Additive Schwarz preconditioner for the finite volume element discretization of symmetric elliptic problems

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1 BIT Numer Mat 206) 56: DOI 0.007/s x Additive Scwarz preconditioner for te finite volume element discretization of symmetric elliptic problems L. Marcinkowski T. Raman 2 A. Loneland 2,3 J. Valdman 4,5 Received: 2 October 204 / Accepted: 26 August 205 / Publised online: 25 September 205 Te Autors) 205. Tis article is publised wit open access at Springerlink.com Abstract A symmetric and a nonsymmetric variant of te additive Scwarz preconditioner are proposed for te solution of a class of finite volume element discretization of te symmetric elliptic problem in two dimensions, wit large jumps in te entries of te coefficient matrices across subdomains. It is sown tat te conver- Communicated by Miciel E. Hocstenbac. L. Marcinkowski was partially supported by te Polis Scientific Grant 20/0/B/ST/079, and Cinese Academy of Science Project: 203FFGA0009-GJHS J. Valdman acknowledges te support of te Project GA3-8652S GA CR). T. Raman acknowledges te support from te NRC troug te DAADppp Project B L. Marcinkowski lmarcin@mimuw.edu.pl T. Raman Talal.Raman@ib.no A. Loneland Atle.Loneland@ii.uib.no J. Valdman jvaldman@prf.jcu.cz Faculty of Matematics, University of Warsaw, Banaca 2, Warsaw, Poland 2 Department of Computing, Matematics and Pysics, Bergen University College, 5020 Bergen, Norway 3 Department of Informatics, University of Bergen, 5020 Bergen, Norway 4 Institute of Matematics and Biomatematics, University of Sout Boemia, Branišovská 3, České Budějovice, Czec Republic 5 Institute of Information Teory and Automation of te ASCR, Pod Vodárenskou věží 4, 8208 Prague, Czec Republic 23

2 968 L. Marcinkowski et al. gence of te preconditioned generalized minimal residual iteration using te proposed preconditioners depends polylogaritmically, in oter words weakly, on te mes parameters, and tat tey are robust wit respect to te jumps in te coefficients. Keywords Domain decomposition Additive Scwarz metod Finite volume element GMRES Matematics Subject Classification 65F0 65N22 65N30 63N55 Introduction We consider te classical finite volume element discretization of te second-order elliptic partial differential equation, were we seek for te discrete solution in te space of standard P conforming finite element functions, or te space of continuous and piecewise linear functions, cf. [7,20]. We furter consider te second-order elliptic partial differential equation to ave coefficients tat may ave large jumps across subdomains. Due to te finite volume discretization, te resulting systems are in general nonsymmetric, wic become increasingly nonsymmetric for coefficients varying increasingly inside te finite elements. All of tese make te problem ard to solve in a reasonable amount of time, particularly in large scale computations. Robust and efficient algoritms for te numerical solution of suc systems are terefore needed. Te design of suc algoritms is often callenging, particularly teir analysis wic is not as well understood as it is for te symmetric system. Te main objective of tis paper is to design and analyze a class of robust and scalable preconditioners based on te additive Scwarz domain decomposition metodology, cf. [30,32], using tem in te preconditioned generalized minimal residual GMRES) iteration based on minimizing te energy norm of te residual, cf. [9,29], for solving te system. Te finite volume element metod or te FVE metod, also known in te literature as te control volume finite element metod or te CVFE, provides a systematic approac to construct a finite volume or a control volume discretization of te differential equations using a finite element approximation of te discrete solution. Te metod as drawn muc interest in te scientific communities over te years. It as bot te flexibility of a finite element metod to easily adapt to any geometry, and te property of a finite volume discretization of being locally conservative, like conservation of mass, conservation of energy, etc. For a quick overview of te existing researc on te metodology, we refer to [ 4,,2,7,20,23,34]. Additive Scwarz metods ave been studied extensively in te literature, see [30,32] and te references terein. Wen it comes to solving second-order elliptic problems, te general focus as been on solving symmetric systems resulting from te finite element discretization of te problem. Despite te growing interest for finite volume elements, te researc on fast metods for te numerical solution of nonsymmetric systems resulting from te finite volume element discretization as been very limited. In particular metods like te domain decomposition, wic 23

3 Additive Scwarz preconditioner for te general FVE 969 are considered among te most powerful metods for large scale computation, ave rarely been tested on finite volume elements. Among te few existing works wic can be found in te literature, are te works of [3,35] based on te overlapping partition of te domain, and te works of [27,33] based on te nonoverlapping partition of te domain. Te latter ones are given witout te convergence analysis. In tis paper, we propose nonoverlapping domain decomposition metods for te finite volume element, wic are based on substructuring, and formulate tem as additive Scwarz metods. We sow tat teir convergence is robust wit respect to jumps in te coefficients across subdomains, and depends poly-logaritmically on te mes parameter wen used as preconditioners in te GMRES iteration. For an overview of important classes of iterative substructuring metods we refer to [32, capters 4 6] for symmetric and positive definite systems, and [32, Capter.3] for teir extension to nonsymmetric systems. We restrict ourselves to problems in 2D; extension to 3D is possible and will be investigated in te future. Also, since te present work is an attempt to develop a first analysis of iterative substructuring type domain decomposition metods for te finite volume element, we limit ourselves to coefficients tat may ave jumps only across subdomain boundaries. However, by modifying te coarse space, e.g. using oscillatory boundary conditions instead of te linear boundary conditions for te basis functions, cf. [8,9,27], or by enricing te coarse space wit eigenfunctions from local eigenvalue problems, cf. [0,4,2,3], it will be possible to extend te metods to effectively deal wit igly eterogeneous coefficients. For te general purpose of designing additive Scwarz preconditioners for a finite volume element discretization in tis paper, we ave formulated an abstract framework wic is ten later used in te analysis of te preconditioners proposed in te paper. Te framework borrows te basic ingredients of te abstract Scwarz framework for additive Scwarz metods, cf. [30,32], wile te analysis follows te work of [9] were additive Scwarz metods were considered for te advection-diffusion problem. Te framework as already been used in two of autors recent papers on te Crouzeix Raviart finite volume element, cf. [24,25], demonstrating its usefulness in te design and analysis of new and effective preconditioners for te finite volume element discretization of elliptic problems. For furter information on domain decomposition metods for nonsymmetric problems in general, we refer to [26,30,32] and references terein. Te paper is organized as follows: in Sect. 2, we present te differential problem, and in Sect. 3, its finite volume element discretization. In Sect. 4, we present te two variants of te additive Scwarz preconditioners and te two main results, Teorems 4. and 4.2. Te complete analysis is provided in te next two sections, te abstract framework in Sect. 5, and te required estimates in Sect. 6. Finally, numerical results are provided in Sect. 7. Trougout tis paper, we use te following notations: x y and w z denote tat tere exist positive constants c, C independent of mes parameters and te jump of coefficients suc tat x cy and w Cz, respectively. 23

