We consider the primal hybrid formulation for second order elliptic problems introduced by Raviart-Thomas and apply the classical iterative method of Uzawa to obtain a non overlapping domain decomposition method that converges geometrically with a mesh independent ratio. The proposed method connects with the Finite Element Tearing and Interconnecting (FETI) method proposed by Farhat-Roux and collaborators. In this research work we use the detailed work on domains with corners developed by Grisvard [6], which clarifies the situation of cross-points, and the direct computation of the duality H−1/2 − H1/2 using the H1/2 scalar product; therefore no consistency error appears

Bernardi, Christine

Chacón Rebollo, Tomás

Chacón Vera, Eliseo

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–8)A domain decomposition method derived from the PrimalHybrid Formulations for 2nd order elliptic problemsC. Bernardi1, T . Chaco´n Rebollo2, E. Chaco´n Vera21Laboratoire Jacques-Louis Lions, Universite´ Paris VI et CNRS, 175 rue du Chevaleret 75013 Paris.E-mail: bernardi@ann.jussieu.fr2 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, E-41080 Sevilla. E-mails: chacon@us.es,eliseo@us.esPalabras clave: Domain Decomposition Methods, Lagrange MultipliersResumenWe consider the primal hybrid formulation for second order elliptic problems in-troduced by Raviart-Thomas [9] and apply the classical iterative method of Uzawato obtain a non overlapping domain decomposition method that converges geometri-cally with a mesh independent ratio. The proposed method connects with the FiniteElement Tearing and Interconnecting (FETI) method proposed by Farhat-Roux andcollaborators [7]-[8]. In this research work we use the detailed work on domains withcorners developed by Grisvard [6], which clarifies the situation of cross-points, andthe direct computation of the duality H−1/2 − H1/2 using the H1/2 scalar product;therefore no consistency error appears.1. IntroductionThe primal hybrid formulation for second order elliptic problems inforce via the Lagran-ge multipliers the continuity of the approximations across interfaces, and this is expressedvia the duality H−1/2 −H1/2, see Raviart-Thomas [9]. Usually, for numerical discretiza-tions, this duality is worked out by means of some projection operator onto the L2 spaceon the interfaces, see Ben Belgacem [3]. In our approach we use Riesz representation andreplace the duality with the H1/2 scalar product that is explicitly computed. As a con-sequence, we have a formulation in terms of a saddle point problem suitable for iterativetechniques, see Bacuta [2].The method presented is similar to the classical Lagrange Finite Element Tearing andInterconnecting (FETI) method proposed by Farhat-Roux and collaborators [7]-[8]. In thisresearch work we use the detailed work on domains with corners developed by Grisvard1C. Bernardi, T . Chaco´n Rebollo, E. Chaco´n Vera[6] which clarifies the situation of cross-points and the direct computation of the dualityH−1/2 −H1/2 using the H1/2 scalar product; therefore no consistency error appears.2. Formulation with Lagrange MultipliersLet Ω ⊂ R2 be a poligonal domain and consider f ∈ L2(Ω). Then, our departureproblem looks for u ∈ H10 (Ω) such that(∇u,∇v)Ω + (u, v)Ω = (f, v)Ω ∀v ∈ H10 (Ω). (1)Assume that Ω is a polygonal bounded domain in R2 with a Lipschitz–continuous boundaryand consider a decomposition without overlapping in polygonal subdomainsΩ = ∪Rr=1Ωr and Ωr ∩ Ωr′ = ∅, 1 ≤ r < r′ ≤ R (2)where each Ωr has a Lipschitz–continuous