Introduction to Domain Decomposition Methods Alfio Quarteroni
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Aachen, EU Regional School Series – February 18, 2013 Introduction to Domain Decomposition Methods Alfio Quarteroni Chair of Modeling and Scientific Computing Mathematics Institute of Computational Science and Engineering Ecole´ Polytechnique F´ed´erale de Lausanne MOX – Modeling and Scientific Computing Mathematics Department Politecnico di Milano 1 MOTIVATION DD can be used in the framework of any discretization method for PDEs (FEM, FV, FD, SEM) to make their algebraic solution more efficient on parallel computer platforms DDM allow the reformulation of a boundary-value problem on a partition of the computational domain into subdomains ⇒ very convenient framework for the solution of heterogeneous or multiphysics problems, i.e. those that are governed by differential equations of different kinds in different subregions of the computational domain. 2 THE IDEA The computational domain Ω where the bvp is set is subdivided into two or more subdomains in which problems of smaller dimension are to be solved. Parallel solution algorithms may be used. There are two ways of subdividing the computational domain: with disjoint subdomains with overlapping subdomains Ω2 Ω2 Ω1 Γ2 Γ1 Ω Ω1 Γ Correspondingly, different DD algorithms will be set up. 3 AN HISTORICAL REMARK The classical “logo” of DDM is [http://www.ddm.org/] Why this symbol? What does it mean? We have to go back to a work of 1869 by Hermann Schwarz... 4 ...who wanted to establish the existence of harmonic functions with prescribed boundary values on regions with non-smooth boundaries: −#u = f in Ω u = g on ∂Ω ! where Ω ∂Ω Schwarz’s idea: split Ωinto two elementary subdomains: Ω2 Ω Ω1 Γ2 Γ1 and set up a suitable iterative method passing information between the subproblems. 5 MODERN EXAMPLES OF SUBDIVISIONS 6 REFERENCES B.F. Smith, P.E. Bjørstad, W.D. Gropp (1996) Domain Decomposition Cambridge University Press, Cambridge. A. Quarteroni and A. Valli (1999) Domain Decomposition Methods for Partial Differential Equations Oxford Science Publications, Oxford. A. Toselli and O.B. Widlund (2005) Domain Decomposition Methods – Algorithms and Theory Springer-Verlag, Berlin and Heidelberg. 7 CLASSICAL ITERATIVE DD METHODS 8 MODEL PROBLEM Consider the model problem: find u :Ω→ R s.t. Lu = f in Ω u =0 on∂Ω ! L is a generic second order elliptic operator. The weak formulation reads: 1 find u ∈ V = H0 (Ω): a(u, v)=(f , v) ∀v ∈ V where a(·, ·)isthebilinearformassociatedwithL. 9 SCHWARZ METHODS Consider a decomposition of Ωwith overlap: Ω2 Ω Ω1 Γ2 Γ1 (0) Given u2 on Γ1,fork ≥ 1: (k) Lu1 = f in Ω1 (k) (k−1) solve u1 = u2 on Γ1 (k) u1 =0 on∂Ω1 \ Γ1 (k) Lu2 = f in Ω2 (k) k u solve u( ) = 1 on Γ 2 (k−1) 2 & u1 (k) u2 =0 on∂Ω2 \ Γ2. 10 Choice of the trace on Γ2: (k) if u1 ⇒ multiplicative Schwarz method (k−1) if u1 ⇒ additive Schwarz method We have two elliptic bvp with Dirichlet conditions in Ω1 and Ω2,andwe (k) (k) wish the two sequences {u1 } and {u2 } to converge to the restrictions of the solution u of the original problem: (k) (k) limk→∞ u = u and limk→∞ u = u . 1 |Ω1 2 |Ω2 The Schwarz method applied to the model problem always converges, with a rate that increases as the measure |Γ12| of the overlapping region Γ12 increases. 11 Example Consider the model problem −u$$(x)=0 a < x < b, u(a)=u(b)=0, ! where Ω2 abγ2 γ1 Ω1 The solution is u =0.Weshowafewiterationsofthemethod: (0) u2 (1) u1 u(1) (2) 2 u1 (2) u2 abγ2 γ1 Clearly, the method converges with a rate that reduces as the length of the interval (γ2,γ1)getssmaller. 12 Remark The Schwarz method requires at each iteration the solution oftwo subproblems of the same kind as those of the original problem. The boundary conditions of the initial problem remain unchanged. Dirichlet-type boundary conditions are imposed across the interfaces. 