4 970 L. Marcinkowski et al. 2 Te differential problem Given Ω, a polygonal domain in te plane, and f L 2 Ω), te purpose is to solve te following differential equation, Ax) u)x) = f x), x Ω, us) = 0, s Ω, were A L Ω)) 2 2 is a symmetric matrix valued function satisfying te uniform ellipticity as follows, α>0 suc tat ξ T Ax)ξ α ξ 2 2 x Ω and ξ R 2, were ξ 2 2 = ξ 2 + ξ 2 2. Furter we consider α equal to wic can be always obtained by scaling te original problem by α. We assume tat Ω is decomposed into a set of disjoint polygonal subdomains {D j } suc tat, in eac subdomain D j, Ax) is continuous and smoot in te sense tat A W, D i ) C Ω, 2.) were C Ω is a positive constant. We also assume tat λ j > 0 suc tat ξ T Ax)ξ λ j ξ 2 2 ξ 2 2 x D j and ξ R 2. Due to A L D j )) 2 2, we ave te following, j > 0 suc tat ν T Ax)ξ j ν 2 ξ 2 x D j and ξ,ν R 2. We also assume tat j λ j. We ten ave λ j u 2 H D j ) D j u T Ax) u dx j u 2 H D j ) u H D j ). 2.2) In te weak formulation, te differential problem is ten to find u H0 Ω) suc tat au,v)= f v) v H0 Ω), 2.3) were 23 au,v)= Ω u T Ax) v dx and f v) = Ω f v dx.

5 Additive Scwarz preconditioner for te general FVE 97 3 Te discrete problem For te discretization, we use a finite volume element discretization were te Eq. 2.3) is discretized using te standard finite volume metod on a mes wic is dual to te primal mes, and were te finite element space or te solution space is defined on te primal mes, see for instance in [6,7,20]; for an overview of finite volume element metods we refer to [23]. Let T = T Ω) be be a sape regular triangulation of Ω, cf.[8] or[6], ereon referred to as te primal mes consisting of triangles {τ} wit te size parameter = max τ T diamτ), and let Ω, Ω, and Ω be te sets of triangle vertices corresponding to Ω, Ω, and Ω, respectively. We assume tat eac τ T is contained in one of D j. Define V as te conforming linear finite element space consisting of functions wic are continuous piecewise linear over te triangulation T, and are equal to zero on Ω. Now, let T = T Ω) be te dual mes corresponding to T. For simplicity we use te so called Donald mes for te dual mes. For eac triangle τ T,letc τ be te centroid, x j, j =, 2, 3 te tree vertices, and m kl = m lk, k, l =, 2, 3tetree edge midpoints. Divide eac triangle τ into tree polygonal regions inside te triangle by connecting its edge midpoints m kl = m lk to its centroid c τ wit straigt lines. One suc polygonal region ω τ,x τ, associated wit te vertex x, as illustrated in Fig., is te region wic is enclosed by te line segments c τ m 3, m 3 x, x m 2, and m 2 c τ, and wose vertices are c τ, m 3, x, and m 2.Nowletω xk be te control volume associated wit te vertex x k, wic is te sum of all suc polygonal regions associated wit te vertex x k, i.e. ω xk = {τ T : x k is a vertex of τ} ω τ,xk Te set of all suc control volumes form our dual mes, i.e. T = T Ω) ={ω x} x Ω. A control volume ω xk is called a boundary control volume if x k Ω. Let V be te space of piecewise constant functions over te dual mes T, wic ave values equal to zero on Ω. We let te nodal basis of V be {φ x } x Ω, were φ x is te standard finite element basis function wic is equal to one at te vertex x and zero at all oter vertices. Analogously, te nodal basis of V is {ψ x} x Ω were ψ x is a piecewise constant function wic is equal to one over te control volume ω x associated wit te vertex x, and is zero elsewere. Fig. Sowing ω τ,x saded region) wic is part of te control volume ω x restricted to te triangle τ. Te control volume is associated wit te vertex x m 23 c τ x 3 m 3 x x 2 m 2 23

6 972 L. Marcinkowski et al. Te two interpolation operators, I and I, are defined as follows. I : CΩ) + V V and I : CΩ) V are given respectively as I v = vx)ψ x and I v = vx)φ x. x Ω x Ω We note ere tat I I v = v for v V,aswellasI I u = u for u V. Let te finite volume bilinear form be defined on V V as a FV : V V R suc tat a FV u,v)= vx i ) A unds u V,v V, x i Ω ω xi or equivalently on V V as a : V V R suc tat a u,v)= a FV u, I v) 3.) for u,v V. We note ere tat a, ) is a nonsymmetric bilinear form in general, wile a, ) is a symmetric bilinear form. Te discrete problem is ten to find u V suc tat a FV u,v)= f v) v V, 3.2) or equivalently a u,v)= f I v) v V. Te problem as a unique solution for sufficiently small, wic follows from te fact tat 5.2) olds, cf. Proposition 6., leading to te fact tat a u, u) is V -elliptic, cf. Lemma 5.2. An error estimate in te norm induced by au,v)can also be given, cf. e.g. [7, 3], We close tis section wit te following remark. Note tat in some cases, te bilinear form a, ) may equal te symmetric bilinear form a, ), as for instance in te case wen te matrix A is piecewise constant over te subdomains {D j }. Tis may not be true if we coose to use a different dual mes or if A is not piecewise constant. In tis paper, we only consider te case wen te bilinear form a, ) is nonsymmetric. 4 An edge based additive Scwarz metod We propose in tis section two variants of te edge based additive Scwarz metod ASM) for te discrete finite volume element problem of te previous section. Tey are constructed in te same way as it is done in te standard case using te classical abstract framework of additive Scwarz metods, cf. [26,30,32]. Te convergence analysis of te metods is owever based on te abstract framework wic will be developed in Sect. 5. Te new framework is an extension of te classical framework, and is based on tree key assumptions. We validate tose assumptions for te proposed metods in Sect. 6, tereby completing te analysis. We assume tat we ave a partition of Ω into te set of N nonoverlapping polygonal subdomains {Ω j } N j=, suc tat tey form a sape regular coarse partition of Ω, as described in [7]. Accordingly, we assume tat te intersection between two subdomains 23

7 Additive Scwarz preconditioner for te general FVE 973 is eiter a vertex, an edge, or te empty set, and tat tere is a fixed number M) of reference polygonal domains E k k M) of diameter one eac, suc tat for eac subdomain Ω j tere is an invertible affine map μ j x) = B j x + b j wic maps Ω j to one of te reference domains E k j) k j) M). Here B j is a 2 2 nonsingular matrix and b j a2 vector, and B j H j, B j H j, were H j = diamω j ). We assume tat eac subdomain Ω k lies in exactly one of te polygonal subdomains {D j } described earlier, and tat none of teir boundaries cross eac oter. Te interface Γ = N Ω k \ Ω, k= wic is te sum of all subdomain edges and subdomain vertices not lying on te boundary Ω, or crosspoints, plays a crucial role in te design of our preconditioner. We also assume tat te primal mes T is perfectly aligned wit te partitioning of Ω, in oter words, no edges of te primal mes cross any edge of te coarse substructure. As a consequence, te coefficient matrix Ax) restricted to a subdomain Ω k is in W, Ω k )) 2 2, and ence cf. 2.)) A W, Ω k ) A W, D j ) C Ω. Eac subdomain Ω k inerits its own local triangulation from te T, denote it by T Ω k ) ={τ T : τ Ω k }.LetV Ω k ) be te space of continuous and piecewise linear functions over te triangulation T Ω k ), wic are zero on Ω Ω k, and let V,0 Ω k ) := V Ω k ) H0 Ω k). Te local spaces are equipped wit te bilinear form a k u,v)= u T Ax) v dx. Ω k We define te local projection operator P k : V Ω k ) V,0 Ω k ) suc tat a k P k u,v)= a k u,v) v V,0 Ω k ) and te local discrete armonic extension operator H k : V Ω k ) V Ω k ) suc tat H k u = u P k u. Note tat H k u is equal to u on te boundary Ω k, and discrete armonic inside Ω k in te sense tat a k H k u,v)= 0 v V,0 Ω k ). 23

8 974 L. Marcinkowski et al. Te local and global spaces of discrete armonic functions are ten defined as W k = H k V Ω k ) and W = H V ={u V : u Ωk = H k u Ωk }, respectively. We now define te subspaces required for te ASM preconditioner, cf. [26,30,32]. For eac subdomain Ω k, te local subspace V k V is defined as V,0 Ω k ) extending it by zero to te rest of te subdomains, i.e. V k ={v V : v Ωk V,0 Ω k ) and v Ω\Ωk = 0} Te coarse space V 0 W is defined as te space of discrete armonic functions wic are piecewise linear over te subdomain edges i.e. linear along eac subdomain edge and discrete armonic inside eac subdomain). Its degrees of freedom are associated wit te subdomain vertices lying inside te domain Ω, i.e. te crosspoints, and ence te dimension equals te cardinality of V = k V k, were V k is te set of all subdomain vertices wic are not on te boundary Ω. For an analogous coarse space, we refer to [32, 5.4] were 3D substructuring metods are considered. Using teir ideas it is possible to extend te metod to 3D. Finally, te local edge based subspaces are defined as follows. For eac subdomain edge Γ kl, wic is te interface between Ω k and Ω l,weletv kl W be te local edge based subspace consisting of functions wic may be nonzero inside Γ kl, but zero on te rest of te interface Γ, and discrete armonic in te subdomains. It is not difficult to see tat te support of V kl is contained in Ω k Ω l. We ave te following decomposition of te finite element spaces W and V.Bot te symmetric and te nonsymmetric variant of te preconditioner use te same decomposition: W =V 0 + V kl, V =W + Γ kl Γ V k = V 0 + V kl + k= Γ kl Γ V k. k= 4.) Note tat te subspaces of W, i.e. V 0 and V kl, Γ kl Γ, are ortogonal to te subspaces V k, k =,...,N, wit respect to te bilinear form a, ). Remark 4. If Ax) is te identity matrix, and {Ω j } j N form a coarse triangulation of te domain Ω, ten te coarse space V 0 will consist of functions tat are continuous and piecewise linear over te coarse triangulation, cf. [32, 5.4.]. In te following, we describe te two preconditioners: te symmetric and te nonsymmetric preconditioner. We will see later in te numerical experiments section tat te two are very similar in teir performance. It is not difficult to see tat tey also ave similar computational complexities. 23

9 Additive Scwarz preconditioner for te general FVE Symmetric preconditioner For te symmetric variant of te preconditioner, we define te coarse space operator and te local subspace operators, T k : V V k, k = 0,,...,N, as at k u,v)= a u,v) v V k, and te local edge based subspace operators T kl : V V kl, Γ kl Γ,as at kl u,v)= a u,v) v V kl. Note tat te bilinear form used to calculate eac of te operations above, tat is on te left and side of te equation, is te symmetric bilinear form a, ). Now, defining our additive Scwarz operator T as te sum of te operators, tat is T = T 0 + T k + T kl, k= Γ kl Γ we can replace te discrete problem 3.2) by te following equivalent preconditioned system of equations, in te operator form, cf. [30, Capter 5]: Tu = g, 4.2) were g = g 0 + N k= g k + Γ kl Γ g kl, g 0 = T 0 u, g k = T k u, k =,...,N, and g kl = T kl u, Γ kl Γ, are te rigt and sides wit u being te exact solution. Note tat te rigt and sides can be calculated witout knowing te exact solution, cf. [30]. Te GMRES iteration is used to solve te preconditioned system 4.2). Te convergence rate is given by te standard GMRES convergence Teorem 5.) wit estimates of its two parameters using Teorem 4.. Teorem 4. Tere exists suc tat, if, ten for any u V atu, Tu) au, u), atu, u) were H = max k H k ) and H k = diamω k ). )) H 2 + log au, u), Te teorem is proved using te abstract framework given in Sect. 5, i.e. by using Teorem 5.2, and te tree propositions in Sect. 6, Propositions 6., 6.2, and 6.3, verifying te tree assumptions required by te framework. 4.2 Nonsymmetric preconditioner Analogously, for te nonsymmetric variant of te preconditioner, te coarse space operator and te local subspace operators, S k : V V k, k = 0,,...,N, are defined as 23

10 976 L. Marcinkowski et al. a S k u,v)= a u,v) v V k, and te local edge based subspace operators, S kl : V V kl, Γ kl Γ,as a S kl u,v)= a u,v) v V kl. Te bilinear form used to calculate eac of te operations above is te nonsymmetric bilinear form a, ). Again, denoting te additive Scwarz operator by S, were S = S 0 + S k + S kl, k= Γ kl Γ te discrete problem 3.2) can be replaced by te equivalent preconditioned system of equations, in te operator form: Su =ĝ, 4.3) were ĝ = ĝ 0 + N k= ĝ k + Γ kl Γ ĝkl, ĝ k = S k u, k = 0,,...,N, and ĝ kl = S kl u, Γ kl Γ, are te rigt and sides wit u being te exact solution. As in te symmetric case, te rigt and sides can be calculated witout knowing te exact solution. Again, te GMRES iteration is used to solve te preconditioned system 4.3), wose convergence rate is given by te standard GMRES convergence Teorem 5.) wit estimates of its two parameters using Teorem 4.2. Teorem 4.2 Tere exists an suc tat, if, ten for any u V asu, Su) au, u), asu, u) )) H 2 + log au, u), were H = max k H k ) and H k = diamω k ). Te teorem is again proved using te abstract framework of Sect. 5, i.e. by using Teorem 5.3, and te tree propositions in Sect. 6, Propositions 6., 6.2, and 6.3, verifying te tree assumptions required by te framework. 5 Te abstract framework In tis section, we formulate an abstract framework for te convergence analysis of additive Scwarz metods for a class of finite volume elements, using te metods as preconditioners in te GMRES iteration. Te framework is based on tree key assumptions wic need to be verified every time a convergence analysis is to be performed. Once te assumptions are verified, te framework can be used to derive estimates for te convergence of te metod under consideration. For two very recent applications of te framework, were te Crouzeix Raviart finite volume element as been considered, we refer to [24,25]. 23

11 Additive Scwarz preconditioner for te general FVE 977 We consider a family of finite dimensional subspaces V indexed by te parameter, an inner product a, ) and its induced norm a := a, ), and a family of discrete problems: Find u V a u,v)= f v) v V, were a u,v)is a nonsymmetric bilinear form. We start by stating te convergence result of te GMRES iteration cf. [28]) for solving a system of equations, in te operator form, Pu = b, were te pair P and b can be eiter te pair T and g or te pair S and ĝ, in our case. Te teorem was originally developed using te l 2 norm, cf. e.g. [5], it owever extends to any Hilbert norm, cf. [9,29]. Te convergence rate is based on te two parameters: te smallest eigenvalue of te symmetric part of te operator and te norm of te operator, respectively apu, u) β = inf u =0 u a 2 and Pu a β 2 = sup. 5.) u =0 u a A standard convergence rate of te GMRES iteration, in te a norm, is given in te following teorem. Teorem 5. Eisenstat et al. [5]) If β > 0, ten te GMRES metod for solving te linear system 5.9) converges for any starting value u 0 V wit te following estimate: ) m/2 g Pu m a β2 β2 2 g Pu 0 a, were u m is te m-t iterate of te GMRES metod. Te abstract framework is based on tree assumptions leading to te two main results of te framework, namely Teorems 5.2 and 5.3 respectively for te symmetric and te nonsymmetric case. In te first assumption, we assume tat te nonsymmetric bilinear form a, ) is a small perturbation of te symmetric bilinear form a, ). Assumption ) For all < 0, were 0 is a constant, E u,v):= a u,v) au,v) converges to zero as tends to zero satisfying te following uniform bound. C E > 0 : < 0 and u,v V, E u,v) C E u a v a, 5.2) were C E is a constant independent of. 23

12 978 L. Marcinkowski et al. Te above assumption is important, also because, it leads to te continuity and te coerciveness of te bilinear form a, ), as sown in te following lemmas. Lemma 5. For any M, 2) tere exists 0 suc tat if < ten te bilinear form a u,v)is uniformly bounded in te a -norm, i.e. u,v V a u,v) M u a v a. 5.3) Proof It follows from 5.2) [Assumption 5)] tat a u,v) = au,v)+ E u,v) + C E ) u a v a + C E ) u a v a. Taking = minm )/C E, 0 ) ends te proof. Lemma 5.2 For any α 0, ), tere exists 0 suc tat if < ten te bilinear form a u,v)is uniformly V -elliptic in te a -norm, i.e. Proof By assumption 5.2) [Assumption )], we ave u V a u, u) α u 2 a. 5.4) au, u) a u, u) + E u, u) a u, u) + C E au, u). If < 0 and C E α, ten and te proof follows. a u, u) C E )au, u) αau, u), Te following two assumptions are associated wit te domain decomposition, were te function space V is decomposed into its subspaces as follows, V = Z 0 + Z k, were Z k V for k = 0,,...,N, wit Z 0 being te coarse space. Tese are te same assumptions used in te abstract Scwarz framework for symmetric positive definite systems, cf. [30,32], te first one ensuring a stable splitting of te function space, wile te second one ensuring a strengtened Caucy Scwarz inequality between te subspaces excluding te coarse space. Assumption 2) For any u V, tere are functions u k Z k, k = 0,,...,N, suc tat te following olds. Tere exists a positive constant C 0 wic may depend on te mes parameters) suc tat k= 23 u = u 0 + u k 5.5) k=

13 Additive Scwarz preconditioner for te general FVE 979 and au k, u k ) C0 2 au, u). 5.6) Assumption 3) For any k, l =,...,N,letɛ kl be te minimal nonnegative constants suc tat au k, u l ) ɛ kl u k a u l a u k Z k, u l Z l. 5.7) Let ρe ) be te spectral radius of te N N symmetric matrix E = ɛ kl ) N k,l=. 5. Symmetric preconditioner For k = 0,,...,N, we define te projection operator T k : V Z k as at k u,v)= a u,v) v Z k. 5.8) Note tat te bilinear form au,v) is an inner product in V, ence T k is a well defined linear operator. Let te additive Scwarz preconditioned operator T : V V be given as T = T 0 + N k= T k, and te original problem be replaced by te preconditioned one: Tu = g, 5.9) were g = g 0 + N k= g k wit g k = T k u for k = 0,,...,N. Note tat T k and T are in general nonsymmetric. Te following lemma is needed for te two main teorems of te framework, namely Teorems 5.2 and 5.3. Lemma 5.3 Let u k Z k for k = 0,,...,N. Ten 2 u k 2 + ρe )) u k a 2, a were ρe ) is te spectral radius of te matrix E = ɛ kl ) N k,l=. Proof We see tat 2 N u k 2 u 0 a u k. a k= a 23

14 980 L. Marcinkowski et al. Using 5.7) and a Scwarz inequality in te l 2 -norm we get 2 u k = k= a au k, u l ) k,l= and te proof follows. ɛ kl u k a u l a k,l= ρe ) N u k a 2 N N u l a 2 = ρe ) u k a 2, k= Te first main teorem of te framework giving estimates of te two parameters of te GMRES convergence, cf. Teorem 5., for te symmetric preconditioner, is stated in te following. l= Teorem 5.2 Tere exists 0 suc tat if < ten were β 2 = 2M + ρe )) and β = α 2 C 2 0 β 2 C E. Proof It follows from Lemma 5.3 tat By 5.3) and 5.8), we get atu, Tu) = T k u k= atu, Tu) β2 2 au, u), 5.0) atu, u) β au, u), 5.) N T k u a 2 = at k u, T k u) = 2 a 2 + ρe )) T k u a 2. a u, T k u) = a u, Tu) M u a Tu a. 5.2) Te upper bound 5.0) ten follows wit β 2 = 2M + ρe )). To prove te lower bound, cf. 5.), we start wit te splitting of u V,cf.5.5), suc tat 5.6) olds. Ten using 5.4), 5.8), a Scwarz inequality, and 5.6), we get αau, u) a u, u) = 23 a u, u k ) = at k u, u k ) T k u a u k a N T k u a 2 N u k a 2 C N 0 T k u a 2 u a.

15 Additive Scwarz preconditioner for te general FVE 98 Tis and 5.2), ten yield α 2 au, u) C 2 0 T k u a 2 = C2 0 a u, Tu). Finally, from te assumption 5.2) [Assumption )] and te upper bound 5.0), we get Hence, a u, Tu) = au, Tu) + E u, Tu) au, Tu) + C E u a Tu a au, Tu) + β 2 C E u 2 a. atu, u) α 2 C 2 0 β 2 C E )au, u). Taking β = α 2 C0 2 β 2 C E ) we get te lower bound in 5.). Remark 5.2 In some cases te constant C 0 in 5.6) may depend on C 0 = C 0 )) ten C 0 ) cannot grow too fast wit decreasing, oterwise β may become negative and our teory would not work, e.g. if ρe ) is independent of wic is usually te case in ASM metods, ten it would be sufficient if lim 0 C0 2 ) = 0, because ten tere exists an 0 suc tat β is positive for any <. 5.2 Nonsymmetric preconditioner For k = 0,,...,N, we define te projection operators S k : V Z k as a S k u,v)= a u,v) v Z k. 5.3) Note tat te bilinear form a u,v) is Z k -elliptic, cf. 5.4), so S k is a well defined linear operator. Now, introducing te additive Scwarz operator S : V V as S = S 0 + N k= S k, we replace te original problem wit Su =ĝ, were ĝ =ĝ 0 + N k= ĝ k wit ĝ k = S k u for k = 0,,...,N. Te second main teorem of te framework giving estimates of te two parameters of te GMRES convergence, for te nonsymmetric preconditioner, is stated in te following. Teorem 5.3 Tere exists 0 suc tat for any <, te following bounds old. asu, Su) β2 2 au, u), asu, u) β au, u) 23

16 982 L. Marcinkowski et al. were β 2 = 2M α + ρe )) and β = spectral radius of te matrix E = ɛ kl ) k,l= N. α3 M 2 C 2 0 β 2 C E, and, as before, ρe ) is te Proof We follow te lines of proof of Teorem 5.2. For te upper bound, we use Lemma 5.3 to see tat asu, Su) 2 + ρe )) Using 5.3), 5.4), and 5.3), we get α as k u, S k u) a S k u, S k u) = S k u a 2. a u, S k u) = a u, Su) M u a Su a. 5.4) Te upper bound ten follows wit β 2 = 2M α + ρe )). For te lower bound, again, we use te splitting 5.5) ofu V suc tat 5.6) olds. Next 5.3), 5.4), 5.5), 5.3), a Scwarz inequality, and 5.6) yield tat αau, u) a u, u) = a u, u k ) = a S k u, u k ) M S k u a u k a M N S k u a 2 N u k a 2 MC N 0 S k u a 2 u a. Combining te estimate above wit 5.4), we get α 2 au, u) M 2 C 2 0 S k u a 2 M2 C0 2 a u, Su). α Finally, using similar arguments as in te proof of Teorem 5.2, we can conclude tat au, Su) α 3 M 2 C 2 0 and te lower bound follows wit β = ) β 2 C E au, u), α3 β M 2 C0 2 2 C E. 6 Tecnical tools In tis section, we present te tecnical results necessary for te proof of Teorems 4. and 4.2. We use te abstract framework developed in te previous section, based on 23

17 Additive Scwarz preconditioner for te general FVE 983 wic, we first sow 5.2) [verifying Assumption )], ten sow tat ρe ) is bounded by a constant [verifying Assumption 3)], and finally give an estimate of C0 2 suc tat 5.5) 5.6) old [verifying Assumption 2)], all of tese being formulated below as Propositions 6., 6.2, and 6.3, respectively. We start wit te proposition wic sows tat 5.2) olds true for te two bilinear forms a, ) and a, ) of 2.3) and 3.), respectively. Proposition 6. It olds tat C E > 0 : < 0 and u,v V, a u,v) au,v) C E u a v a, were C E is a constant independent of and te jumps of te coefficients across D j s, but may depend on C Ω in 2.). Te statement of tis proposition is formulated in [7, 3] witout proof. However, as stated in te paper, it can be proved by following te lines of te proof of Lemma 3. in [6]. Te next two lemmas are well known, and are given ere witout proofs. Te first lemma is te extension teorem for discrete armonic functions, cf. e.g. [5, Lemma 5.]. Te second lemma gives an estimate of te H /2 00 Γ kl) norm of a finite element function wic is zero on Ω k \Γ kl by its H /2 seminorm and L norm, cf. e.g. [22, Lemma 4.]. Let Γ kl, be te set of nodal points tat are on te open edge Γ kl common to Ω k and Ω l. Lemma 6. Discrete extension teorem) Let u W k, ten u H Ω k ) u H /2 Ω k ). Lemma 6.2 Let u, u W k be suc tat u x) = ux) for x Γ kl, and u Ωk \Γ kl = 0, ten u 2 H /2 00 Γ kl) u 2 H /2 Γ kl ) + + log Hk )) u 2 L Γ kl ) In te following we present additional set of tecnical lemmas, tey are given ere wit proofs. Te first one is a simple result wic will be used to estimate te H seminorm of functions from te coarse space V 0. Lemma 6.3 For u V 0 and C being an arbitrary constant, te following olds, i.e. u 2 H Ω k ) x V k ux) C 2, were V k is te set of all vertices of Ω k wic are not on Ω. 23

18 984 L. Marcinkowski et al. Proof Note tat u Ωk = x V k ux)φ x Ωk, were φ x is a discrete armonic function wic is equal to one at x, zero at V k \{x}, and linear along te edges Γ kl Ω k. Tus, for any constant C, weave u 2 H Ω k ) = u C 2 H Ω k ) x V k ux) C 2 φ x 2 H Ω k ) x V k ux) C 2. Te last inequality follows from using te fact tat a discrete armonic function as te minimal energy of all functions taking te same values on te boundary, and ten by applying te standard estimate of H -seminorm of te coarse nodal basis function. Definition 6. Let I H : V V 0 be a coarse interpolant defined using function values at te vertices V as follows. For u V, I H u V 0 implies tat I H ux) = ux) x V. Lemma 6.4 For any u V, te following olds, i.e. I H u 2 H Ω k ) + log Hk )) u 2 H Ω k ). Proof From Lemmas 6.3 and te discrete Sobolev like inequality, cf. e.g. Lemma 7 in [30] or Lemma 4.5 in [32], we get I H u 2 H Ω k ) + log Hk )) ) Hk 2 u C 2 L 2 Ω k ) + u C 2 H, Ω k ) for any constant C. A scaling argument and a quotient space argument complete te proof. Lemma 6.5 Let Γ kl Ω k be an edge, and u kl W k and u V be functions suc tat u kl x) = ux) I H ux) for x Γ kl,, and u kl is zero on Ω k \Γ kl. Ten, we ave u kl 2 H /2 + log 00 Γ kl) Hk )) 2 u 2 H Ω k ) 6.) Proof Note tat u kl equals to u I H u on Γ kl and is zero on Ω k \Γ kl, and ence by Lemma 6.2, we get u kl 2 H /2 u I 00 Γ H u 2 kl) H /2 Γ kl ) + + log Hk )) u I H u 2 L Γ kl ). 6.2) Te first term can be estimated using te standard trace teorem, a triangle inequality and Lemma 6.4 as follows, )) u I H u 2 H /2 Γ kl ) u 2 H Ω k ) + I H u 2 H Ω k ) Hk + log u 2 H Ω k ). 6.3) 23