boundary. We describe ∂Ωr in terms of its edgesvia∂Ωr = Γr,0 ∪ Γr,1 ∪ ... ∪ Γr,Jr (3)where Γr,0 = ∂Ωr ∩ ∂Ω such that ∂Ω = ∪Rr=1Γr,0 and assume that Ωr ∩Ωs is either empty,a single point or a full edge Γr,s. On each Γr,j we consider the classical Hilbert space oftraces H1/200 (Γr,j) and its dual space H−1/200 (Γr,j), see Adams [1]. We call skeleton of Ω,and denote it by E , the set of all interfaces in ΩE = ∪Ii=1Γi (4)where Γi = Γi,0 for i = 1, ..., R describe the boundary ∂Ω, and for i ≥ R + 1 we setΓi = Γr,j for some r, j ≥ 1. Green’s formulae on polygonal domains will be usedLemma 1 (Grisvard [6]) When O ⊂ R2 is a poligonal domain and ∂O = ∪Jj=1Γj, thenH2(O) is dense on E = {u ∈ H1(O); ∆u ∈ L2(Ω)}. The mapping u 7→ ∂nju|Γj has aunique continuous extension from E to H−1/200 (Γj) dual space of H1/200 (Γj). Moreover, foreach u ∈ E and v ∈ H1(O) such that v|Γj ∈ H1/200 (Γj) we have−(∆u, v)O = (∇u,∇v)O −J∑j=1< ∂nju, v >−1/2,00,Γj . (5)Also, using that D(O)d is dense on H(div;O), for any ~q ∈ H(div;O), we have nj · ~q ∈H−1/200 (Γj) and for any v ∈ H1(Ω) with v|Γj ∈ H1/200 (Γj)(~q,∇v)O + (div(~q), v)O =J∑j=1< nj · ~q, v >−1/2,00,Γj . (6)2A domain decomposition method derived from the Primal Hybrid FormulationsNext, on each Ωr we consider the classical Hilbert spaceH1b (Ωr) = {vr ∈ H1(Ωr); vr = 0 on ∂Ωr ∩ ∂Ω} (7)with scalar product (ur, vr)1,Ωr = (ur, vr)Ωr + (∇ur,∇vr)Ωr the dense subspace Wr ofH1b (Ωr) given byWr = {u ∈ H1b (Ωr); u|Γr,j ∈ H1/200 (Γr,j), j = 1, ...,Kr},needed to apply Green’s formulae, the global, defined on Ω, Hilbert spaceX = {v ∈ L2(Ω); vr = v|Ωr ∈ H1b (Ωr), r = 1, ..., R} ≈R∏r=1H1b (Ωr) (8)with scalar product and norm given by(u, v)X =R∑r=1(ur, vr)1,Ωr , ‖v‖2X = (v, v)X =R∑r=1‖vr‖21,Ωr ,∀v, u ∈ X (9)and finally X0 ≈∏Rr=1Wr that is also a dense subspace of X. We also need the HilbertspaceH0(div; Ω) = {~q ∈ L2(Ω)d; div(~q) ∈ L2(Ω), nr,0 · ~q = 0 en Γr,0, 1 ≤ r ≤ R} (10)where nr,j · ~q ∈ H−1/200 (Γr,j), j = 1, ...,Kr, consider Tr =∏Krj=1H−1/200 (Γr,j) and M givenbyM = {~µ ∈R∏r=1Tr;µr,j = nr,j · ~q, for some ~q ∈ H0(div; Ω)}. (11)Let b :M ×X 7→ R given for v ∈ X0, ~λ ∈M byb(~λ, v) =I∑i=R+1< λi, vs − vt >−1/2,00,Γi (12)when Ωs ∩ Ωt = Γi and extended by density to all v ∈ X. ThenH10 (Ω) = {v ∈ X; b(~λ, v) = 0, ∀~λ ∈M}.Define the bilinear form a : X ×X 7→ R given bya(u, v) =R∑r=1{(∇ur,∇vr)Ωr + (ur, vr)Ωr} =R∑r=1∫Ωr{∇ur · ∇vr + ur vr} dx. (13)Then, the primal hybrid formulation for Poisson problem (1) consists in looking for apair (u,~λ) ∈ X ×M such thata(u, v) + b(~λ, v) =R∑r=1(f, vr)Ωr , ∀v ∈ X (v|Ωr = vr) (14)b(~µ, u) = 0, ∀~µ ∈M. (15)3C. Bernardi, T . Chaco´n Rebollo, E. Chaco´n VeraTheorem 1 If u ∈ H10 (Ω) solves the Dirichlet problem (1) then there exists a unique(u,~λ) ∈ X ×M that solves problem (14)-(15). If (u,~λ) ∈ X ×M solves (14)-(15) thenu ∈ H10 (Ω) and solves the Dirichlet problem (1). Moreover, for i = R+ 1, ..., Iλi = −∂niu ∈ H−1/200 (Γi). (16)Next, via Riesz representation we identifyH−1/200 (Γi) (dual space ofH1/200 (Γi)) withH1/200 (Γi),write the duality in terms of the scalar product in H1/200 (Γi), identifyM with its dual spaceM ′ and define b :M ×X 7→ R given for any v ∈ X0, ~λ ∈M byb(~λ, v) =I∑i=R+1(λi, vs − vt)1/2,00,Γi (17)when Ωs∩Ωt = Γi and extended by density to all v ∈ X. Then, the formulation of Poissonproblem (1) that we shall use is: Find a pair (u,~λ) ∈ X ×M such thata(u, v) + b(v,~λ) =R∑r=1(f, vr)Ωr , ∀ v ∈ X, (18)b(u, ~µ) = 0, ∀ ~µ ∈M. (19)Thanks to Theorem 1 this is equivalent to (1) but it also is whithin the saddle pointproblems