Some Domain Decomposition Methods for Discontinuous Coefficients - PowerPoint PPT Presentation

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 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, 2005. Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 2 / 38

Geometrically Nonconforming Subdomain Partition Ω = ∪ N i =1 Ω i Ω i disjoint shaped regular polygonal subdomains of diameter O ( H i ) T h i (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-2011 3 / 38

Problem of Interest Continuous PDE. Find u ∈ H 1 0 (Ω) such that � � � N ∀ v ∈ H 1 a ( u , v ) := ( ρ i ∇ u , ∇ v ) dx = fvdx =: f ( v ) 0 (Ω) Ω i Ω i =1 Conforming FEM. V h (Ω) ⊂ H 1 0 (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. ˆ V 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 ∈ ˆ V h (Ω) such that � � a h ( u , v ) := ˆ ( ρ ∇ u τ τ ∈T h (Ω) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 4 / 38

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 ρ (Ω) � H 2 | u | 2 L 2 H 1 ρ (Ω) | I H u | 2 ρ (Ω) � | u | 2 H 1 H 1 ρ (Ω) ◮ First inequality not always possible with constant independently of ρ i � � u � 2 ρ (Ω) := ( ρ u , u ) dx L 2 Ω � | u | 2 ρ (Ω) := ( ρ ∇ u , ∇ u ) dx H 1 Ω 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 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 P 1 nonconforming: � V H (Ω) �⊂ H 1 0 (Ω) ◮ Broken weighted H 1 − stability and weighted L 2 − approximation | � I H u | � ρ, H (Ω) � | u | H 1 ρ (Ω) H 1 � u − � I H u � L 2 ρ (Ω) � H | u | H 1 ρ (Ω) ◮ � I H u averages u on faces (edges) of Ω i � N � | u | 2 ρ, H (Ω) := ( ρ i ∇ u , ∇ u ) dx � H 1 Ω i i =1 Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 6 / 38

Nonconforming FEM (2D Local Analysis) m 1 , m 2 , m 3 middle points of edges E 1 , E 2 , E 3 of Ω i u m 1 , ¯ ¯ u m 2 , ¯ u m 3 edge averages u m 1 | 2 + | ¯ | � I H u | 2 u m 1 | 2 H 1 (Ω i ) � | ¯ u m 2 − ¯ u m 3 − ¯ Note that � � � 2 � � E 2 ( u − ¯ u m 1 ) ds � � u m 1 | 2 = � H − 1 � u − ¯ u m 1 � 2 � | ¯ u m 2 − ¯ � � L 2 ( E 2 ) � E 2 1 ds � � H − 2 � u − ¯ u m 1 � 2 u m 1 | 2 L 2 (Ω i ) + | u − ¯ H 1 (Ω i ) By Friedrich’s inequality, the H 1 − stability in Ω i holds u − � I H u has zero average on edges of Ω i so Friedrich holds on Ω i H 1 (Ω i ) ≤ 2 H 2 � � � u − � L 2 (Ω i ) � H 2 | u − � H 1 (Ω i ) + | � I H u � 2 I H u | 2 | u | 2 I H u | 2 H 1 (Ω i ) 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 P 1 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 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 P 1 : � u − u ( V ) � L 2 ρ (Ω V ) for u ∈ V H (Ω V ) ρ (Ω V ) � H | u | H 1 e does not hold in � ◮ CR: broken weighted Poincar´ I H (Ω V ) Quasi-monotone coefficients: holds for � I H (Ω V ) and H 1 (Ω V ) 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 := ∂ Ω i ∩ ∂ Ω j Let ¯ u i , ¯ u j element averages on ∂ Ω i and ∂ Ω j Define the � I H u by ρ β ρ β β ∈ [1 j i u ij ,β = ¯ u i + ¯ u j , ¯ 2 , ∞ ] ρ β i + ρ β ρ β i + ρ β j j Denote m ij , m ik , m il middle points of E ij , E ik , E il Let ¯ u m ij , ¯ u m ik , ¯ u m il be edge averages of u on E ij , E ik , E il Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 9 / 38

CR: Element Averages (cont.) Broken weighted H 1 − stability � u ik ,β | 2 � u ik ,β | 2 ≤ 3 ρ i u m ij | 2 + | ¯ u m ik | 2 + | ¯ ρ i | ¯ u ij ,β − ¯ | ¯ u ij ,β − ¯ u m ij − ¯ u m ik − ¯ � � 2 ρ β ρ β u m ij | 2 = ρ i j i ρ i | ¯ u ij ,β − ¯ (¯ u i − ¯ u m ij ) + (¯ u j − ¯ u m ij ) ρ β i + ρ β ρ β i + ρ β j j � � 2 ρ β j ρ i ≤ ρ j if β ≥ 1 / 2 ρ β i + ρ β j Weighted L 2 − stability ρ i � u − � L 2 (Ω i ) � ρ i H 2 | u − � H 1 (Ω i ) + ρ i H | u − � I H u � 2 I H u | 2 I H u | 2 L 2 ( E ij ) ρ β ρ β u − � j i I H u = ( u − ¯ u i ) + ( u − ¯ u j ) ρ β i + ρ β ρ β i + ρ β j j Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 10 / 38

CR: Face Coarse Spaces V H and � � V H �⊂ � 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 � H u equal to ( � I E I H u )( m ij ) at every middle point of the fine triangulation on E ij Step 2: Discrete harmonic extension to define � I 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-2011 11 / 38

CR-Face Coarse Basis Functions E ij := Ω i ∩ Ω j Face coarse basis functions θ E ij . On fine nodes, let: ◮ θ E ij : 1 on E ij ◮ θ E ij := 0 on all subdomain edges but E ij ◮ θ E ij : discrete harmonic extension inside Ω i and Ω j ◮ θ E ij := 0 elsewhere Note that the support of θ E ij is Ω i ∪ Ω j What is � I E H ? � � I E H u = u E θ E ¯ Edges E ∈T H (Ω) ˆ H is the range of � V E I E H Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 12 / 38

CR: NN Coarse Basis Functions For each subdomain Ω i define � ρ β i θ i = θ E ij ρ β i + ρ β E ij ⊂ ∂ Ω i j ext Support of θ i is Ω : union of all Ω j sharing an edge with Ω i i � I NN given by H N � � I NN = ¯ u i θ i H i =1 Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 13 / 38

CR: Face Coarse Space (cont.) Note that � I H u and � I E H u have the same average on each coarse edge E ij It is possible to show: ◮ Weighted L 2 − approximation � u − � H u � 2 ρ (Ω i ) � H 2 | u | 2 I E L 2 � H 1 ρ, h (Ω i ) ◮ Broken weighted H 1 − stability � � 1 + log H | � I E H u | 2 | u | 2 ρ, h (Ω i ) � � � H 1 H 1 h ρ, h (Ω i ) ◮ Broken weighted H 1 − norm � � N � | u | 2 ρ, h (Ω) := ( ρ i ∇ u , ∇ u ) dx H 1 � τ i =1 τ ∈T h (Ω i ) ◮ � I NN H : the bounds are similar, however, | u | 2 | u | 2 replaced by � H 1 H 1 � ρ, h (Ω ext ρ, h (Ω i ) ) i Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 14 / 38