19 Additive Scwarz preconditioner for te general FVE 985 For any constant C, we note tat u I H u = u C I H u C) on Γ kl, and since I H u is a linear function along Γ kl, I H u L Γ kl ) u L Γ kl ). Hence te L norm of u I H u in 6.2) can be estimated as follows, u I H u 2 L Γ kl ) u C 2 L Γ kl ) + I H u C) 2 L Γ kl ) u C 2 L Γ kl ) )) Hk ) + log Hk 2 u C 2 L 2 Ω k ) + u C 2 H, Ω k ) were C is an arbitrary constant. Te last inequality is due to te discrete Sobolev like inequality, cf. Lemma 7 in [30] or Lemma 4.5 in [32]. Finally, a scaling argument and a quotient space argument yield u I H u 2 L Γ kl ) + log Hk )) u 2 H Ω k ). Te above estimate togeter wit te estimates 6.3) and 6.2), complete te proof. A standard coloring argument bounds te spectral radius, and is given ere in our second proposition. Proposition 6.2 Let E be te symmetric matrix of Caucy Scwarz coefficients, cf. 5.7), for te subspaces V k,v l, and V kl,k, l =,...,N, of te decomposition 4.). Ten, ρe ) C, were C is a positive constant independent of te coefficients and mes parameters. Te tird and final proposition gives an estimate of te C0 2 old for any u V. suc tat 5.5) 5.6) Proposition 6.3 For any u V tere exists u k V k k = 0,,...,N and u kl V kl suc tat u = u 0 + N k= u k + Γ kl Γ u kl and au 0, u 0 ) + au k, u k ) + au kl, u kl ) k= Γ kl Γ were H = max N k= H k wit H k = diamω k ). + log )) H 2 au, u), Proof We first set u 0 = I H u V 0, cf. Definition 6.. Next, let u k V k for k =,...,N, be defined as P k u Ωk on Ω k, be extended by zero to te rest of Ω. Now define w = u u 0 k u k. Note tat w is discrete armonic inside eac subdomain Ω k, since u 0 is discrete armonic in te same way, and te sum w + u 0 ) Ωk = u Ωk P k u Ωk = H k u Ωk 23

20 986 L. Marcinkowski et al. is in fact a function of W k. Moreover, wx) = ux) I H ux) = 0 x V. Consequently, w can be decomposed as follows, w = u kl, Γ kl Γ were u kl V kl, wit u Γkl = w Γkl. We now prove te inequality by considering eac term at a time. For te first term, by Lemma 6.4, we see tat au 0, u 0 ) N k u 0 2 H Ω k ) = k I H u 2 H Ω k ) k= + log k= )) H N a k u, u) = k= + log )) H au, u). 6.4) For te second term, since P k is te ortogonal projection in a k u,v), we get au k, u k ) = k= a k P k u Ωk, P k u Ωk ) k= a k u Ωk, u Ωk ) = au, u).6.5) And, for te last term, let Γ kl Γ be te edge wic is common to bot Ω k and Ω l. Note tat u kl V kl as support bot in Ω k Ω l. By Lemma 6., we note tat k= au kl, u kl ) = a s u kl, u kl ) s u kl 2 H Ω s ) s=k,l s=k,l s=k,l s u kl 2 H /2. 00 Γ kl) Utilizing Lemma 6.5 for s = k if k l oterwise we take s = l), and 2.2), we get 23 s=k,l s u kl 2 H /2 00 Γ kl) + log )) H 2 k u 2 H Ω k ) )) H 2 + log a k u, u).

21 Additive Scwarz preconditioner for te general FVE 987 Combining te last two estimates, we get Γ kl Γ au kl, u kl ) Γ kl Γ s=k,l + log + log H )) H 2 a s u, u) )) 2 au, u). 6.6) Te proof ten follows by summing 6.4), 6.5), and 6.6) togeter. 7 Numerical experiments In tis section, we present some numerical experiments sowing te performance of te proposed metods. We consider our model problem to be defined on te domain Ω = [0, ] [0, ] wit te rigt and side f equal to, and apply te finite volume element discretization and te proposed additive Scwarz preconditioners for te solution. Te domain is divided into H H equally sized square subdomains of size H H, allowing equal number of subdomains in eac direction, eac of wic is ten furter divided into H H small equally sized squares of size, again wit equal number of small squares in eac direction. We get te primal mes by slicing eac small square into two triangles in a regular fasion, as sown in te figures. All our numerical results ave been obtained in Matlab, employing te preconditioned GMRES algoritm based on te a -norm minimization, wic is obtained by replacing te standard l 2 inner product wit te a, ) inner product in te original algoritm, see for instance [29]. For te stopping criteria, on te oter and, for simplicity, we only look at te l 2 -norm of te residual. We let te algoritm to run until te l 2 norm of te initial residual is reduced by a factor of 0 6 in eac test, i.e., until r i 2 / r were r 0 and r i respectively are te initial and te i-t residual vector. Te number of iterations to converge, and an estimate of te smallest eigenvalue of te symmetric part of te preconditioned operator P, tat is te smallest eigenvalue of 2 P t + P ), wic is also te most critical parameter of te two describing te GMRES convergence, are presented in te tables below. Our experiments ave sown tat te second parameter, wic is te norm of te operator, is a constant independent of te mes parameters H and, and te coefficient A, wic is in agreement wit our analysis. For te first numerical experiment we test te dependency of te iteration count and te smallest eigenvalue on te mes parameters and H, were te coefficient A is equal to 2 + sinπ x) sinπy). Te results are reported in Table for te symmetric preconditioner. We observe a mild decrease in te value of te smallest eigenvalue, wit a slow increase in te iteration count, as te subdomain problem size H increases, suggesting a poly-logaritmic dependence as predicted in our teory. In te second numerical experiment we perform similar tests, owever, wit a sligtly faster varying A, namely A = 2 + sin0π x) sin0πy). Te results are reported in Tables 2 and 3 respectively for te symmetric and te nonsymmetric variant 23

22 988 L. Marcinkowski et al. Table Iteration numbers and estimates of te smallest eigenvalue in parenteses) for te symmetric preconditioner for different values of and H H e ) e ) e ) e ) e ) e ) e ) 6 2.4e ) e ) e ) e ) 9.72e ) e ) e ) e ) e ) 2.28e ) 20.70e ) e ) e ) e ) Here A = 2 + sinπ x) sinπy) Table 2 Iteration numbers and estimates of te smallest eigenvalue in parenteses) for te symmetric preconditioner for different values of and H H e ) e ) 3 4.3e ) e ) e ) e ) e ) 23.6e ) e ) e ) e 2) 27.7e ) 22.94e ) e ) 5.53e ) e 2) e 2) 26.4e ) e ) e ) 5.57e ) Here A = 2 + sin0π x) sin0πy) Table 3 Iteration numbers and estimates of te smallest eigenvalue in parenteses) for te nonsymmetric preconditioner for different values of and H H e ) e ) e ) e ) e ) e ) e ) 23.62e ) e ) e ) e 2) 27.7e ) 22.95e ) e ) 5.54e ) e 2) e 2) 26.4e ) e ) e ) 5.57e ) Here A = 2 + sin0π x) sin0πy) of te preconditioner. In bot cases, we observe convergence beavior wic are similar to te one in te first experiment. We also note tat te performances of te two variants of te preconditioner are almost identical. 23

