on smoothed agglomeration techniques: (H/δ)2

• bound. [TW]

For elasticity: coarse basis functions Ih(φi ∗ RBMs). RBMs are the rigid body motions. H/δ bound. [S2]

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 32 / 38

PU Coarse Spaces with Harmonic Overlap

PU-ASHO: Restrict φi to the Γδ

i := ∪N i=1∂Ωδ i and extend discrete

harmonically elsewhere. We get H/δ bounds. [S3] Enhanced PU-Coarse Space: The bounds can be improved by considering coarse spaces based on Ih(VEnh ∗ φi). In [S1]: ”For each subdomain Ωδ

i we let the space VEnh be defined as the vector space

generated by few lowest finite element eigenmodes associated to

• perator aΩδ

i without assuming Dirichlet boundary condition on ∂Ωδ

i “

PU-ASHO for discontinuous coefficients: For nodes x on ∂Ωi define φi(x) = ρβ

i /( j ρβ j ). For nodes x on Γδ i , if inside Ωi define φi(x) = 1,

if outside φi(x) = 0. The remaining nodes use discrete harmonic

• extension. [S1]. We get a (1 + H/h + (log H/h)(log δ/h)) bounds,

the same bound as in RASHO [S4] PU-OBDD where the local problems are Neumann solvers on

• verlapping subdomais [S5]

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References

S1 M. Sarkis. “Partition of unity coarse spaces: enhanced versions, discontinuous coefficients and applications to elasticity”. Domain decomposition methods in science and engineering, Natl. Auton.

• Univ. Mex., Mexico, 2003, pp. 149-158.

S2 M. Sarkis. “A coarse space for elasticity: partition of unity rigid body motions coarse space”. Applied Mathematics and Scientific Computing (Dubrovnik, 2001), Kluwer/Plenum, New York, 2003, pp 261-273. S3 M. Sarkis. “ Partition of unity coarse spaces and Schwarz methods with harmonic overlap”. Lect. Notes Comput. Sci. Eng. (Zurich, 2001), Springer-Verlag, Vol. 23, 2002, pp. 75-92. S4 X-C. Cai, M. Dryja and M. Sarkis. Restricted additive Schwarz preconditioner with harmonic overlap for symmetric positive definite linear systems. SIAM J. Numer. Anal., 41(4), 2003, pp. 1209-1231. S5 OBDD: overlapping balancing domain decomposition methods and generalizations to Helmholtz equations. LNCSE, vol. 55, Springer, 2006, pp. 317-324.

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Galvis-Efendiev PU-Generalized Eigenvectors

Overlapping regions Ω

′ i : union of all Ωk sharing a vertex Vi

For each Ω′

i :, the enhanced space V i Enh : is defined as the span of all

the eigenvectors associated to eigenvalues smaller than O(H) Aiv(i,k) = λ(i,k)Miv(i,k) Coarse Space := Span(i,k){Ih(φiv(i,k))} Let Pi

Enh be the a-projection from Vh(Ω′ i) to V i Enh(Ω′ i)

Key idea: (forany heterogeneous positive coefficients ρ) u − Pi

Enhu2 L2

ρ(Ω′ i) H2|u|2

H1

ρ(Ω′ i)

|Pi

Enhu|2 H1

ρ(Ω′ i) ≤ |u|2

H1

ρ(Ω′ i)

Variants: Mandel et al., Galvis-Efendiev et al., Nataf-Scheichl et al.

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 35 / 38

References

• M. Brezina, C. Heberton, J. Mandel, and P. Vanek, “An iterative

method with convergence rate chosen a priori”, UCD/CCM Report 140, April 1999. An earlier version presented at the 1998 Copper Mountain Conference on Iterative Methods, April 1998.

• J. Galvis and Y. Efendiev, “Domain Decomposition Preconditioners

for Multiscale Flows in High-Contrast Media”, Multiscale Model.

• Simul. Volume 8, Issue 4, pp. 1461-1483 (2010).
• J. Galvis and Y. Efendiev, Domain decomposition preconditioners for

multiscale flows in high contrast media. Reduced dimension coarse

• spaces. Multiscale Model. Simul. Volume 8, Issue 5, pp. 1621-1644

(2010).

• N. Spillane, F. Nataf, V. Dolean, P. Hauret, C. Pechstein, and R.

Scheichl and N. Spillane, “Abstract Robust Coarse Spaces for Systems of PDEs via Generalized Eigenproblems in the Overlaps. Preprint NuMa Report No. 2011-07 (2011).

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 36 / 38

A Few References on Heterogeneous Coefficients

I.G. Graham, P. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numerische Mathematik 106:589-626, 2007

• R. Scheichl and E. Vainikko, Additive Schwarz with

Aggregation-Based Coarsening for Elliptic Problems with Highly Variable Coefficients, Computing 80(4):319-343, 2007

• J. Van lent, R. Scheichl and I.G. Graham, Energy Minimizing Coarse

Spaces for Two-level Schwarz Methods for Multiscale PDEs, Numerical Linear Algebra with Applications 16(10):775-799, 2009

• C. Pechstein and R. Scheichl, Weighted Poincare inequalities,

submitted 10th December 2010, NuMa-Report 2010-10, Institute of Computational Mathematics, Johannes Kepler University Linz, 2010.

Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 37 / 38

References on Heterogeneous Coefficients (cont)

• A. Klawonn and O. Rheinbach, Robust FETI-DP methods for

heterogeneous three dimensional elasticity problems , Comput. Meth.

• Appl. Mech. Engrg., Vol. 196, pp. 1400-1414, January 2007
• C. Pechstein and R. Scheichl, Analysis of FETI Methods for

Multiscale PDEs - Part II: Interface Variation, Numerische Mathematik 118 (3):485-529, 2011

• M. Dryja and M. Sarkis. Boundary Layer Technical Tools for

FETI-DP Methods to Heterogeneous Coefficients. In Domain Decomposition Methods in Science and Engineering XIX, Huang, Y.; Kornhuber, R.; Widlund, O.; Xu, J. (Eds.), Volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, 2011, pp.205-212

• M. Dryja and M. Sarkis. Additive average Schwarz methods for

discretization of elliptic problems with highly discontinuous

• coefficients. Computational Methods in Applied Mathematics, Vol 10

(2), pp. 164-176, 2010.

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