Some Domain Decomposition Methods for Discontinuous Coefficients

Transcription

1 Some Domain Decomposition Methods for Discontinuous Coefficients Marcus Sarkis WPI RICAM-Linz, October 31, 2011 Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

2 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 overlap), 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, Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

3 Geometrically Nonconforming Subdomain Partition Ω = N i=1 Ω i Ω i disjoint shaped regular polygonal subdomains of diameter O(H i ) T hi (Ω i ) shape regular triangulations In this talk: geometrically conforming partitions and matching meshes The Ω i are simplices (to drop later) with constant coefficient ρ i Subdomains, faces, edges are open sets Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

4 Problem of Interest Continuous PDE. Find u H0 1 (Ω) such that a(u, v) := N (ρ i u, v)dx = Ω i i=1 Ω fvdx =: f (v) v H 1 0 (Ω) Conforming FEM. V h (Ω) H0 1 (Ω) : continuous piecewise linear functions in Ω. Find u V h (Ω) such that a(u, v) = f (v) v V h (Ω) Nonconforming FEM. ˆV h (Ω): piecewise linear functions and continuous at the middle points of the edges of T h (Ω) and zero at the middle points of the edges of T h ( Ω). Find u ˆV h (Ω) such that â h (u, v) := (ρ u τ T h (Ω) τ Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

5 Conforming FEM P 1 conforming: V H (Ω) H 1 0 (Ω) Weighted L 2 approximation and weighted H 1 stability u I H u 2 L 2 ρ (Ω) H2 u 2 H 1 ρ (Ω) I H u 2 H 1 ρ (Ω) u 2 H 1 ρ (Ω) First inequality not always possible with constant independently of ρi u 2 L 2 ρ (Ω) := (ρu, u)dx u 2 Hρ 1 (Ω) := (ρ u, u)dx Ω Ω Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

6 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 Poincaré-Friedrich type inequalities inside each Ω i Step 3: establish local H 1 stability and L 2 approximation in Ω i Step 4: establish global weighted H 1 stab. and L 2 appr. in Ω Crouzeix-Raviart P1 nonconforming: VH (Ω) H 1 0 (Ω) Broken weighted H1 stability and weighted L 2 approximation ÎHu Ĥ1 ρ,h (Ω) u H 1 ρ (Ω) u ÎHu L 2 ρ (Ω) H u H 1 ρ (Ω) ÎH u averages u on faces (edges) of Ω i u 2 := N Ĥρ,H 1 (Ω) i=1 Ω i (ρ i u, u)dx Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

7 Nonconforming FEM (2D Local Analysis) m 1, m 2, m 3 middle points of edges E 1, E 2, E 3 of Ω i ū m1, ū m2, ū m3 edge averages ÎHu 2 H 1 (Ω i ) ū m 2 ū m1 2 + ū m3 ū m1 2 Note that ū m2 ū m1 2 E = 2 (u ū m1 )ds E 2 1ds 2 H 1 u ū m1 2 L 2 (E 2 ) H 2 u ū m1 2 L 2 (Ω i ) + u ū m 1 2 H 1 (Ω i ) By Friedrich s inequality, the H 1 stability in Ω i holds u ÎHu has zero average on edges of Ω i so Friedrich holds on Ω i u ÎHu 2 L 2 (Ω i ) H2 u ÎHu 2 H 1 (Ω i ) 2H2 ( u 2 H 1 (Ω i ) + ÎHu 2 H 1 (Ω i ) ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

8 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 P 1 In certain cases, discrete Poincaré-Friedrich-type inequalities with constants independently of coefficients are more difficult to obtain than for conforming elements Let V be a coarse vertex of T H (Ω). Let Ω V be the union of all Ω j touching V (let us assume there are 6 Ω j ) Coefficients (M, 1, 1, M, 1, 1) (anti-clockwise orientation). M very large Conforming P1 : u u(v ) L 2 ρ (Ω V ) H u H 1 ρ (Ω V ) for u V H (Ω V ) CR: broken weighted Poincaré does not hold in ÎH(Ω V ) Quasi-monotone coefficients: holds for ÎH(Ω V ) and H 1 (Ω V ) The preconditioning analysis for CR more complicated than conforming Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

9 CR: Boundary Element Averages How to define the values at the subdomain edges? Let us take an edge E ij := Ω i Ω j Let ū i, ū j element averages on Ω i and Ω j Define the ĨHu by ū ij,β = ρ β i ρβ i + ρ β j ū i + ρ β i ρβ j + ρ β j ū j, β [ 1 2, ] Denote m ij, m ik, m il middle points of E ij, E ik, E il Let ū mij, ū mik, ū mil be edge averages of u on E ij, E ik, E il Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

10 CR: Element Averages (cont.) Broken weighted H 1 stability ρ i ū ij,β ū ik,β 2 3ρ i ( ūij,β ū mij 2 + ū mij ū mik 2 + ū mik ū ik,β 2) ρ i ū ij,β ū mij 2 = ρ i ( ρ i ( ρ β i Weighted L 2 stability ρ β i ρ β j ρ β i + ρ β j + ρ β (ū i ū mij ) + ρβ j j ρ β i + ρ β (ū j ū mij ) j ) 2 ρ j if β 1/2 ) 2 ρ i u ĨHu 2 L 2 (Ω i ) ρ ih 2 u ĨHu 2 H 1 (Ω i ) + ρ ih u ĨHu 2 L 2 (E ij ) u ĨHu = ρ β i ρβ i + ρ β j (u ū i ) + ρβ j ρ β i + ρ β j (u ū j ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

11 CR: Face Coarse Spaces V H and ṼH V h Continuity at the middle points of the coarse edges ( V H ) does not imply continuity at the middle points of the fine edges ( V h ) Edge coarse space (in 2D) and Face coarse space (in 3D) Step 1: On each subdomain edge E ij : make Î E H u equal to (ÎHu)(m ij ) at every middle point of the fine triangulation on E ij Step 2: Discrete harmonic extension to define Î E H u inside Ω i The resulting coarse space can be constructed easily for general Ω i Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

12 CR-Face Coarse Basis Functions E ij := Ω i Ω j Face coarse basis functions θ Eij. On fine nodes, let: θ Eij : 1 on E ij θeij := 0 on all subdomain edges but E ij θeij : discrete harmonic extension inside Ω i and Ω j θeij := 0 elsewhere Note that the support of θ Eij is Ω i Ω j What is Î E H? Î E H u = Edges E T H (Ω) ū E θ E ˆV E H is the range of Î E H Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

13 CR: NN Coarse Basis Functions For each subdomain Ω i define θ i = E ij Ω i ρ β i β ρi + ρ β j θ Eij Support of θ i is Ω ext i Î NN H given by : union of all Ω j sharing an edge with Ω i Î NN H N = ū i θ i i=1 Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

14 CR: Face Coarse Space (cont.) Note that ÎHu and Î E H u have the same average on each coarse edge E ij It is possible to show: Weighted L 2 approximation u Î E H u 2 L 2 ρ (Ω i ) H2 u 2 Ĥ 1 ρ,h (Ω i ) Broken weighted H1 stability ( Î H E u log H ) u 2 Ĥρ,h 1 (Ω i ) h Ĥρ,h 1 (Ω i ) Broken weighted H1 norm u 2 Ĥ 1 ρ,h (Ω) := N i=1 τ T h (Ω i ) τ (ρ i u, u)dx ÎH NN : the bounds are similar, however, u 2 Ĥ 1 ρ,h (Ω i ) replaced by u 2 Ĥ 1 ρ,h (Ωext i ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

15 References (Two-level) Two-level: ˆV h = ˆV E(NN) H + N i=1 ˆV 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 An earlier version in: Two-level Schwarz methods for P 1 nonconforming finite elements and discontinuous coefficients. Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Number 3224, Part 2, pages , Hampton VA, 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 / 38

16 References (Multi-level) Decomposition: ˆV hl = ˆV E(NN) h L + ˆV hl + L l=0 V h l (conforming spaces) E(NN) Bilinear forms: â h on ˆV h L, diag. of â h on ˆV hl, a-bpx on L l=0 V h l Order of (1 + log 2 H/h) bounds independent of coefficients M. Sarkis, Multilevel methods for P 1 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 M. Sarkis, Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. PhD Dissertation, Technical Report 671, Courant Institute, September Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

17 P 1 Conforming FEM and Discrete Sobolev Inequalities V H (Ω) H0 1(Ω): P 1 conforming FEM in the coarse triangulation V h (Ω) V H (Ω): P 1 conforming FEM in the fine triangulation Pointwise interpolation I H : V h (Ω) V H (Ω), defined by (I H u)(x) = u(x), x vertices of the Ω i Discrete Sobolev inequalities u 2 L (Ω i ) (1 + log H h ) u 2 H 1 (Ω i ) in 2D u 2 L (Ω i ) H h u 2 H 1 (Ω i ) in 3D Same estimates hold for u u(x) 2 L (Ω i )... u 2 H 1 (Ω i ) x Ω i Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

18 Pointwise Interpolation: L 2 and H 1 Poincaré Inequalities L 2 approximation for 2D. For u V h (Ω i ) u I H u 2 L 2 (Ω i ) (1 + log H h )H2 u 2 H 1 (Ω i ), u V h(ω i ) H 1 stability for 2D I H u 2 H 1 (Ω i ) (1 + log H h ) u 2 H 1 (Ω i ), u V h(ω i ) In 3D, replace (1 + log H h ) by H h Estimates done locally in Ω i, hence, holds for weighted norms in Ω H/h factor in 3D (not so good) Not trivial to extend to general Ω i and discontinuous coefficients Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

19 Discrete Sobolev Inequalities for Conforming FEM (3D) Let u V be the value of u V h (Ω i ) at a vertex V of Ω i u u V 2 L 2 (Ω i ) (H h )H2 u 2 H 1 (Ω i ) Let ū E be the edge average of u V h (Ω i ) on an edge E of Ω i u ū E 2 L 2 (Ω i ) (1 + log H h )H2 u 2 H 1 (Ω i ) Let ū F be the face average of u V h (Ω i ) on a face F of Ω i u ū F 2 L 2 (Ω i ) H2 u 2 H 1 (Ω i ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

20 Coarse Basis Functions (3D) Similar construction as we did for the nonconforming case I B H u = V u V θ V + E ū E θ E + F ū F θ F θv := 1 at vertex V and zero at remaining interface nodes θ E := 1 at nodes on edge E and zero at remaining interface nodes θ F := 1 at nodes on face F and zero at remaining interface nodes θ V, θ E and θ F : discrete harmonic inside the subdomains H 1 norm of the basis functions θ V 2 H 1 (Ω i ) h θ E 2 H 1 (Ω i ) H θ F 2 H 1 (Ω i ) (1 + log H h )H Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

21 Stability and Approximation The θ V, θ E and θ F form a partition of unity on Ω i I B H u = V (ū u V )θ V + E (ū ū E )θ E + F (ū ū F )θ F Let us bound for instance (ū ū F )θ F H 1 (Ω i ), the others are similar (ū ū F )θ F 2 H 1 (Ω i ) ū ū F 2 (1 + log H h )H ū ū F 2 u ū 2 + u ū F 2 1 ) ( u H 3 ū 2 L 2 (Ω i ) + u ū F 2 L 2 (Ω i ) H 1 stability I B H u 2 H 1 (Ω i ) (1 + log H h ) u 2 H 1 (Ω i ) L 2 approximation u I B H u 2 L 2 (Ω i ) H2 u I B H u 2 H 1 (Ω i ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

22 Subspace Decomposition Global space V. Example: V = V h (Ω) H 1 0 (Ω) Global bilinear form a(u, v) for u, v V. Example: a(u, v) = (ρ u, v)dx Local spaces V i, i = 1 : N. Example: V i = V h (Ω) H0 1 (Ω δ i ) Ω Local bilinear forms a i (u i, v i ) for u i, v i V i. Example: a i (u i, v i ) = (ρ u i, v i )dx Ω δ i Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

23 Subspace Decomposition (cont) Coarse space V 0. Example: V 0 = V B h or V H h or V NN h Coarse bilinear form a0 (u 0, v 0 ) for u 0, v 0 V 0. Example: a 0 (u 0, v 0 ) = a(u 0, v 0 ) Extension operators R T i : V i V. Examples: Local spaces: u = R T i u i is the zero extension H 1 0 (Ωδ i ) to H1 0 (Ω) Coarse space: u = R T 0 u 0 is the identity since V 0 V Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

24 Abstract Theory of Schwarz Methods Additive Schwarz T as = ( N i=0 RT i A 1 i R i )A = (B)A Lower bound: C0 2 a(u, u) a(t asu, u) Lions Lemma: Find a C 0 > 0 such that, for any u V, there exists a decomposition u = N i=0 RT i u i such that N a i (u i, u i ) C0 2 a(u, u) i=0 Upper bound: a(t as u, u) ωρ(e)a(u, u) Inexact solvers: Find an ω where for any i = 0 : N and ui V i a(r T i u i, R T i u i ) ωa i (u i, u i ) Strengthened Cauchy-Schwarz: Find a upper bound for the spectral radius of E = {ɛ ij } i,j=0:n a(r T i u i, R T j u j ) ɛ ij a(r T i u i, R T i u i ) 1/2 a(r T j u j, R T j u j ) 1/2 Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

25 Decomposition Partition of Unity φ i (x) V h (Ω) for the overlapping subdomains Ω δ i supp(φ i ) Ω δ i 0 φ i (x) 1, x Ω δ i N φ i (x) = 1, i=1 φ i C/δ, Decomposition of u = u 0 + N i=1 u i x Ω 1 i N u 0 = I B h u. Let w = u u 0 ui = I h (φ i w) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

26 Lower bound estimation We need to estimate a(u 0, u 0 ) + It follows from Hρ(Ω) 1 stability N a i (u i, u i ) i=1 a(u 0, u 0 ) = Ih B u 2 Hρ 1(Ω) (1 + log H h ) u 2 Hρ 1(Ω) = (1 + log H )a(u, u) h Easy to see: I h (φ i w) 2 w φ i 2 + φ i w 2 (ρ I h (φ i w), I h (φ i w))dx 1 δ 2 ρw 2 + ρ w 2 dx Ω δ i Ω δ i Ω δ i (1 + log H h )(H δ )2 u 2 Hρ(Ω 1 ext i ) + (1 + log H h ) u 2 Hρ(Ω 1 ext i ) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

27 Small Overlapping Technique To improve the factor (H/δ) 2 to H/δ Note that the support of φ i is a δ-layer near Ω i Divide the support of φ i in pieces: the layer inside Ω i, and the layers Ω δ i Ω j. Note that in each of these layers, ρ is constant We can estimate the L 2 norm in each layer by the L 2 and H 1 seminorms on the corresponding subdomain the layer belongs 1D: Estimate w 2 L 2 (0,δ) in terms of w 2 L 2 (0,H) and w 2 H 1 (0,H) Use Fundamental Theorem of Calculus and Cauchy-Schwarz: w 2 (x) w 2 (y) + H w 2 H 1 (0,H) x, y (0, H) Integrate x in (0, δ) and y in (0, H) H w 2 L 2 (0,δ) δ w 2 L 2 (0,H) + H2 δ w 2 H 1 (0,H) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

28 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, 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 M. Dryja and O. B. Widlund. Domain Decomposition Algorithms with Small Overlap, SIAM J. Sci. Stat. Comput., Vol. 15, No. 3, May 1994, 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, Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

29 Primal Iterative Substructuring Methods Eliminate all the interior variables inside of the Ω i The idea: replace the local overlapping Dirichlet solvers by nonoverlapping Neumann solvers Local spaces V i := V h (Ω i ) with vertex, edge, face, or/and the whole boundary average constraint Local bilinear forms a i (u i, v i ) = (ρ i u i, v i )dx Ω i Extension operator R T i : u i u Nodes x Ωi define u(x) = u i (x)(ρ β i )/( j ρβ j ) Nodes x Ωj \ Ω i 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 / 38

30 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, 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, Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

31 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) See also [TW] book Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

32 Partition of Unity Coarse Spaces Choose φ i as coarse basis functions ū i : Average of u in Ω δ i and define I PU H N u = ū i φ i i=1 For constant coefficients: H/δ bound. [S3] Obs: For φ i based on smoothed agglomeration techniques: (H/δ) 2 bound. [TW] For elasticity: coarse basis functions I h (φ i RBMs). RBMs are the rigid body motions. H/δ bound. [S2] Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

33 PU Coarse Spaces with Harmonic Overlap PU-ASHO: Restrict φ i to the Γ δ 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 I h (V Enh φ i ). In [S1]: For each subdomain Ω δ i we let the space V Enh be defined as the vector space generated by few lowest finite element eigenmodes associated to operator a Ω δ without assuming Dirichlet boundary condition on Ω δ i i PU-ASHO for discontinuous coefficients: For nodes x on Ω i define φ i (x) = ρ β i /( j ρβ j ). For nodes x on Γδ i, if inside Ω i define φ i (x) = 1, if outside φ i (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 overlapping subdomais [S5] Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

34 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 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 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 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 S5 OBDD: overlapping balancing domain decomposition methods and generalizations to Helmholtz equations. LNCSE, vol. 55, Springer, 2006, pp Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

35 Galvis-Efendiev PU-Generalized Eigenvectors Overlapping regions Ω i : union of all Ω k sharing a vertex V i For each Ω i :, the enhanced space VEnh i : is defined as the span of all the eigenvectors associated to eigenvalues smaller than O(H) A i v (i,k) = λ (i,k) M i v (i,k) Coarse Space := Span (i,k) {I h (φ i v (i,k) )} Let PEnh i be the a-projection from V h(ω i ) to V Enh i (Ω i ) Key idea: (forany heterogeneous positive coefficients ρ) u P i Enh u 2 L 2 ρ(ω i ) H2 u 2 H 1 ρ(ω i ) P i Enh u 2 H 1 ρ(ω i ) u 2 H 1 ρ(ω i ) Variants: Mandel et al., Galvis-Efendiev et al., Nataf-Scheichl et al. Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

36 References M. Brezina, C. Heberton, J. Mandel, and P. Vanek, An iterative method with convergence rate chosen a priori, UCD/CCM Report 140, April An earlier version presented at the 1998 Copper Mountain Conference on Iterative Methods, April J. Galvis and Y. Efendiev, Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media, Multiscale Model. Simul. Volume 8, Issue 4, pp (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 (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). Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

37 A Few References on Heterogeneous Coefficients I.G. Graham, P. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numerische Mathematik 106: , 2007 R. Scheichl and E. Vainikko, Additive Schwarz with Aggregation-Based Coarsening for Elliptic Problems with Highly Variable Coefficients, Computing 80(4): , 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): , 2009 C. Pechstein and R. Scheichl, Weighted Poincare inequalities, submitted 10th December 2010, NuMa-Report , Institute of Computational Mathematics, Johannes Kepler University Linz, Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38

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 , January 2007 C. Pechstein and R. Scheichl, Analysis of FETI Methods for Multiscale PDEs - Part II: Interface Variation, Numerische Mathematik 118 (3): , 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 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 , Marcus Sarkis (WPI) Domain Decomposition Methods RICAM / 38