23 Additive Scwarz preconditioner for te general FVE 989 Eac diagonal subdiagonal) in te Tables, 2 and 3, corresponds to a fixed subdomain problem size H. As we can see from te entries along eac suc diagonal subdiagonal), tat bot te eigenvalue estimates and te iteration counts remain almost uncanged suggesting tat bot preconditioners are algoritmically scalable, in te sense tat, for a fixed subdomain size H, te number of GMRES iterations is asymptotically) independent of te number of subdomains processors). In te tird numerical experiment, we consider an example were A is discontinuous across subdomains, wic is given by A = α 2+sin0π x) sin0πy)) wit α being a constant in eac subdomain. Te mes parameters are H = /8 and = /64. We assign te parameter α in te coefficient A, te value wite subdomain) or te value ˆα red or saded subdomain) in a ceckerboard fasion as depicted in Fig. 2. Number of iterations and an estimates of te smallest eigenvalues for different values of ˆα varying jumps) are reported in Table 4 sowing tat te convergence is independent of te jumps in te coefficient supporting our analysis. Again, we see an identical performance of te two variants of te algoritm. In our fourt numerical experiment, we examine two different cases, were in te first te coefficients ave jumps inside subdomains as depicted in Fig. 3, and in te second te jumps are along subdomain boundaries as depicted in Fig. 4. Teir results Fig. 2 Ceckerboard distribution of A, wit A = α 2 + sin0π x) sin0πy)), wereα = ˆα in te red saded) subdomains and oterwise color figure online) Table 4 Corresponding to Fig. 2: iteration numbers and estimates of te smallest eigenvalue for different values of ˆα, wit fixed mes sizes = /64 and H = /8 ˆα Symmetric Nonsymmetric e ) 23.62e ) e ) 26.6e ) e ) 27.60e ) e ) 27.60e ) e ) 27.60e ) e ) 27.60e ) e ) 27.60e ) 23

24 990 L. Marcinkowski et al. Fig. 3 Distribution of A wit cannels and inclusions in te interior of subdomains. A = α 2 + sin0π x) sin0πy)), wereα =ˆα in te red saded) regions and elsewere color figure online) Fig. 4 Distribution of A wit jumps along te subdomain boundaries. A = α 2 + sin0π x) sin0πy)), were α =ˆα in te red saded) regions and elsewere color figure online) are presented in Tables 5 and 6, respectively. As we can see from te table entries, bot preconditioners are robust wit respect to inclusions and cannels inside subdomains, but not if te cannels cross subdomain boundaries. Tis is not unexpected, as our coarse space include only functions tat are linear along subdomain boundaries. Terefore any variations tat are along te subdomain boundaries, te coarse problem cannot capture tem tat easy. For problems wit inclusions and cannels bot inside and across subdomains, it may be enoug to replace te coarse space wit te so called multiscale coarse space, cf. [8,9,27], or in extreme cases to enric it wit eigenfunctions corresponding to bad eigen modes of some local eigenvalue problems, cf. [4,2,3]. Tese are subjects for future investigation. Acknowledgments Te autors would like to tank te anonymous reviewers for teir valuable comments and suggestions wic ave been extremely elpful in improving te paper. 23

25 Additive Scwarz preconditioner for te general FVE 99 Table 5 Corresponding to Fig. 3: iteration numbers and estimates of te smallest eigenvalue for different values of ˆα, wit fixed mes sizes = /64 and H = /8 ˆα Symmetric Nonsymmetric e ) 23.62e ) e ) 27.05e ) e 2) e 2) e 2) e 2) e 2) e 2) e 2) e 2) e 2) e 2) Table 6 Corresponding to Fig. 4: iteration numbers and estimates of te smallest eigenvalue for different values of ˆα, wit fixed mes sizes = /64 and H = /8 ˆα Symmetric Nonsymmetric e ) 23.62e ) e ) 28.2e ) e 2) e 2) e 3) e 3) e 4) e 4) e 5) e 5) e 6) e 6) Open Access Tis article is distributed under te terms of te Creative Commons Attribution 4.0 International License ttp://creativecommons.org/licenses/by/4.0/), wic permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to te original autors) and te source, provide a link to te Creative Commons license, and indicate if canges were made. References. Bi, C., Cen, W.: Mortar finite volume element metod wit Crouzeix Raviart element for parabolic problems. Appl. Numer. Mat. 58, ) 2. Bi, C., Ginting, V.: A residual-type a posteriori error estimate of finite volume element metod for a quasi-linear elliptic problem. Numer. Mat. 4, ) 3. Bi, C., Ginting, V.: Finite-volume-element metod for second-order quasilinear elliptic problems. IMA J. Numer. Anal. 33), ) 4. Bi, C., Rui, H.: Uniform convergence of finite volume element metod wit Crouzeix Raviart element for non-selfadjoint and indefinite elliptic problems. J. Comput. Appl. Mat. 200, ) 5. Bjørstad, P.E., Widlund, O.B.: Iterative metods for te solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 236), ) 6. Braess, D.: Finite Elements, 3d edn. Cambridge University Press, Cambridge 2007). doi:0.07/ CBO Teory, fast solvers, and applications in elasticity teory. Translated from te German by Larry L. Scumaker 7. Brenner, S.C.: Te condition number of te Scur complement in domain decomposition. Numer. Mat. 832), ) 8. Brenner, S.C., Scott, L.R.: Te Matematical Teory of Finite Element Metods. Texts in Applied Matematics, vol. 5, 3rd edn. Springer, New York 2008). doi:0.007/

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