Bath Institute For Complex Systems

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1 BICS Bath Insttute for Complex Systems Analyss of FETI methods for multscale PDEs Part II: Interface varaton C. Pechsten and R. Schechl Bath Insttute For Complex Systems Preprnt 7/09 (2009)

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3 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION CLEMENS PECHSTEIN AND ROBERT SCHEICHL 2 Abstract. In ths artcle we gve a new rgorous condton number estmate of the fnte element tearng and nterconnectng (FETI) method and a varant thereof, all-floatng FETI. We consder the scalar ellptc equaton n a two- or three-dmensonal doman wth a hghly heterogeneous (multscale) dffuson coeffcent. Ths coeffcent s allowed to have large jumps not only across but also along subdoman nterfaces and n the nteror of the subdomans. In other words, the subdoman parttonng does not need to resolve any jumps n the coeffcent. Under sutable assumptons, we can show that the condton numbers of the one-level and the all-floatng FETI system are robust wth respect to strong varatons n the contrast n the coeffcent. We get only a dependence on some geometrc parameters assocated wth the coeffcent varaton. In partcular, we can show robustness for so-called face, edge, and vertex slands n hgh-contrast meda. As a central tool we prove and use new weghted Poncaré and dscrete Sobolev type nequaltes that are explct n the weght. Our theoretcal fndngs are confrmed n a seres of numercal experments. Keywords FETI doman decomposton fnte element method multscale problems precondtonng varyng and hgh contrast coeffcents weghted Poncaré nequaltes Mathematcs Subject Classfcaton (2000). Introducton 65F0, 65N22, 65N30, 65N55 In recent years, the detaled and fast smulaton of bologcal, physcal or engneerng processes has become an almost standard demand. Often such problems are posed on complex geometres and nvolve hghly heterogeneous (often non-lnear) materal parameters. As a consequence, the development of effcent and robust parallel solvers for heterogeneous meda has been a very actve area of research, specfcally n the settng of multscale solvers, and n the doman decomposton and multgrd communtes [2, 3, 7, 8, 3, 4, 5, 6, 7, 29, 30, 3, 32, 33, 34, 35, 40, 4, 42]. In ths paper, we are concerned wth the convergence of a varant of the fnte element tearng and nterconnectng (FETI) doman decomposton method n the context of heterogeneous (multscale) problems, for the partcular case that we have large jumps n the coeffcent not only across, but also along the subdoman nterfaces. As such ths paper s a contnuaton of the work n [29, 28], where we have shown the robustness of one-level FETI methods wth respect to coeffcent varaton n the subdoman nterors, and of [30], where we have extended these results also to dual prmal FETI methods. See also [2, 26, 27]. Date: Aprl 22, Insttute of Computatonal Mathematcs, Johannes Kepler Unversty, Altenberger Str. 69, 4040 Lnz, Austra, phone: (+43) 732 / , E-mal: clemens.pechsten@numa.un-lnz.ac.at 2 Department of Mathematcal Scences, Unversty of Bath, Bath BA2 7AY, Unted Kngdom, phone: (+44) 225 / , E-mal: r.schechl@maths.bath.ac.uk The frst author has been supported by the Austran Scence Fund (FWF) under grant P9255-N8.

4 2 C. PECHSTEIN AND R. SCHEICHL In the bgger context, the work follows some earler work on the robustness of two-level Schwarz-type doman decomposton methods for heterogeneous meda by Graham et al. [4], Graham and Schechl [5, 6], as well as Schechl and Vankko [35]. FETI methods are robust doman decomposton methods for solvng fnte element dscretsatons of partal dfferental equatons (PDEs) wth excellent parallel scalablty propertes. They belong to the class of dual teratve substructurng methods and were ntroduced by Farhat and Roux []. For an extensve lterature revew on the analyss of FETI methods see our frst paper [29]. Assumng that the coeffcents of the PDE are constant n each subdoman, Klawonn and Wdlund [9] proved (based on poneerng work by Mandel and Tezaur [23]) that the spectral condton number of the precondtoned FETI system s bounded by (.) C ( + log(h/h)) 2, where the constant C n (.) s ndependent of possble jumps n the coeffcents across subdoman nterfaces when a specal scalng of the precondtoner s appled. Here, as usual, H and h denote the subdoman dameter and the mesh wdth, respectvely, and C s ndependent of H and h, as well. Ths bound (.) was also shown to hold true for FETI-DP methods and for the related balancng Neumann-Neumann and BDDC methods [9, 20, 22]. An excellent account of all these results can be found n the recent monograph [39] by Tosell and Wdlund. We also refer to [5] where t s shown that ths bound s sharp. Fnally, we menton also that untl recently, certan regularty assumptons on the subdomans had to be made. In the artcle [8] (see also [9]) the authors were able to weaken these assumptons sgnfcantly and to treat also qute rregular subdomans n two dmensons, as they appear when decomposng unstructured meshes wth graph parttoners. However, all the above mentoned analyses assume that the coeffcents of the PDE are pecewse constant wth respect to the subdoman parttonng. The man focus of the present work s the analyss of FETI methods for hghly heterogeneous multscale problems,. e., n the case of coeffcent jumps that are not algned wth the subdoman nterfaces and/or vary strongly wthn a subdoman, partcularly for the case that jumps occur along the nterface. It has already been observed numercally by several authors (see e. g. [7, 2, 3, 32]) that a smple generalsaton of the scalng employed by Klawonn and Wdlund n [9] leads to robustness of the FETI method even n ths case. In the followng, we restrct ourselves to the model ellptc problem (.2) (α u) = f, n a bounded polygonal or polyhedral doman Ω R d, d = 2 or 3, subject to sutable boundary condtons on the boundary Ω. The coeffcent α(x) may vary over many orders of magntude n an unstructured way on Ω. It s not surprsng that (.) also holds n ths case but n general wth C = C(α),. e., wth some possble loss of α-robustness. In our recent artcle [29] we have shown that the dependence on α s restrcted to the varaton of α(x) n the vcnty of subdoman nterfaces (wthn each subdoman). More precsely, f Ω,η denotes the boundary layer of wdth η of any of the subdomans Ω, and f α(x) α(y) for all x, y Ω,η, then C(α) (H/η) 2, ndependent of the varaton of α(x) n the remander Ω \Ω,η of each subdoman and ndependent of any jumps of α(x) across subdoman nterfaces. The hdden constant depends on the local varaton of α(x) n Ω,η. Of course ths constant blows up when a coeffcent jump appears along a subdoman nterface, but numercal experments ([2, 29, 32]) ndcate that the condton number of the precondtoned FETI system does not blow up, at least when α jumps only a few tmes along each subdoman nterface.

5 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 3 In the present work, we extend these results and gve a rgorous analyss showng that one-level FETI and a varant of one-level FETI, the all-floatng (or total) FETI method [0, 24, 25, 27], are α-robust n the followng sense: () Let us assume that each subdoman boundary layer Ω,η can be subdvded nto two connected subregons Ω (),η, Ω(2),η such that α(x) α(y) for all x, y Ω(k),η, separately for each k =, 2. Then the constant n (.) s bounded by C(α) (H/η) β, where the exponent β equals d or d, dependng on the sze of the nterface between Ω (),η and Ω(2),η. The bound s agan completely ndependent of the values of α(x) n the subdoman nterors, but also of the contrast of the value of α(x) n Ω (),η and n Ω (2),η. () Let us assume that each subdoman boundary layer Ω,η can be subdvded nto M connected subregons Ω (),η,..., Ω(M),η such that α(x) α(y) for all x, y Ω (k),η, agan separately for each k =,..., M. In addton, we assume that we can extend each of the subregons Ω (k),η to the nteror of Ω such that all the extended subregons Ω (k) have one common nterface X, that may be a vertex, a lne, or a surface, and such that for each pont x Ω (k) we have α(x) mn (k) y Ω α(y),. e., α(x) n the nteror,η s essentally larger than α(x) n the boundary layer n each of the extensons. If the nterface X between the extended subregons n each subdoman Ω s at least an edge (resp. vertex) n three (resp. two) dmensons, then C(α) (H/η) 2 ( + log(h/h)). Our central theoretcal tools to prove ths robustness are new weghted Poncaré and dscrete Sobolev type nequaltes that are explct n the weght α and n the geometrcal parameters H and η. To the best of our knowledge these nequaltes have not appeared n the lterature before. The dfference to many exstng nequaltes n weghted norms s that our constants are explct n the weght, and often even ndependent thereof. Note however that other weghted Poncaré type nequaltes have been proved by Xu and Zhu [42] (based on [4]) and n a recent artcle by Galvs and Efendev [2]. The remander of ths artcle s structured as follows. Secton 2 starts wth some prelmnares. Secton 3 s devoted to weghted Poncaré and dscrete Sobolev type nequaltes. In Secton 4 we descrbe our generalsaton of the one-level and the all-floatng FETI method to multscale PDEs and gve the statements of our key results. The proofs of these results are then gven n Secton 5 where we ntroduce some addtonal techncal tools needed n our analyss. We fnsh wth some numercal experments that confrm our theoretcal results n Secton Prelmnares Let Ω R d (wth d = 2 or 3) be a connected, open, and bounded doman wth Lpschtz boundary Ω. We consder the followng model problem: Fnd u H (Ω), u Ω = g D such that (2.) α(x) u(x) v(x) dx = f(x) v(x) dx v H0 (Ω), Ω Ω