13 NONOVERLAPPING DECOMPOSITION We partition now the domain Ωin two disjoint subdomains: Ω2 Ω Ω1 Γ The following equivalence result holds. Theorem The solution u of the model problem is such that u = ui for i =1, 2, |Ωi where ui is the solution to the problem Lui = finΩi ui =0 on ∂Ωi \ Γ ! with interface conditions ∂u1 ∂u2 u1 = u2 and = on Γ ∂nL ∂nL 14 ∂/∂nL is the conormal derivative. Example The problem −div (µ∇u)=f in Ω u =0 on∂Ω ! is equivalent to −div (µi ∇ui )=fi in Ωi u =0 on∂Ωi u1 = u2 on Γ ∂u ∂u µ 1 = µ 2 on Γ 1 ∂n 2 ∂n Remark The proof of the theorem can be done using the weak formulationofthe problem. We refer to Lemma 1.2.1 in [QV99]. 15 DIRICHLET-NEUMANN METHOD (0) Given u2 on Γ, for k ≥ 1solvetheproblems: (k) Lu1 = f in Ω1 (k) (k−1) u1 = u2 on Γ (k) u1 =0 on∂Ω1 \ Γ (k) Lu2 = f in Ω2 ∂u(k) ∂u(k) 2 = 1 on Γ ∂nL ∂nL (k) u2 =0 on∂Ω2 \ Γ The equivalence theorem guarantees that when the two sequences (k) (k) {u1 } and {u2 } converge, their limit will be necessarily the solution to the exact problem. The DN algorithm is therefore consistent. However, its convergence is not always guaranteed. 16 Example Let Ω= (a, b), γ ∈ (a, b), L = −d 2/dx 2 and f =0.Ateveryk ≥ 1the DN algorithm generates the two subproblems: (k) $$ −(u1 ) =0 a < x <γ (k) (k−1) u1 = u2 x = γ (k) u1 =0 x = a (k) $$ −(u2 ) =0 γ<x < b (k) $ (k) $ (u2 ) =(u1 ) x = γ (k) u2 =0 x = b. The two sequences converge only if γ>(a + b)/2: (0) u2 (0) u2 u(2) γ a+b 2 ab2 a+b γ ab2 (1) (1) 17 u2 u2 AvariantoftheDNalgorithmcanbesetupbyreplacingthe Dirichlet condition in the first subdomain by (k) (k−1) (k−1) u1 = θu2 +(1− θ)u1 on Γ using a relaxation parameter θ>0. In such a way it is always possible to reduce the error between two subsequent iterates. In the previous example, we can easily verify that, by choosing (k−1) u1 θopt = − (k−1) (k−1) u2 − u1 the algorithm converges to the exact solution in a single iteration. More in general, there exists a suitable value 0 <θmax < 1suchthat the DN algorithm converges for any possible choice of the relaxation parameter θ in the interval (0,θmax ). 18 NEUMANN-NEUMANN ALGORITHM Consider again a partition of Ωinto two disjoint subdomains and denote by λ the (unknown) value of the solution u on their interface Γ: λ = ui on Γ(i =1, 2) Consider the following iterative algorithm: for any given λ(0) on Γ, for k ≥ 0andi =1, 2, solve the following problems: (k+1) Lui = f in Ωi (k+1) (k) ui = λ on Γ (k+1) ui =0 on∂Ωi \ Γ (k+1) Lψi =0 inΩi ∂ψ(k+1) ∂u(k+1) ∂u(k+1) i = 1 − 2 on Γ ∂n ∂n ∂n (k+1) ψi =0 on∂Ωi \ Γ with (k+1) (k) (k+1) (k+1) λ = λ − θ σ1ψ1|Γ − σ2ψ2|Γ where θ is a positive acceleration parameter,' while σ1 and( σ2 are two positive coefficients. 19 ROBIN-ROBIN ALGORITHM For every k ≥ 0solvethefollowingproblems: (k+1) u1 = f in Ω1 (k+1) u1 =0 on∂Ω1 \ Γ ∂u(k+1) ∂u(k) 1 + γ u(k+1) = 2 + γ u(k) on Γ ∂n 1 1 ∂n 1 2 then (k+1) Lu2 = f in Ω2 (k+1) u2 =0 on∂Ω2 \ Γ ∂u(k+1) ∂u(k+1) 2 + γ u(k+1) = 1 + γ u(k+1) on Γ ∂n 2 2 ∂n 2 1 (0) where u2 is assigned and γ1, γ2 are non-negative acceleration parameters that satisfy γ1 + γ2 > 0. (k) (k+1) Aiming at parallelization, we could use u1 instead of u1 . 20 THE STEKLOV-POINCAREINTERFACE´ EQUATION 21 MULTI-DOMAIN FORMULATION OF POISSON PROBLEM AND INTERFACE CONDITIONS We consider now the model problem: −#u = f in Ω u =0 on∂Ω. ! For a domain partitioned into two disjoint subdomains,wecanwritethe equivalent multi-domain formulation (ui = u|Ωi , i =1, 2): −#u1 = f in Ω1 u1 =0 on∂Ω1 \ Γ −#u2 = f in Ω2 u =0 on∂Ω \ Γ 2 2 u1 = u2 on Γ ∂u ∂u 1 = 2 on Γ. ∂n ∂n 22 Remark On the interface Γwe have the normal unit vectors n1 and n2: Ω1 n1 Γ n2 Ω2 There holds: n1 = −n2 on Γ. We denote n = n1 so that ∂ ∂ ∂ = = − on Γ. ∂n ∂n1 ∂n2 23 THE STEKLOV-POINCAREOPERATOR´ Let λ be the unknown value of the solution u on the interface Γ: λ = u|Γ Should we know a priori the value λ on Γ,wecouldsolvethefollowing two independent boundary-value problems with Dirichlet condition on Γ (i =1, 2): −#wi = f in Ωi w =0 on∂Ω \ Γ i i wi = λ on Γ. 24 With the aim of obtaining the value λ on Γ, let us split wi as follows ∗ 0 wi = wi + ui , ∗ 0 where wi and ui represent the solutions of the following problems (i =1, 2): ∗ −#wi = f in Ωi ∗ wi =0 on∂Ωi ∩ ∂Ω ∗ wi =0 on Γ and 0 −#ui = 0 in Ωi 0 ui =0 on∂Ωi ∩ ∂Ω 0 ui = λ on Γ. 25 ∗ The functions wi depend solely on the source data f ∗ ⇒ wi = Gi f where Gi is a linear continuous operator 0 ui depend solely on the value λ on Γ 0 ⇒ ui = Hi λ where Hi is the so-called harmonic extension operator of λ on the domain Ωi .