framework, see Girault-Raviart [5], which allows the use of different methodsfor computing the solution. Also, the analysis at the continuous level is reproduced in thediscrete version of the saddle point problems as a simple consequence of the finite elementextension theorems, see for instance Bernardi-Maday-Rapetti [4].3. Domain decomposition methodsA rephrasing of the problem in terms of functional operators will clarify what we do.To fix ideas we work with Ω split up slicewise into two subdomains. Let B : X 7→M givenby Bv = (v1)|Γ − (v2)|Γ , i.e., the jump of v across the interface Γ, set R : X ′ 7→ X as theRiesz isomorphism associated with the scalar product a(·, ·) on X and F : X 7→ R givenby < F, v >=∑2r=1(f, vr)Ωr . Then, our saddle point problem looks for (u, λ) ∈ X ×Msuch thatR−1 u+B′λ = F on X ′ (20)B u = 0 on M, (21)where B′ is the transpose operator to B. Then, u = R(F −B′λ)⇒ Bu = BRF −BRB′λand (using Bu = 0) from here we have the dual problem associated to the saddlepoint problem(BRB′)λ = BRF on M. (22)Thanks to the inf-sup condition the operator BRB′ is symmetric positive definite,see Bacuta [2]. Now the resolution of (22) via an iterative method is possible; we propose4A domain decomposition method derived from the Primal Hybrid Formulationsthe use of the iterative method of Richardson, which amounts to the classicalUzawa’s Method.Given ρ > 0 and λ0 ∈M , for m = 0, 1, 2, 3, ... setrm = BRF − (BRB′)λm = B um, using (??) (23)λm+1 = λm + ρ rm (24)which unfolds from (20)-(21) as2∑r=1(um,r, vr)1,Ωr =2∑r=1(f, vr)Ωr − (λm, v1 − v2, )1/2,00,Γ, ∀v ∈ X, (25)and update λm+1 = λm + ρ(um,1 − um,2). (26)Following standard convergence results, see Bacuta [2] and references therein, we havegeometric convergence for this iterative process by simply blocking to zero the testfunctions alternatively on each subdomainTheorem 2 The iterative process:Given ρ > 0 and λ0 ∈M , find for m ≥ 0 um ∈ X via(∇um,1,∇v1)Ω1 + (um,1, v1)Ω1 = (f, v1)Ω1 − (λm, v1)1/2,00,Γ, ∀v1 ∈ X1,(∇wm,2,∇v2)Ω2 + (um,2, v2)Ω2 = (f, v2)Ω2 + (λm, v2)1/2,00,Γ, ∀v2 ∈ X2,and update λm+1 = λm + ρ(um,1 − um,2) on Γis a non overlapping domain decomposition method geometrically convergent witha ratio of convergence independent of the mesh size.The drawback that this method presents is how to fix the optimal parameter ρ > 0. Inthe numerical experiments that we present the value of ρ has been tuned easily by handthanks to the great speed of convergence that the method exhibits.For a method that has no need of fixing any parameter we could use the applicationof the Conjugate Gradient Method which is the core of the FETI methods.4. Numerical experimentsWe compute on a non convex domain with three subdomains. We use a Galerkinapproximation with P1 Lagrange finite elements on a uniform triangular mesh size h of Ωand its restriction to each of the Ωi for i = 1, 2, 3. The numerical results show a geometricrate of convergence with a mesh independent ratio as the theory predicts.We set Ω = (−1, 1)2 \ {(−1, 0)× (−1, 0)} and decompose it into three squares so thatour interfaces are Γ1 = {0} × (0, 1) and Γ2 = (0, 1)× {0}. Then we solve−∆u = 1, on Ω, u = 0 on ∂Ω.We take λ0 = (0, 0) and stop iterating for m = niter(h) such that‖um+1h − umh ‖X‖umh ‖X=(∑3i=1∫Ωi|∇(um+1i,h − umi,h)|2 dx)1/2(∑3i=1∫Ωi|∇umi,h|2 dx)1/2≤ 10−7; (27)5C. Bernardi, T . Chaco´n Rebollo, E. Chaco´n Vera0 2 4 6 8 10 12 14 16 18 200.150.20.250.30.350.40.450.5Error ratio 0 2 4 6 8 10 12 14 16 1800.511.52Uzawa h=1/4Uzawa h=1/8Uzawa h=1/16Uzawa h=1/32CG h=1/4CG h=1/8CG