6 4 C. PECHSTEIN AND R. SCHEICHL for gven functons f L 2 (Ω) and g D H /2 (Ω). For smplcty, we choose Drchlet boundary condtons on the whole of Ω. However, all of our work can easly be generalsed to Neumann boundary condtons on a part of Ω. The doman Ω decomposes nto nonoverlappng subdomans {Ω } =,...,N such that Ω = N (2.2) Ω. = The subdoman boundares Ω are assumed to be Lpschtz. We defne the skeleton Γ S, the nterface Γ, and the subdoman nterfaces Γ j by (2.3) Γ S := Ω, Γ := Γ S \ Ω, Γ j := ( Ω Ω j ) \ Ω. The subdoman dameters are denoted by H := dam Ω. As n our prevous work [29] we need some regularty assumptons on the subdoman partton. Defnton 2. (regular doman). For d = 2 (or 3), let D R d be a bounded contractble doman wth a smply-connected Lpschtz boundary. D s called a regular doman, f t can be decomposed nto a conformng coarse mesh of shape-regular trangles (tetrahedra). Whenever consderng a famly of regular domans, such as parttons nto subdomans, we mplctly assume that the number of smplces formng an ndvdual subdoman s unformly bounded. Defnton 2.2 (shape parameter). For a smplex T, we defne the shape parameter ρ(t ) to be the radus of the largest nscrbed ball. The shape parameter of a regular doman D s defned as ρ(d) := mn ρ(t ), where {T } s are the smplces accordng to Defnton 2.. s Defnton 2.3 (shape-regular partton). Let D be an open doman n R 2 or R 3. A partton of D nto regular subdomans {D } =,...,N, such that D = N = D, s called shape-regular, f ρ(d ) dam D, and D D j = dam D dam D j, j =,..., N. Assumpton A. The subdomans {Ω } form a non-overlappng shape-regular partton of Ω, and the underlyng coarse mesh (cf. Defnton 2.) s conformng. We ntroduce the followng topologcal sets smlar to [39, Defnton 4.]. Defnton 2.4. The skeleton Γ S s the dsjont unon of subdoman faces, regarded as open and connected sets (not necessarly planar), whch are shared by two subdomans or by one subdoman and the outer boundary Ω, subdoman edges, also regarded as open and connected (but not necessarly straght), shared by at least two subdomans, such that the closure of all edges forms the boundares of the faces, subdoman vertces, where at least two subdoman edges meet. We denote by F the set of subdoman faces, E the set of subdoman edges, V the set of subdoman vertces on Ω. These are further subdvded nto faces, edges, and vertces on the nterface Ω Γ, denoted by F Γ, EΓ, and VΓ, respectvely, and nto faces, edges, and vertces on the outer (Drchlet) boundary Ω Ω, denoted by F D, E D, and V D, respectvely.

7 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 5 η Ω remander Ω, η (boundary layer) Fgure. Boundary layer Ω,η of the subdoman Ω, cf. Defnton 2.5. We consder smplcal trangulatons T on Ω whch are quas-unform and shape-regular. The local mesh parameter s denoted by h. We requre that the trangulatons match on the subdoman nterfaces. Note that these assumptons (together wth Assumpton A) mply that H H j and h h j f Γ,j. The set of nodes of the mesh on the local boundary Ω s denoted by Ω h, and smlarly we defne Γh and Γ h j to be the sets of nodes on the nterface Γ and on the subdoman nterface Γ j, respectvely. A typcal node wll be denoted by x h. For the dscretsaton of (2.) we use contnuous pecewse lnear fnte elements. We denote by V h (Ω), V h (Ω ) and V h ( Ω ) the spaces of contnuous pecewse lnear functons (wth respect to the mesh) on the doman Ω, on a subdoman Ω and on the local boundary Ω, respectvely. Note that these spaces do not ncorporate the essental boundary condtons on Ω. Wthout loss of generalty we assume that the gven Drchlet trace g D s n V h ( Ω). Also, wthout loss of generalty we assume that the coeffcent α s pecewse constant on the elements of the trangulaton. Our analyss wll requre some noton of a boundary layer near subdoman nterfaces. Therefore we need the followng defnton whch s closely related to the one n [4]. Defnton 2.5 (dscrete boundary layer). Let {,..., N} and η > 0. We defne the dscrete boundary layer of Ω to be the open set Ω,η such that Ω,η := { τ : τ T, dst(τ, Ω ) η },. e., the set of all ponts whch have at most dstance η from the boundary Ω extended to a unon of elements. An llustraton of ths defnton s gven n Fg.. Defnton 2.6 (η -regular). The dscrete boundary layer Ω,η s called η -regular f there exsts a shape-regular partton Ξ,η := { } ω,,..., ω,s of Ω,η nto non-overlappng, regular (n the sense of Defnton 2.) patches ω,j wth dam ω,j η, such that () the ntersecton of ω,j wth Ω s non-empty and equal to the unon of a set of faces of the smplces formng the patch ω,j, and () the ntersecton of ω,j wth an edge E E s the unon of edges of the smplces formng the patch ω,j. Assumpton A2. For each {,..., N} the parameter η > 0 s chosen such that () Ω,η s η regular, () η η j, f Γ j, and () the meshes nduced by the patches match on the subdoman nterfaces.

8 6 C. PECHSTEIN AND R. SCHEICHL For the sake of smplcty, we make no dfference between functons on dscrete spaces and ther vector representatons wth respect to the standard nodal bass, as well as between operators and ther matrx representatons wth respect to the same bass. Smlarly, we dentfy any dscrete space X wth ts dual space X. On the subdoman Ω, we can assemble the local fnte element stffness matrx K and group t wth respect to the unknowns on the subdoman boundary (subscrpt B) and the nteror (subscrpt I), ( ) K,BB K (2.4) K =,BI. K,IB K,II Snce none of the spaces V h (Ω ) ncorporates essental boundary condtons, each of the local operators K s only postve sem-defnte wth ker K = span{ Ω }, where Ω denotes the constant functon on Ω. We defne the Schur complement S of K,II n K by (2.5) S = K,BB K,BI [K,II ] K,IB. Note, that the applcaton of S means actually solvng a Drchlet boundary value problem on the subdoman Ω. Snce S s symmetrc postve semdefnte, t defnes a semnorm, (2.6) v S := S v, v /2 for v V h ( Ω ), that obeys the mnmsng property { } (2.7) v 2 S = mn α(x) ṽ(x) 2 dx : ṽ V h (Ω ), ṽ Ω = v. Ω We denote by H,α v the functon ṽ V h (Ω ) for whch the mnmum s attaned, and we call ths functon the dscrete α-harmonc extenson of v from V h ( Ω ) to V h (Ω ). The Galerkn projecton of (2.) onto the space V h (Ω) (whch does not nclude the essental boundary condtons) leads to the followng constraned lnear system. Fnd ũ V h (Ω), u Ω = g D such that (2.) (substtutng ũ for u and ṽ for v) holds for all test functons ṽ V h (Ω), ṽ Ω = 0, n short (2.8) K ũ = f. The global stffness matrx K and the load vector f can be assembled from (parts of) the local contrbutons K and f, respectvely. Non-homogeneous Drchlet boundary condtons can be treated wth standard homogensaton technques. FETI methods are specal doman decomposton methods to solve system (2.8) n parallel. The common dea of these methods s to decouple the system subdoman-wse and to enforce the contnuty of ũ across the subdoman nterfaces by Lagrange multplers λ. There are varous strateges to elmnate the prmal varables and to desgn parallel precondtoners for the dual system n λ; these are the one-level, dual-prmal, and all-floatng or total FETI methods, see [0, 39], as well as [24, 25] for further results developed n the closely related area of boundary elements (BETI). To smplfy the presentaton and the proofs we wll follow manly the all-floatng approach where the Drchlet boundary condtons are also ncorporated va Lagrange multplers. However, the results carry over also to the more classcal one-level FETI approach, albet wth one addtonal assumpton. We wll come back to ths below. See [27, 30] for detals on how the proof technques can also be extended to dual-prmal methods.

9 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 7 3. Weghted Poncaré and dscrete Sobolev nequaltes The crucal new theoretcal tools needed for our analyss below are weghted Poncaré and dscrete Sobolev nequaltes. As we have seen n [29], the robustness of FETI methods s not affected by varatons of the coeffcent n the nteror of each subdoman Ω. Ths s the reason why n [29] we generalsed the Poncaré and dscrete Sobolev nequaltes to the dscrete boundary layer Ω,η. Here we go one step further and prove certan weghted versons of Poncaré and dscrete Sobolev nequaltes on the dscrete boundary layer Ω,η n order to allow also for some coeffcent varaton along the nterfaces between subdomans. 3.. Weghted Poncaré nequalty two coeffcent regons. Let us consder a sngle subdoman Ω and let us fx η > 0 accordng to Assumpton 2. To state our weghted Poncaré nequalty we further decompose the dscrete boundary layer Ω,η nto two nonoverlappng connected subregons Ω (),η and Ω (2),η such that (3.) Ω,η = Ω (),η Ω (2),η, see also Fgure 2. Let Γ (2),η Ω (),η Ω (2),η. be the larger of the connected components of the nterface Defnton 3.. If Ω,η s η regular, we say that the parttonng (3.) s compatble, f each of the subregons Ω (k),η can be parttoned nto a unon of the patches ω,j n Defnton 2.6, such that the nterface Γ (2),η s the unon of faces of the patches. On each of the subregons Ω (k),η (3.2) such that α (k),η α(x) α (k),η we set α (k),η := mn α(x), x Ω (k),η for all x Ω (k),η. α (k),η := max α(x), x Ω (k),η Lemma 3.2 (weghted Poncaré nequalty). Let η > 0 and let Ω,η be η regular. Suppose that the parttonng (3.) s compatble and u H (Ω,η ) wth Γ (2) u ds = 0. Then,,η α(x) u(x) 2 dx C,η α(x) u(x) 2 dx Ω,η Ω,η wth η 2 C,η := ( H η ) d α (k),η max k=,2 α (k),,η n general. If d = 3 and the area of the nterface fulfls Γ (2),η H η the nequalty holds wth C,η = ( H η ) 2 α (k),η max k=,2 α (k).,η Proof. Let k {, 2}. Snce Ω,η s η regular and the parttonng (3.) s compatble, we can partton Ω (k),η nto a unon of patches from the set Ξ,η (cf. Defnton 2.6) denoted by {ω (k),j }, and

10 8 C. PECHSTEIN AND R. SCHEICHL Ω (), η η Ω (), η H H Ω (2), η Ω (2), η Fgure 2. Dfferent settngs of the subregons Ω (k),η (Left: Γ (2),η H η ; Rght: Γ (2),η η 2). n three dmensons η 2 u 2 L 2 (Ω (k),η ) = j η 2 u 2 L 2 (ω (k),j ). Let Λ (k) {ω (k),j (3.3) := Ω (k),η Ω. Then, by Defnton 2.6, Λ (k) s the unon of faces γ (k),j of the patches }. Hence, by a standard Fredrchs type nequalty on each patch ω(k), we have η 2 u 2 L 2 (Ω (k),η ) j { u 2 + } u 2 H (ω (k),j ) η L 2 (γ (k),j ),j = u 2 H (Ω (k),η ) + η u 2 L 2 (Λ (k) ). Now, on each of the parts Λ (k) we can apply a generalsed Poncaré nequalty of the type proved n [29], namely ( u 2 H ) β u 2 (3.4) u H (Ω (k) η L 2 (Λ (k) ) η H (Ω (k),η ),η ), u ds = 0, Γ (2),η wth β = d n general, and β = 2 f d = 3 and Γ (2),η H η. A proof of (3.4) s gven n the Appendx. Substtutng (3.4) nto (3.3) and usng (3.2) we fnally get η 2 Ω,η α(x) u(x) 2 dx k=,2 α (k),η ( H η ) β u 2 H (Ω (k),η ) ( H ) β α (k),η max η k=,2 α (k) α(x) u(x) 2 dx.,η Ω,η Remark 3.3. () Frst of all note that the Poncaré constant n Lemma 3.2 only depends on the local varaton of α on each subregon Ω (k),η of Ω,η, and s completely ndependent of the values and the varaton of α n the nteror of Ω, or of the contrast between the two regons Ω (),η and Ω (2),η. In partcular, f α(x) s constant on each of the regons Ω (k),η then C,η = (H /η ) β (wth β dependng on the sze of the nterface Γ (2),η ) and the Poncaré constant s completely ndependent of the range of α. () Note that f η > H /2 then Ω,η = Ω and so Lemma 3.2 also holds on all of Ω. In ths case the nequalty reduces to α(x) u(x) 2 dx C,H α(x) u(x) 2 dx. Ω Ω H 2

11 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 9 () Also f one of the subregons, say Ω (2),η, s empty, we can set Γ (2),η := Ω, and the nequalty reduces to ( α(x) u(x) 2 H ) () 2 α,η dx α(x) u(x) 2 dx. Ω,η η Ω,η η 2 α (),η (v) Note that our new proof dea (see Secton 5) uses only arguments on patches nstead of subdomans. Therefore we can even generalse Lemma 3.2 to non-regular subdomans whose boundary layers are η -regular and where each subdoman face (resp. edge) stll touches O((H /η ) d ) patches. Then, n order to prove (3.4) (see the Appendx) we need the followng assumptons for each k =, 2. (a) There exsts a shape-regular partton of Ω (k),η nto regular patches of dameter O(η ), (b) each par of such patches must be connectble va a path contanng at most O(H /η ) patches, and (c) n three dmensons, the overlappng argument as stated n the proof of [29, Lemma 4.3] must be applcable. In fact these assumptons can only be volated f the regons Ω (k),η are rather badly shaped. Obvously, the above concept can be extended n a straghtforward way to M > 2 subregons Ω (k),η by ntroducng M functonals.however, snce we wll only have one degree of freedom per subdoman n our coarse space (see Lemma 5.2 below), ths wll be of no use here. Instead we prove a weghted dscrete Sobolev type nequalty n the next subsecton. Weghted Poncaré nequaltes wth more than one functonal (based on [4]) were used by Xu and Zhu [42] recently to prove coeffcent robustness of geometrc multgrd n the case of pecewse constant coeffcents that are resolved by the coarse meshes. A very smlar weghted Poncare nequalty to the one we gave here (wth one functonal) was also recently proved by Galvs and Efendev [2] and used n the analyss of two-level overlappng Schwarz Weghted dscrete Sobolev nequalty multple coeffcent regons. Let us agan consder a sngle subdoman Ω, but now let {Ω (k) } k=,...,m be a non-overlappng shape-regular parttonng of (all of) Ω nto M connected subregons,. e. (3.5) Ω = Ω ()... Ω (M ). such that the ntersecton of all subregons (3.6) X := M k= s a vertex, an edge, or a face of all the subregons Ω (k). Note that ths restrcts us to M = O() and that the dameter of each of the subregons Ω (k) s O(H ), see Fgure 3, left. Now, let η > 0 and let Ω (k),η n Ω (k). For each Ω (k),η, let α (k),η := Ω (k) and α (k) take larger values n the nteror of Ω.,η Ω (k) Ω,η,. e. the part of the dscrete boundary layer be as defned n (3.2). Note, however, that α may Lemma 3.4 (weghted dscrete Sobolev nequalty). Let η > 0 and let Ω,η be η regular. Suppose that the parttonng (3.5) s compatble and that (3.7) α(x) α (k),η x Ω (k) k =,..., M.

12 0 C. PECHSTEIN AND R. SCHEICHL H Ω (3) Ω () Ω (2) X * Ω (4) η Ω (5) Ω (2) Ω () Fgure 3. Left multple coeffcent regons, admssble for Lemma 3.4, n grey: Ω (k),η, rght beam type example, see Remark 3.5() Then, for all u V h (Ω ) wth X u ds = 0, f X X s a vertex, we have wth η 2 Ω,η α(x) u(x) 2 dx C,η C,η := σ(h, h ) H η s a face or an edge, and u(x ) = 0, f Ω α(x) u(x) 2 dx M α (k),η max k= α (k),η and f X s a face n three dmensons or an edge n two dmensons, σ(h, h ) := ( + log(h /h )) f X s an edge n three dmensons or a vertex n two dmensons, H /h f X s a vertex n three dmensons. Proof. The proof s dentcal to that of Lemma 3.2. However, nstead of (3.4) we use the followng dscrete Sobolev nequalty on each of the subregons Ω (k), (3.8) H u 2 L 2 (Λ (k) ) σ(h, h ) u 2 H (Ω (k) ). Ths nequalty follows from [39, Lemma 4.5, Lemma 4.2, and Sect. 4.6.]. Remark 3.5. () Note that as long as the coeffcent α fulfls condton (3.7) the constant n Lemma 3.2 only depends on the local varaton of α on each subregon Ω (k) but not on the contrast between the regons Ω (k). In partcular f α s constant on each of the regons Ω (k) then the Poncaré constant s completely ndependent of α. () Next we would lke to ntroduce the concept of an artfcal coeffcent n order to establsh a better understandng of Lemma 3.4. Suppose that we can fnd an artfcal coeffcent α art (x) and M regons Ω (k), wth the common nterface X, stays n the same range as α art restrcted to Ω (k),η. such that α art restrcted to Ω (k) Then, Lemma 3.4 holds for α = α art. However, t stll holds, f α s made larger n the subdoman nteror,. e. f α(x) α art (x) for all x Ω \ Ω,η. () The beam type example n Fgure 3, rght, cannot be treated wth the weghted Poncaré nequalty n Lemma 3.2 because the beam s not connected n the boundary layer. However, t can be treated wth Lemma 3.4 snce the ntersecton of the two subregons contans a face. If the thckness of the beam s O(η ) one can even derve

13 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION analogous estmates beng explct n the aspect rato H /η usng the framework gven n the Appendx. The followng stronger result follows mmedately from the proof of Lemma 3.4. Corollary 3.6. The statement of Lemma 3.4 stll holds f the subregons Ω (k) overlap and condton (3.7) holds. Also, ther unon does not have to be the whole of Ω as long as the entre boundary layer Ω,η remans covered and nequalty (3.8) holds. Remark 3.7. () The overlappng case can be seen as follows. For each subregon Ω (k) we can fnd an artfcal coeffcent α (k) art whch stays n the same range as when restrcted to Ω (k),η. The real coeffcent α must now be larger than each α (k) art,. e., for a pont x common to more than one subregon, α(x) must be larger than the maxmum of the artfcal coeffcents at x. Then by carefully addng up the separate nequaltes (3.8) on these subregons, one easly concludes the weghted dscrete Sobolev nequalty of Lemma 3.4 by lftng the artfcal coeffcents to α. Note that we can even choose artfcal subregons Ω (k),art that overlap n the boundary layer. Ths way we can treat the plate type example, shown n Fgure 4, left, where a regon Ω () wth a larger coeffcent cuts through a regon wth a smaller coeffcent. To see ths we set Ω (),art = Ω() and Ω (2),art = Ω. Then the nequalty follows agan by a lftng argument. () In contrast to the beam example n Remark 3.5() t makes a bg dfference whether the coeffcent n the plate s larger or smaller than n the surroundng regon. In the latter case our theory does not apply and we cannot guarantee that the Poncaré constant n Lemma 3.4 s ndependent of the contrast. () To dscuss the second asserton n the Corollary 3.6, suppose now that the unon forms just a part of Ω and let Ω (R) denote the remander that s located n the nteror of Ω, see Fgure 4, rght. Then the coeffcent n Ω (R) can be chosen arbtrarly, n partcular arbtrarly small, and the Poncaré constant of the subregons Ω (k) n Lemma 3.4 s ndependent of the values of α n Ω (R). The dfference compared to () n ths remark s that n the current settng the nclusons do not separate regons of larger coeffcents from each other. We pont out that the shapes of Ω (k) may of course nfluence the Poncaré constant as well. If we generalse to subregons that can be parttoned nto regular patches of dameter O(η ), we can work out smlar results usng the framework gven n the Appendx, but possbly wth a dfferent dependency on H /η. We would also lke to pont to the recent artcles [9, 8] where nequaltes related to (3.8) were proved for qute rregular subregons n two dmensons wth only weak dependence on ther shapes. Ω (k) 4. FETI methods for multscale ellptc PDEs Let us now descrbe the classcal one-level and the all-floatng FETI method. Followng [39, Sect. 6.3], we ntroduce separate unknowns u V h (Ω ) on the subdomans and denote by u = [u,..., u N ] the dscontnuous approxmaton of ũ n N = V h (Ω ). The contnuty of the soluton s enforced by constrants of the form (4.) u (x h ) u j (x h ) = 0 for x h Γ h j.

14 2 C. PECHSTEIN AND R. SCHEICHL Ω () η H Ω () η H Ω () * X Ω (R) Ω (3) Ω (R) Ω (4) η Ω (2) Ω (5) Fgure 4. Coeffcent dstrbutons related to Corollary 3.6. Left plate type example, see Remark 3.5(), rght nclusons Ω (R), Note that at nodes where more than two subdomans meet ths ntroduces redundances. In ths work we consder only fully redundant constrants,. e., the full set of possble constrants s used. In the all-floatng formulaton (cf. [0, 25]), we ncorporate the Drchlet boundary condtons by addtonal constrants of the form (4.2) u (x h ) g D (x h ) = 0 for x h Ω h Ω. Note that unlke the contnuty constrants (4.), the Drchlet constrants (4.2) act completely locally on each subdoman. Let N C denote the total number of constrants n (4.) and (4.2) and set U := R N C. Then we can wrte (4.) and (4.2) compactly as (4.3) N B u = b U, or equvalently, B u = b. = The operators B : V h (Ω ) U can be represented by sgned Boolean matrces. The full jump operator B : N = V h (Ω ) U s defned as B := [B,..., B N ]. The entres of the vector b U that correspond to the constrants n (4.2) contan the values g D (x h ). All other entres of b are zero. By a smple energy mnmsaton argument, t follows from the above that solvng (2.8) s equvalent to fndng u = [u,..., u N ] and λ U satsfyng the saddle pont system K 0 B u f... (4.4).. 0 K N BN u N =. f N. B B N 0 λ b System (4.4) s unquely solvable up to addng elements from ker B to λ f and only f the block K := dag (K ) s SPD on ker B, or equvalently, ker K ker B = {0}. Ths condton s true whenever the Drchlet boundary s non-empty. We refer to U as the space of Lagrange multplers. The more classcal one-level FETI formulaton leads to a very smlar saddle pont system. See [29, 39] for detals. Recall that each of the operators K s only postve sem-defnte and ker K = span{ Ω }. We ntroduce operators R : R V h (Ω ) : ξ ξ Ω such that range R = ker K. Let K denote some pseudonverse of K. Then, under the compatblty condton f B λ range K,

15 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 3 we can elmnate the unknowns u from (4.4),. e. (4.5) u = K [f B λ] + R ξ, for some ξ = [ξ ] N =. Fnally, usng that range K = ker R and wth the abbrevatons K := dag (K ), f := [f,..., f N ], R := dag (R ), K := dag (K ), F := B K B, G := B R, d := B K f b, e := R f, we arrve at the dual formulaton (see [39] for detals): Fnd (λ, ξ) U R N such that ( ) ( ) ( ) F G λ d (4.6) G =. 0 ξ e In practce, ths saddle pont system s solved usng a projecton (4.7) P : U ker G U such that P := I Q G (G Q G) G, where a careful choce of the SPD scalng operator Q : U U wll be crucal to render the method robust to coeffcent varaton (see (4.4) (4.5) below). Introducng the subspace (4.8) V := {λ U : B z, λ = 0 z ker K} = ker G = range P and usng the projecton operator the saddle pont system (4.6) can be reduced to solvng (4.9) P F λ = P (d F λ 0 ), for λ V, where λ 0 = Q G(G Q G) e. The orgnal varables λ and ξ can then be recovered from the relatons λ = λ 0 + λ and ξ = (G Q G) G Q (F λ d). Several thngs are worth mentonng. Frst note that G Q G s the Galerkn projecton of B Q B onto ker K. Snce ker K ker B = {0}, the operator G Q G s nvertble as long as Q s SPD on range G. Furthermore, snce equaton (4.9) s SPD on the subspace V modulo ker B, t can be solved usng a projected precondtoned conjugate gradent method. The actual soluton u can fnally be recovered usng (4.5). Note that even f λ s only unque up to an element from ker B, the soluton u s always unque (see e. g. [27] for a more detaled dscusson). The crucal ngredents that wll make the method robust wth respect to varyng coeffcents are the choce of Q and of the precondtoner M. In the sequel we present sutable choces for Q and M, generalsng the method analysed by Klawonn and Wdlund and buldng on the results n [29, 5.2 & 5.3]. 4.. Choce of Q and M. We follow [29, 5.2]. In order to defne M we need to ntroduce scalng operators D for the Boolean matrces B on the space U of Lagrange multplers. For each subdoman Ω and for each x h Ω, we defne the pontwse weght (4.0) α (x h ) := max α τ, τ ω (x h ) where ω (x h ) Ω s the (local) patch of all elements n T that contan node x h (see [29, Fg. 5]). Furthermore, we defne the weghted countng functons [ δ α (x h ) α k (x )] h for x h Ω h (xh (4.) ) :=, k N (x h ) 0 for x h Γ h S \ Ωh, where N (x h ) := {k : x h Ω k }, the ndex set of the subdomans sharng node x h Γ h. Each functon δ ( ) can be nterpreted as a fnte element functon on the skeleton Γ S, and the unon

16 4 C. PECHSTEIN AND R. SCHEICHL of all these functons provdes a partton of unty on the skeleton, cf. [39, Secton 6.2.]. Now, to defne D, let λ j (x h ) denote the component of λ U whch corresponds to the constrant (of type (4.)) at an nterface node x h Γ h j and let λ D(x h ) denote the component of λ correspondng to the constrant (of type (4.2)) at a Drchlet node x h Ω h Ω. Let D : U U be the dagonal matrx such that { (D λ) j (x h ) := δ j (xh )λ j (x h ) for x h Γ h j, (4.2) (D λ) D (x h ) := λ D (x h ) for x h Ω h Ω. Then our FETI precondtoner s chosen to be N ( M S 0 (4.3) := D B 0 0 = ) B D. where the S are the Schur complements of the stffness matrces K defned n (2.5). The lnear operator Q whch appears n the projecton P n (4.7) s (usually) also set to be a dagonal matrx wth { (Q λ)j (x h ) := mn( α (x h ), α j (x h )) q (x h ) λ j (x h ) for x h Γ h j, (4.4) (Q λ) D (x h ) := α (x h ) q (x h ) λ D (x h ) for x h Ω h Ω, where, n three dmensons, ( + log(h q (x h /h )) h2 H ) := f x h les on a subdoman face, (4.5) h f x h les on a subdoman edge or vertex. In two dmensons, q (x h ) = ( + log(h /h )) h /H for edges and q (x h ) = for vertces. Note that q (x h ) q j (x h ) for neghbourng subdomans snce H H j and h h j. If the coeffcent α s pecewse constant wth respect to the subdomans, our choces of M and Q concde wth the ones gven n [25] (whch bulds on [9]). In each step of the projected precondtoned conjugate gradent method, we have to apply P F and P M. Therefore, the man ngredents of FETI methods are local Drchlet and regularsed Neumann solves on the subdomans Ω as well as a coarse solve wth the operator G Q G, whch s sparse, see [29, 39]. We assume that these types of problems can be handled by drect solvers. We remark that f we do not mpose the Drchlet boundary condtons by Lagrange multplers but ncorporate them n the local spaces V (Ω ), we obtan the standard one-level FETI method, cf. [29, 39]. There, some of the operators K are regular, thus some of the kernel spannng operators R are not needed, and the dmenson of the coarse space gets smaller, but otherwse the setup and the choce of Q and M s the same Man result. We are now ready to state the man result of ths paper and present new condton number bounds for the standard one-level and for the all-floatng FETI method. The proofs are postponed to Secton 5. Before we gve these results we need one fnal techncal assumpton on the local varaton of α(x) (see also Remark 4.4 below). Assumpton A3. Let {,..., N} and let ω,j Ξ,η be one of the patches coverng Ω,η n Defnton 2.6. We assume that α (x h ) s constant on each face f and on each edge e of ω,j Ω,.e. we assume no coeffcent varaton locally on any of the patch (boundary)

17 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 5 faces or edges n any of the subdoman boundary layers. (Recall that faces and edges are always consdered to be open,.e. they do not contan ther boundary nodes.) Theorem 4. (all-floatng FETI). Let {η } and α(x) be such that Assumptons A, A2 and A3 hold. On each subdoman let C,η be the mnmum of the values of C,η n Lemma 3.2 and n Lemma 3.4 wth compatble parttonngs (3.) and (3.5), respectvely, such that (3.7) holds. Then the condton number κ of the precondtoned all-floatng FETI system descrbed above satsfes κ max N H j j= η j N max = ( C,η ( + log(h /h )) 2). The hdden constant s ndependent of H, η, h and N, as well as of the contrast n the coeffcent α. It does depend on the local varatons of α on each patch ω,j (see also Remark 4.4). The constants C,η depend on () max k α (k),η /α (k),η, () a power of H /η, and possbly () a logarthmc or lnear term n H /h (see Secton 3 for detals). Proof. Postponed to Secton 5. Corollary 4.2 (one-level FETI). Let the assumptons of Theorem 4. hold. In addton, let us assume that, for each subdoman Ω that touches the Drchlet boundary, the subregons Ω (k),η n (3.) or Ω (k) n (3.5) all touch the Drchlet boundary at least n a subdoman edge (resp. vertex) n three (resp. two) dmensons. Then the condton number of the one-level FETI system also satsfes ( κ C,η ( + log(h /h )) 2), max N H j j= η j N max = Proof. Follows from Theorem 4.. For some detals see the fnal paragraph Secton 5. Remark 4.3. () Frst, we would lke to llustrate ths result for a specal case. If (a) η can be chosen n the order of H, (b) α(x) s constant (or only mldly varyng) n the subregons Ω (k),η, and (c) the nterface X from Lemma 3.4 can be chosen to be at least an edge n three dmensons, then κ ( + log(h /h )) 3 at worst. Moreover, f the number of subregons M = 2 for all subdomans Ω, then Lemma 3.2 apples and the cubc dependence s reduced to a quadratc one. () We would lke to emphasse once more that n case we use Lemma 3.2 the estmates are totally ndependent of the values of α(x) n the subdoman nterors Ω \ Ω,η, see also [29, Remark 3.5]. () Condton (3.7) n Lemma 3.4 s n accordance wth the theory gven n [29] that better estmates can be acheved, f the coeffcent n the nteror of each subdoman s larger than near ts boundary. (v) It can be seen from the proof of Theorem 4. n Secton 5 that Assumpton A s only needed for the weghted Poncaré and dscrete Sobolev type nequaltes n Secton 3. In that sense, we could relax A to the weaker assumpton that the boundary layer Ω,η of each subdoman s η -regular, see also Remark 3.3(v). In vew of [8], even weaker assumptons mght be possble. (4.6) (v) The extra factor max j H j η j κ n Theorem 4. and Corollary 4.2 can be elmnated,.e. ( C,η ( + log(h /h )) 2), max N =

18 6 C. PECHSTEIN AND R. SCHEICHL (4.7) f we set Q = M or f we assume a pror nformaton on α(x) and redefne q n (4.5) to be ( + log(h q (x h /h )) h2 η ) := f x h les on a subdoman face, h f x h les on a subdoman edge or vertex, n three dmensons (wth sutable modfcatons n two dmensons). Note that ths choce requres an a pror knowledge of the parameter η for each subdoman Ω. However, we beleve that usng techncal trcks and a dscrete weghted Sobolev type nequalty for edges, one should be able to get estmate (4.6) also for the orgnal choce of Q, but we dd not pursue ths ssue further. Remark 4.4. Assumpton A3 ensures that the functons δ j n (4.) are constant on boundary edges and faces of the boundary layer patches ω,j n Defnton 2.6. We can drop Assumpton A3, f we choose α (x h ) := max j: x h ω,j α L (ω,j ) nstead of (4.0). Ths choce for α (x h ) requres even more a pror knowledge on the coeffcent than n Remark 4.3(v) and s not very sutable for mplementatons. However, we beleve that the statements of Theorem 4. and Corollary 4.2 can also be proved for the orgnal choce of α (x h ) n (4.0) under a sutable smoothness assumpton on α (x h ),.e. α (x h ) α (y h ) x h y h α (x h ), η where x h and y h are two neghbourng nodes on a boundary face (resp. edge) of one of the boundary patches ω,j. Ths assumpton bascally excludes rapd oscllatons n α(x) on the boundary of any of the patches, but allows for large jumps between patches. 5. Proof of Theorem 4. As n our prevous work [29] we gve the proof for the three-dmensonal case; the twodmensonal case s analogous. Secton 5. ntroduces an abstract framework on the operator level, smlar to usual FETI proofs, cf. [39, Sect. 6.3]. Secton 5.2 ntroduces some techncal tools whch we need specfcally for the case of varyng coeffcents. Fnally, we prove the crucal estmate for the projecton operator defned n (5.2) below. In contrast to the proof dea outlned n [29, Sect. 4] our followng proof reduces the crucal bounds to standard results on patches ω,j (cf. Defnton 2.6) rather than on subdomans. 5.. Abstract framework. Frst, we defne the spaces N W := V h ( Ω ), W = W, n whch we wll carry out the analyss, we wrte S : W W, and we defne S : W W by S := dag (S). We recall that each S nduces the semnorm w S := S w, w /2, cf. (2.6), and that w + c S = w S for any constant c R. On the product space W we defne = ( N ) /2 w S := w 2 S for w W. =

19 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 7 Furthermore, we defne the space (5.) V := {µ U : B z, Q µ = 0 z ker K} = range P, whch can be shown to be somorphc to the space V defned n (4.8). In the followng we assume that U = range B,. e., ker B = {0}. The general case, where we have to work n the factor spaces modulo ker B, follows then from ths specal case; for detals see e. g. [27, 39]. Fnally we defne the operator (5.2) P D := [ B D B... B ND N B N ], whch can be shown to be a projecton fulfllng B P D = B. In other words, I P D s a projecton to the functons that are contnuous across the subdoman nterfaces and that fulfl the homogeneous Drchlet condtons on Ω. Analogously to [9] we can show that { (P D w) (x h j N (x ) = h ) δ j (xh ) [ w (x h ) w j (x h ) ] for x h Ω h (5.3) Γ, w (x h ) for x h Ω h Ω. In the followng we wll regard P D as an operator mappng W to W, because B acts only on degrees of freedom on the subdoman boundares. Smlarly, we wll occasonally regard B as a mappng from W to U, and B : U W. Note also that ker S = span{ Ω }. As a second mportant dentty we have (5.4) B M B = P D S P D. Usng the fact that P D s a projecton, t can be shown that the precondtoner P M s SPD as a mappng from V to V as long as Q s SPD on range G = B (ker K); for detals see e. g. [9, 27]. Therefore, P M has a well-defned SPD nverse M : V V. To show our bound on κ we show the spectral bounds (5.5) wth M λ, λ F λ, λ C M λ, λ λ V, C := ( N H ) j max j= η j N max = ( C,η ( + log(h /h )) 2). The lower bound n (5.5) can be shown by algebrac arguments ndependently of our partcular choces of Q and D followng [39, Theorem 6.5]. Usng smlar algebrac arguments the upper bound can be reduced to an estmate n the space W. Before we gve that estmate we need the followng lemma. Lemma 5.. For any w W, there exsts a unque z w ker S such that B(w + z w ) V. Moreover, B z w Q B w Q, where µ Q := µ, Q µ /2. The unque element z w s explctly gven by z w = argmn B(w + z w ) Q = R (G Q G) G Q B w, z ker K whch shows that the mappng w z w s lnear. Furthermore, w z w s a projecton onto ker S,. e., the pecewse constant functons, and ths projecton s orthogonal wth respect to the nner product nduced by B Q B. Proof. The proof follows drectly from [39, Lemma 6.2].

20 8 C. PECHSTEIN AND R. SCHEICHL As shown n [39, Sect. 6.3], the crucal estmate (5.6) P D (w + z w ) 2 S C w 2 S w W. mples the upper bound n (5.5). In order to make our weghted Poncaré and Sobolev type nequaltes from Lemma 3.2 and Lemma 3.4 applcable, we defne for each =,..., N the lnear functonal g : W R by Γ (2),η Γ (2) H,α w ds, f Lemma 3.2 s appled,,η g (w ) := X X H,α w ds, f Lemma 3.4 s appled and X s a face or an edge, (H,α w )(X ), f Lemma 3.4 s appled and X s a vertex, dependng on whch lemma we wsh to use, and the subspaces (5.7) W := { w W : g (w ) = 0 }, W := N = W. Above, H,α w denotes the dscrete α-harmonc extenson of w, see equaton (2.7). Lemma 5.2. Inequalty (5.6) holds for all w W f (5.8) P D (w + z w ) 2 S C w 2 S w W. Proof. Frst, by Lemma 5., z y = y for all y ker S, and usng the lnearty we obtan the nvarance w + z w = (w + y) + z w+y w W, y ker S. Secondly, we have the nvarance w + y S = w S for all w W and y ker S. Thus both sdes of nequalty (5.6) are nvarant f we add a y from ker S to w. Fnally, for all w W we can fnd a unque y w ker S such that w y w W by choosng (y w ) := g (H,α w ) Techncal tools. To conclude our proof we only need to show nequalty (5.8). If α(x) s constant on each subdoman Ω, ths nequalty s shown usng an addtve splttng nto terms correspondng to subdoman faces, edges, and vertces, whch s motvated from formula (5.3). In our case, however, the functons δ j are n general no longer constant on such subdoman faces or edges, ndeed they can have arbtrary large jumps. Therefore, we need a fner splttng nto terms correspondng to faces, edges, and vertces of the patches formng Ω,η, cf. Defnton 2.6. We denote by F = {F } the set of subdoman faces, E = {E} the set of subdoman edges, V = {V } the set of subdoman vertces of Ω, and by F = {f} the set of patch faces, E = {e} the set of patch edges, V = {v} the set patch of vertces wth respect to patches from Ξ,η whch are part of the subdoman boundary Ω. Recall that F Γ, EΓ, VΓ, and F D, E D, V D denote the subsets of faces, edges, and vertces lyng on Γ and Ω, respectvely. Correspondngly, we defne the subsets F Γ, EΓ, VΓ and F D, ED, and

21 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 9 V D. For convenence we also defne X := F E V, and X := F E V. For each x X we defne the unon of touchng patches ω (x) by (5.9) ω (x) := ω,j. x ω,j Note, that by Defnton 2., ω (x) s a regular doman. Wthout loss of generalty we assume that the patches are algned wth the trangulaton T. If they are not we can use the Scott-Zhang quas-nterpolaton operator [37] as we dd n [29]. Smlar to [39, Secton 4.6] we defne the fnte element cut-off functons ϑ v V h (Ω ) as beng at the vertex v, and zero on all other nodes. ϑ e V h (Ω ) as beng at the nodes on the (open) edge e, and zero on all other nodes, ϑ f V h (Ω ) as beng at the nodes on the (open) face f, zero on all nodes n Ω \ (ω (f) f), and dscrete harmonc nsde of ω (f). θ v, θ e, and θ f V h ( Ω ) as the traces of ϑ v, ϑ e, and ϑ f, respectvely. To be more exact we would have to wrte ϑ,v, θ,v, etc., but the doman ndex wll always be clear from the context and s therefore skpped. Throughout the whole secton we make use of the nodal nterpolator I h onto V h (Ω ) (resp. V h ( Ω )) whch s contnuous n the H -semnorm (resp. H /2 -semnorm) and n the L 2 -norm for quadratc functons. See [39, Lemma 3.9]. By defnton the functons θ x provde a partton of unty on Ω n the sense that (5.0) I h (θ x u) = u u W. x X The followng lemma states that cuttng a functon u by one of the functons ϑ x costs only low energy. Lemma 5.3. For x X let ω,j Ξ,η be an arbtrary patch such that x ω,j. Then, { I h (ϑ x u) 2 H (ω (x)) ( + log(η /h )) 2 u 2 H (ω,j ) + } η 2 u 2 L 2 (ω,j ). Proof. For a face f there s only one such patch,. e., ω (f) = ω,j. Due to [39, Lemma 4.24], { I h (ϑ f u) 2 H (ω (f)) ( + log(η /h )) 2 u 2 H (ω,j ) + } η 2 u 2 L 2 (ω,j ). For an edge e, [39, Lemma 4.6 and Lemma 4.9] yeld { I h (ϑ e u) 2 H (ω (e)) u 2 L 2 (e) ( + log(η /h )) u 2 H (ω,j ) + } η 2 u 2 L 2 (ω,j ). For a vertex v recall that ϑ v s the nodal bass functon assocated wth v. We easly obtan I h (ϑ v u) 2 H (ω (v)) h u(v) 2 u 2 L 2 (e), for some edge e E wth v e and e ω,j. From here we can contnue as above. The next lemma states that the cut-off functons ϑ x can be used to estmate the energy norm by a decomposton nto patches.

22 20 C. PECHSTEIN AND R. SCHEICHL η Ω ω () x x ω j,l η j Ω j Fgure 5. Illustraton of the operator Π j x from Lemma 5.5. Lemma 5.4. For a functon u W, we have u 2 S α(x) (I h (ϑ x ũ))(x) 2 dx, x X ω (x) where ũ s an arbtrary extenson of u from Ω to V h (Ω,η ). The hdden constant s ndependent of the number of patches n Ω,η. Proof. Frst, we convnce ourselves that the functon v := x X I h (ϑ x ũ) s an extenson of u from Ω to Ω wth ts support n Ω,η, and that v V h (Ω ). Ths s true because each node n Ω h belongs to a unque x X and ϑ x vanshes on Ω h \ x, and the supports of the ϑ x le entrely n Ω,η. Usng the mnmum property (2.7) of the Schur complement, we have u 2 S α(x) v(x) 2 dx. Ω,η Snce () each ϑ x s only supported n ω (x) whch conssts only of a fnte number of patches, () each patch s contaned n fntely many supports ω (x) and () the supports ω (x) have fnte overlap, we can conclude that u 2 S α(x) v(x) 2 dx, x X ω (x) whch proves the desred statement. The last tool s nspred by [8]. For an llustraton see Fgure 5. Lemma 5.5. Let x X X j and let ω j,l be a patch wth x ω j,l. Then there exsts an operator Π j x : V h (ω j,l ) V h (ω (x)) such that for all u V h (ω j,l ), (5.) Πx j u 2 H (ω (x)) + η 2 (Π j x u) x = u x, Π j x u 2 L 2 (ω (x)) u 2 H (ω j,l ) + η 2 u 2 L 2 (ω j,l ), the operator s even contnuous n the L 2 -norm and the H -semnorm, separately.

23 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 2 Proof. Let U be the open connected unon of ω (x) and ω j,l such that dam U η. In a frst step we construct a contnuous extenson operator E : H (ω j,l ) H (U) makng use of the fact that ω j,l s Lpschtz, and for the tme beng assumng that dam ω j,l =. Followng Sten [38] (see e. g. also [, p. 46ff]), there exsts an extenson operator Ẽ : H (ω j,l ) H (R d ) wth (Ẽu) ω j,l = u and Ẽu L 2 (R d ) u L 2 (ω j,l ), Ẽu H (R d ) u H (ω j,l ). Usng the mean value u := ω j,l ω j,l u dx we defne E : H (ω j,l ) H (U) : u [ Ẽ(u u) ] U + u. Obvously, (Eu) ωj,l = u. Ths operator s stable n L 2, as we have Eu L 2 (U) Ẽ(u u) L 2 (R d ) + u L 2 (U) u u L 2 (ω j,l ) + u L 2 (U) u L 2 (ω j,l ), where n the last step we used Cauchy s nequalty and the fact that U ω j,l. By usng Poncaré s nequalty on ω,l we can also conclude the stablty n the H -semnorm, Eu H (U) Ẽ(u u) H (R d ) + u H (U) u u H (ω j,l ) u u H (ω j,l ). Usng a smple dlaton argument the two above equatons reman also vald when the dameter of ω,l dffers from one, and the constants nvolved depend only on the shapes of ω j,l and U. In a second step we use the quas-nterpolaton operator Π h : H (U) V h (U) ntroduced by Scott and Zhang [37] (see also [6] and [8]). Ths operator s smlar to the one by Clément but averages on manfolds nstead of patches, whch makes t eventually possble to preserve boundary values. In our current settng we can choose the averagng manfolds such that (Π h u) x = u x u H (U), u ωj,l V h (ω j,l ), Π h u 2 H (U) u 2 H (U), Πh u 2 L 2 (U) u 2 L 2 (U) u H (U). Defnng Π j x u := (Π h Eu) ωj,l we meet the requrements of the lemma The P D estmates. In ths subsecton we show nequalty (5.8) by estmatng P D w 2 S and P D z w 2 S separately. For compact notaton we defne the α-weghted (sem)norms ( /2 ( u L 2 (D),α := α(x) u(x) dx) ) /2 (5.2) 2, u H (D),α := α(x) u(x) 2 dx for a generc doman D. D Lemma 5.6. Let the assumptons of Theorem 4. hold. Then, { P D w 2 S max N C,η ( + log(η /h )) 2} w 2 S w W. j= Proof. Due to Assumpton A3, the functons α defned on Ω h are pecewse constant wth respect to (patch) faces f F Γ and edges e E Γ, and are thus equal to a constant α x on each x X Γ. As a consequence, the functons δ share the same property, and we defne D

24 22 C. PECHSTEIN AND R. SCHEICHL δ x analogously. Let now be fxed. Usng dentty (5.3) and the partton of unty property (5.0) we obtan that (P D w) 2 S = δ j x Ih( ) θx (w w j ) + I h (5.3) (θ x w ) 2 S j N x x X Γ Let Ω j be one of the neghbours of Ω. For a fxed x X j, large jumps n α(x) can occur n ω (x). We can, however, always fnd one patch ω,k Ξ,η such that ω,k ω (x) and ( α(x) ) (5.4) α L (ω (x)) α L (ω,k ) sup α x,y ω,k α(y) x. In other words, ω,k s the patch n the unon ω (x) of patches touchng x where the maxmum s attaned (locally). Obvously, the estmate above depends only on the local varaton n α(x) but not on the contrast of α(x) between the subregons Ω (k) Let the operator Π j x (5.5) x X D : V h (ω j,l ) V h (ω (x)) be defned accordng to Lemma 5.5. Wth the defntons w j := H j,α w j for j =,..., N, w x j := Π j x w j for x X X j, we have by constructon that ( w j ) Ωj = w j and ( w j x ) x = (w j ) x for all j. From the defnton (5.3), we see that the functon v := δ j x Ih( ϑx ( w w x ) (5.6) j ) + I h (ϑ x w ) j N x x X Γ s an extenson of (P D w) from W to V h (Ω ). By Lemma 5.4 we can conclude that (P D w) 2 S ( δ ) 2 j,x α [ I h ( ϑ x ( w w j x ) )] 2 dx + (5.7) Usng (5.4), we have (5.8) ψ,x x X Γ j N x + x X D ω (x) x X D,η. } {{ } =:ψ j,x ω (x) α (I h (ϑ x w )) 2 dx. } {{ } =:ψ,x ( α(x) ) sup α x,y ω,k α(y) x I h (ϑ x w ) 2 H (ω (x)) x X D. Wth the same arguments and the elementary nequalty (5.9) α (x h ) ( δ j (xh ) ) 2 mn ( α (x h ), α j (x h ) ) x h Γ h j, cf. [39, Sect ], we obtan that for all x X Γ and j N x, ( α(x) ψ j,x sup x,y ω,k α(y) (5.20) ( α(x) sup x,y ω,k α(y) ) (δ j x ) 2 α x I h (ϑ x ( w w x j )) 2 H (ω (x)) ) { } α x I h (ϑ x w ) 2 H (ω (x)) + α j x I h (ϑ x w j x ) 2 H (ω (x)).

25 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 23 From here on, for a smpler presentaton, we wll not make the dependency on the local varatons sup x,y ω,k α(x)/α(y) explct anymore, but hde them usng the symbolsm. The combnaton of (5.7), (5.8), and (5.20) then yelds (5.2) (P D w) 2 S α x I h (ϑ x w ) 2 H (ω (x)) + α j x I h (ϑ x w j x ) 2 H (ω (x)). x X j N x \{} For later purposes we ntroduce the followng constructon. For a fxed x X X j, we choose patches ω,m Ξ,η and ω j,l Ξ j,ηj such that (5.22) α x α j x ( α(x) sup x,y ω,m α(y) ( α(x) sup x,y ω j,l α(y) x X Γ ) α(x) x ω,m, ) α(x) x ω j,l,. e., we choose the patches where α x and α j x are attaned. Note that m and l depend on x. We contnue now to further estmate (5.2). The terms n the frst sum of ths expresson are estmated usng Lemma 5.3, whch yelds { α x I h (ϑ x w ) 2 H (ω (x)) α x ( + log(η /h )) 2 w 2 H (ω,m ) + } η 2 w 2 L 2 (ω,m ) (5.23) { ( + log(η /h )) 2 w 2 H (ω,m ),α + } η 2 w 2 L 2 (ω,m ),α, where n the last lne we used (5.22). The second sum n (5.2) s estmated usng Lemma 5.3 and Lemma 5.5. For a fxed x X Γ and j N x \ {} we obtan (usng that w j x = Πj x w j ) (5.24) α j x I h (ϑ x w x j ) 2 H (ω (x)) { α j x ( + log(η /h )) 2 Π j x w j 2 H (ω (x)) + η 2 α j x ( + log(η /h )) 2 { w j 2 H (ω j,l ) + η 2 j ( + log(η /h )) 2 { w j 2 H (ω j,l ),α + η 2 j w j 2 L 2 (ω j,l ) w j 2 L 2 (ω j,l ),α } Π j x w j 2 L 2 (ω (x)) } }, where n the last lne we used agan (5.22). Combnng estmates (5.2), (5.23), and (5.24), we obtan wth a fnte summaton argument that (P D w) 2 S { ( + log(η /h )) 2 w j 2 H (Ω j,ηj ),α + } η 2 w j 2 L 2 (Ω j,ηj ),α, j N where N = {j : Ω Ω j } s ndex set of neghbours of Ω (ncludng ). Snce w j Wj, the functon w j = H j,α w j satsfes g j ( w j ) = 0. Hence, Lemma 3.2 and Lemma 3.4 yeld (5.25) (P D w) 2 S Cj,η j ( + log(η /h )) 2 w j 2 H (Ω j ),α. j N Fnally, usng that h j h, η j η, w j H (Ω j ),α = w j Sj, and that each subdoman has only fntely many neghbours, we obtan the statement of Lemma 5.6.

26 24 C. PECHSTEIN AND R. SCHEICHL Lemma 5.7. Let the assumptons of Theorem 4. hold and suppose that Q s chosen accordng to (4.4) (4.5). Then, ( P D z w 2 N H ) k N { S max max Cj,η k= η k j= j ( + log(h k /h k )) 2} w 2 S w W. Proof. Usng equaton (5.4) and Lemma 5. we have P D z w 2 S = B z w 2 Q B w 2 Q = P D w 2 S, and the prevous Lemma 5.6 mples the desred statement. For the other two cases, recall that z w ker S,. e., z w s constant on each subdoman. We denote these constant components by z. For a moment, let be fxed. Wth the same arguments as n the proof of Lemma 5.6, the functon v := δ j x I h (ϑ x (z z j )) + I h (ϑ x z ) j N x x X Γ s an extenson of (P D z w ) from W to V h (Ω ). An applcaton of Lemma 5.4 yelds (P D z w ) 2 S (δ j x ) 2 I h (ϑ x (z z j )) 2 H (ω (x)),α + I h (ϑ x z ) 2 H (ω (x)),α x X Γ j N x x X D = (δ j x ) 2 ϑ x 2 H (ω (x)),α z z j 2 + ϑ x 2 H (ω (x)),α z 2. x X j N x As n the proof of Lemma 5.6, we do not make explct the dependency on the local varatons sup x,y ω,k α(x)/α(y). Usng (5.4),. e., α L (ω (x)) α x, and due to [39, Lemma 4.6, Lemma 4.25] we have x X D x X D ϑ f 2 H (ω (x)),α ϕ,f := α f ( + log(η /h )) η f F, ϑ e 2 H (ω (x)),α ϕ,e := α e η e E, ϑ v 2 H (ω (x)),α ϕ,v := α v h v V. Obvously, for each edge e (and for each vertex v) we can fnd a face f wth e f (resp. an edge e wth v e) such that ϕ,e ϕ,f and ϕ,v ϕ,e, by choosng the one touchng the patch wth the effectvely largest coeffcent. Snce each patch face contans only a fnte number of patch edges and vertces, and due to the elementary nequalty (5.9) we obtan (P D z w ) 2 S mn(ϕ,x, ϕ j,x ) z z j 2 + (5.26) X X Γ X Γ j + X X D x X, x X x X, x X ϕ,x z 2, where the x n the nner sums are of the same knd (face/edge/vertex) as the X n the outer sums. We treat subdoman face, edge, and vertex contrbutons separately.

27 (5.27) ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 25 Snce each face f F contans O((η /h ) 2 ) nodes, we can bound the subdoman face contrbutons n (5.26) from above by h 2 mn( α f, α j f ) ( + log(η /h )) z z j 2 η f F, f F x h f h }{{} F F Γ F Γ j + F F D f F f F q (x h ) H /η h 2 α f ( + log(η /h )) z 2, η x h f h }{{} q (x h ) H /η where f h s the set of nteror nodes on f. Snce each edge e E contans O(η /h ) nodes, we can bound the subdoman edge contrbutons n (5.26) from above by E E Γ EΓ j + E E D e E, e E mn( α e, α j e ) x h e h e E, e E x h e h α e h }{{} z 2, =q (x h ) h }{{} z z j 2 = q (x h ) where e h s the set of nteror nodes on e. Smlarly, the subdoman vertex contrbutons from (5.26) can be bounded by V V Γ VΓ j mn( α (V ), α j (V )) h }{{} z z j 2 + =q (V ) V V D α (V ) h }{{} z 2. =q (V ) Accordng to [9] we observe that the expressons z z j and z are components of B z w. Collectng all terms approprately and usng the defnton (4.4) of Q, we can conclude by Lemma 5. that (P D z w ) 2 S ( N H ) k max B z w 2 Q k= η k ( N H ) k max B w 2 Q. k= η k In the followng we splt B w 2 Q nto face, edge, and vertex terms, wth r x j := mn( α x, α j x ) q x B w 2 Q = x h x h N { = x X Γ XΓ j r x j + x X D r x }, ( w (x h ) w j (x h ) ) 2, r x := α x q x and treat patch face, edge, and vertex terms agan separately. x h x h w (x h ) 2, Recall that q f = ( + log(h /h )) h 2 /H. Due to the quas-unformty assumpton on the mesh on Ω we obtan for the patch face terms that rj f mn( α f, α j f ) ( + log(h /h )) w w j 2 L H 2 (f), }{{} w 2 L 2 (f ) + w j 2 L 2 (f ) r f α f ( + log(h /h )) H w 2 L 2 (f).

28 26 C. PECHSTEIN AND R. SCHEICHL (5.28) In the followng we use constructon (5.22) for x = f,. e., α f α ω,m and α j f α ωj,l. A standard Sobolev norm equvalence (cf. [39, Sect. A.4]) yelds η w 2 L 2 (f) H,α w 2 H (ω,m ) + η 2 H,α w 2 L 2 (ω,m ), and analogously, the L 2 -norm of w j on f s bounded n terms of the scaled H -norm on ω j,l. Combnng these wth the estmate before we obtan after summaton that r f j + r f e E Γ EΓ j f F Γ FΓ j f F D η ( + log(h /h )) H j N { H j,α w j 2 H (Ω j,ηj ),α + η 2 j For the patch edge terms we obtan by smlar arguments that H j,α w j 2 L 2 (Ω j,ηj ),α r e j α e w 2 L 2 (e) + α j e w j 2 L 2 (e), re α e w 2 L 2 (e). Due to [39, Lemma 4.6] and constructon (5.22) for x = e, we have { w j 2 L 2 (e) ( + log(η j /h j )) H j,α w j 2 H (ω j,l ) + } ηj 2 H j,α w j 2 L 2 (ω j,l ), and the analogous estmate for w 2 L 2 (e). obtan rj e + ( + log(η /h )) e E D r e j N }. Collectng and summng all terms we { H j,α w j 2 H (Ω j,ηj ),α + η 2 j H j,α w j 2 L 2 (Ω j,ηj ),α The patch vertex terms can be estmated trvally by patch edge terms. Combnng all the estmates, notcng that H,α w 2 H (Ω,η ),α w 2 S, η /H, and usng Lemma 3.2 we obtan ( (P D z w ) 2 N H ) k S max Cj,η k= η j ( + log(h j /η j )) w j 2 S j w W, =,..., N, k j N whch drectly mples the statement of Lemma 5.7. Combnng Lemma 5.6 and Lemma 5.7, we fnally obtan nequalty (5.8). To obtan the mproved result n Remark 4.3(v) for the choce Q = M we can use the proof of [39, Lemma 6.4] nstead of Lemma 5.7 and avod to ntroduce the extra factor max N k= H k/η k. On the other hand, f Q s chosen accordng to (4.7),.e. usng some a pror nformaton on η, then the factors H /η and η /H n (5.27) and (5.28) (resp.) dsappear and agan we can avod to ntroduce the extra factor max N k= H k/η k. The proof of Corollary 4.2 requres no new deas. For non-floatng subdomans, one can use standard Fredrchs or dscrete Poncaré-Fredrchs nequaltes on the subregons Ω (k) usng that the functon w vanshes at least on an edge. If M = 2 for all subdomans Ω, the quadratc bound s obtaned usng the standard trck of ntroducng a face average, as t s descrbed e. g. n [9, 29]. }.

29 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION Numercal results In ths secton we would lke to confrm our new theoretcal results. We pont out that results for nteror sland coeffcents as well as for nterface varaton are already contaned n [29, Secton 5] and n [30]. Note n partcular, that our new theory explans the robustness of one-level FETI n case of the nonlnear magnetostatcs wth large nterface varaton n [29, Secton 5.4], see also [28]. In all our computatons we used PARDISO [36] as sparse drect solver for the subdoman problems. 6.. Edge slands. In Example we choose Ω to consst of 6 squares wth an sland coeffcent cuttng through a subdoman edge (cf. Fg. 6, left). In order to rule out symmetres we have shfted the centre of the sland to the rght of the subdoman nterface. In Fgure 6 and n what follows, H denotes the subdoman wdth and η denotes the characterstc geometrc scale of the coeffcent sland. We set the coeffcent to outsde the sland (shaded regon) and to a constant α I nsde. We mpose Drchlet boundary condtons on the entre boundary Ω and choose a pecewse constant rght-hand sde. In Tables and 2 we dsplay number of PCG teratons (to acheve a relatve resdual reducton of 0 8 ) and condton numbers (estmated by Lanczos method) for the cases α I = 0 +5 and α I = 0 5. η η η η η H H Fgure 6. Coeffcent dstrbuton. Left Example (edge sland), rght Example 2 (crosspont sland) H η H h = = 4 7 (9) 7 (9) 20 (22) 22 (23) 8 22 (9) 23 (24) 25 (25) 26 (26) 6 22 (23) 24 (25) 26 (28) 28 (30) (25) 25 (27) 27 (29) 29 (3) (27) 27 (30) 30 (32) 3 (33) (3) 36 (36) 37 (37) (39) 46 (42) (49) Table. Example : number of CG teratons; sland coeffcent wth α I = 0 +5, n brackets: α I = 0 5.

30 28 C. PECHSTEIN AND R. SCHEICHL H η H h = = (7.0) 6.9 (6.9) 0. (0.).9 (2.0) (.0) 8.4 (3.3) 0. (5.7).9 (8.3) (20.8) 9.0 (25.0) 0.5 (29.4) 2.2 (22.5) (39.0).7 (47.3) 2.8 (55.7) 4. (64.3) (72.0) 9. (88.2) 9.9 (04.9) 20.8 (2.7) (63.2) 35.7 (95.8) 36.5 (229.) (363.6) 68.8 (429.0) (800.5) Table 2. Example : estmated condton numbers; sland coeffcent wth α I = 0 +5, n brackets: α I = 0 5. FETI for edge sland coeffs FETI for edge sland coeffs condton number e+ e+2 e+3 e+4 e+5 condton number e- e-2 e-3 e-4 e H/eta H/eta Fgure 7. Example : estmated condton numbers; H/h = 52, varyng rato H/η and varyng magntude of the jump α I. Left α I >, rght α I < 6.2. Cross pont slands. In Example 2 we choose the coeffcent dstrbuton sketched n Fg 6, rght. Agan we set α to a constant α I n the sland (the shaded square), and elsewhere. Note that here, the wdth/heght of the sland remans of fxed sze H. H η H h = = 4 7 (7) 9 (8) 20 (20) 2 (22) 8 9 (9) 2 (20) 22 (2) 22 (23) 6 20 (9) 2 (22) 23 (23) 24 (25) (22) 24 (24) 26 (25) 27 (28) (22) 29 (26) 30 (28) 32 (30) (29) 36 (32) 39 (36) (37) 50 (39) (47) Table 3. Example 2: number of CG teratons; sland coeffcent wth α I = 0 +5, n brackets: α I = 0 5.

31 ANALYSIS OF FETI METHODS FOR MULTISCALE PDES PART II: INTERFACE VARIATION 29 H η H h = = (8.06) 8.92 (9.77) 0.64 (.63) 2.5 (3.64) (.99) 0.45 (4.44) 2.3 (7.03) 4.34 (9.78) (9.83) 2.62 (23.95) 4.60 (28.9) 6.75 (32.58) (34.9) 6.85 (4.7) 8.63 (49.38) (57.8) (59.9) (74.) 28.8 (88.67) (03.38) (32.6) (60.58) (89.4) (29.42) (346.84) (635.45) Table 4. Example 2: estmated condton numbers; sland coeffcent wth α I = 0 +5, n brackets: α I = 0 5. FETI for crosspont sland coeffs FETI for crosspont sland coeffs condton number e+ e+2 e+3 e+4 e+5 condton number e- e-2 e-3 e-4 e H/eta H/eta Fgure 8. Example 2: estmated condton numbers; H/h = 52, varyng rato H/η and varyng magntude of the jump α I. Left α I >, rght α I < 6.3. Standard one-level vs. all-floatng FETI. In Example 3 we consder a coeffcent sland cuttng through a doman that touches the Drchlet boundary, cf. Fg. 9, left. As before, we mpose Drchlet boundary condtons on the whole of Ω and choose α = α I = const nsde the shaded square, and α = elsewhere. As one can see n Table 5, the standard one-level FETI method s not robust when α I >, whereas the all-floatng method remans robust. The reason why the number of PCG teratons stays small n all cases (even when the condton number blows up) s probably due to the fact that we have only consdered one coeffcent sland. Ths s related to spectral clusterng effects for doman decomposton precondtoners, whch are explaned n [3]. The fact that one-level FETI s fully robust for α I < s not contradctng our theory and s perfectly explaned by Corollary 3.6 and Remark 3.7. To see ths let Ω be any of the subdomans that supports the coeffcent sland and let Ω (I) be the part of the sland n Ω. Then, choosng Ω (),art := Ω \Ω (I), Ω (2),art := Ω and η := H /2 Corollary 3.6 apples wth σ(h, h ) = (snce the nterface X s an edge, cf. Remark 3.7), and therefore we have robustness due to Corollary 4.2.

32 30 C. PECHSTEIN AND R. SCHEICHL αi Fgure 9. Left coeffcent dstrbuton n Example 3, rght coeffcent dstrbuton and subdoman parttonng n Example 4 std. one-level all-floatng std. one-level all-floatng α I t cond t cond α I t cond t cond Table 5. Example 3: teraton numbers and condton number estmates, standard one-level vs. all-floatng FETI, H/h = Mult-valued coeffcents. In Example 4 we choose 6 subdomans and the coeffcent dstrbuton shown n Fg. 9, rght. Agan we choose Drchlet boundary condtons on the whole of Ω. On the lower left subdoman the coeffcent takes four dfferent values (, 0 3, 0 5, and 0 7 ) and vares n a non-quasmonotone way. We repeat ths pattern on the other subdomans but slghtly ncrease the coeffcent values n order to rule out perodcty. Table 6 shows the teraton numbers and estmated condton numbers for standard one-level and all-floatng FETI. As we expect, the all-floatng method s robust n α. The reason for the robustness of one-level FETI s that we can always choose artfcal subregons and coeffcents n order to connect any of the four subregons to the Drchlet boundary (see Corollary 3.6 and Remark 3.7). H/h std. one-level 25 (28.27) 27 (36.30) 29 (43.94) 30 (50.69) 32 (56.49) all-floatng 26 (20.38) 30 (28.90) 33 (37.07) 34 (44.8) 37 (50.3) Table 6. Example 4: number of PCG teratons, estmated condton numbers n brackets