h=1/16CG h=1/32Figura 1: Decrease error ratio given by (29) on the L-shaped domain test using Uzawa’sMethod and Conjugate Gradient Method (CG).we compute the errors and their decrease ratio given for m ≥ 0 byeuh(h,m) = ‖uh − umh ‖X = (3∑i=1∫Ωi|∇(uh − umi,h)|2 dx)1/2 (28)r(h,m) =euh(h,m+ 1)euh(h,m), r(h) ≈ l´ımmr(h,m) (29)where uh is the P1 solution computed on the whole domain Ω. For Uzawa’s Method wefound by performing few several tests that ρ ≈ 0.12 seems to be the closest value to theoptimal one. The results are shown in Table 1.1/h 4 8 16 32#iterations 17 17 17 21ρ ≈ 0.12 ≈ 0.12 ≈ 0.12 ≈ 0.12r(h) ≈ 0.43.. ≈ 0.43... ≈ 0.43... ≈ 0.44...Table 1: Uzawa’s Method: Number of iterations, values of ρ and of r(h) obtainedwith the inverted L-shape domain for different values of h.We also computed the solution using the Conjugate Gradient Method. In Figure 1, Forboth of the iterative methods proposed, we show the ratio between consecutive errors.Figure 2 shows the Galerkin solution computed in Ω and Figure 3 shows the solutioncomputed right after the first iteration of Uzawa’s Method. We see the lack of jumps onthe two interfaces.AcknowledgmentsResearch work partially supported by Ministerio de Educacion y Ciencia with publicfunds: proyect MTM2006-01275.6A domain decomposition method derived from the Primal Hybrid Formulations0.00729360.0218810.0364680.0510550.0656430.080230.0948170.10940.123990.13858Iso Surfaces of     Aproximated  Solution   with   h=1/8. GiDxyzFigura 2: Approximated solution computed with standard Galerkin P1 finite elements onthe whole domain and with h = 1/8.7C. Bernardi, T . Chaco´n Rebollo, E. Chaco´n Vera0.00733650.0220090.0366820.0513550.0660280.0807010.0953740.110050.124720.13939Iso Surfaces of    Aproximated   Solution   with   ddm.   for   h=1/8. GiDxyzFigura 3: Computed solution after the first iteration with h = 1/8 using Uzawa’s Method.Referencias[1] Adams, R. A., Sobolev Spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.[2] Bacuta, C., A unified approach for Uzawa algorithm SIAM J. Numer. Anal., Vol. 44, No. 6, pp.-2633-2649, 2006.[3] Ben Belgacem, F., The Mortar finite element method with Lagrange multipliers Numerische Mathe-matik, 84:173-197, 1999.[4] C. Bernardi, Y. Maday and F. Rapetti, Discre´tisations variationnelles de proble`mes aux limiteselliptiques. Mathe´matiques & Applications, 45. Springer-Verlag, New-York, 2002.[5] Girault, V. and Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory andAlgorithms. Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986.[6] Grisvard, P., Singularities in Boundary value problems. Recherches en Mathe´matiques Applique´es22, Masson, 1992.[7] C. Farhat, F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solutionalgorithm, Int. J. Numer. Methods Engrg. 32 (1991) 1205-1227.[8] C. Farhat, F.-X. Roux,An unconventional domain decomposition method for an efficient parallelsolution of large-scale finite element systems SIAM J. Sci. Statist. Comput., 13, 379-396 (1992)[9] Raviart, P.A. and Thomas, J.-M., Primal Hybrid Finite Element Methods for second order elipticequations Math Comp, Vol 31, number 138, pp.- 391-413, 1977.8

A domain decomposition method derived from the Primal Hybrid Formulations for 2nd order elliptic problems

https://idus.us.es/xmlui/bitstream/handle/11441/34023/Domain%20decomposition%20method.pdf?sequence=1&isAllowed=y

A domain decomposition method derived from the Primal Hybrid Formulations for 2nd order elliptic problems

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla