TR DUAL-PRIMAL FETI METHODS FOR LINEAR ELASTICITY. September 14, 2004

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1 TR DUAL-PRIMAL FETI METHODS FOR LINEAR ELASTICITY AXEL KLAWONN AND OLOF B WIDLUND Septeber 14, 2004 Abstract Dual-Prial FETI ethods are nonoverlapping doain decoposition ethods where soe of the continuity constraints across subdoain boundaries are required to hold throughout the iterations, as in prial iterative substructuring ethods, while ost of the constraints are enforced by Lagrange ultipliers, as in one-level FETI ethods The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained, which are independent of even large changes in the stiffnesses of the subdoains across the interface between the A theoretical analysis is provided and condition nuber bounds are established which are unifor with respect to arbitrarily large jups in the Young s odulus of the aterial and otherwise only depend polylogarithically on the nuber of unknowns of a single subdoain Key words doain decoposition, Lagrange ultipliers, FETI, preconditioners, elliptic systes, elasticity, finite eleents AMS subject classifications 65F10,65N30,65N55 1 Introduction We will consider iterative substructuring ethods with Lagrange ultipliers for the elliptic syste of linear elasticity The algoriths belong to the faily of dual-prial FETI (finite eleent tearing and interconnection) ethods which was introduced for linear elasticity probles in the plane in [8] and then extended to three diensional elasticity probles in [9] In dual-prial FETI (FETI-DP) ethods, soe continuity constraints on prial displaceent variables are required to hold throughout the iterations, as in prial iterative substructuring ethods, while ost of the constraints are enforced by the use of dual Lagrange ultipliers, as in the older one-level FETI algoriths The prial constraints should be chosen so that the local probles becoe invertible They also provide a coarse proble and they should be selected so that the iterative ethod converges rapidly We also wish to use relatively few, and effective, prial constraints since the they represent a global part of the preconditioner which is relatively difficult to parallelize More recently, the faily of algoriths for scalar elliptic probles in three diensions was extended and a theory was provided in [15, 16]; see also [24, Section 64] It was shown that the condition nuber of the dual-prial FETI ethods can be bounded polylogarithically as a function of the diension of the individual subregion probles and that the bounds can otherwise be ade independent of the nuber of subdoains, the esh size, and jups in the coefficients In the case of the elliptic syste of partial differential equations arising fro linear elasticity, essential changes in the selection of the prial constraints have to be ade in order to obtain the sae quality bounds for elasticity probles as in the scalar case Special ephasis will be given to developing robust condition nuber estiates with bounds which are independent of arbitrarily large jups of the aterial coefficients For benign coefficients, without large jups, it is sufficient to select an appropriate set of edge averages as Fachbereich Matheatik, Universität Duisburg-Essen, Capus Essen, Universitätsstraße 3, D Essen, Gerany E-ail: klawonn@athuni-essende, URL: Courant Institute of Matheatical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA E-ail: widlund@csnyuedu, URL: This work was supported in part by the US Departent of Energy under Contracts DE-FG02-92ER25127 and DE-FC02-01ER

2 2 prial constraints to obtain good bounds, whereas for arbitrary coefficient distributions, additional prial first order oents and constraints at soe of the vertices are also required We note that there is extensive and ongoing experiental work on dual-prial FETI ethods for linear elasticity; cf [11] The results obtained so far, using the algoriths developed in this paper, are quite proising We note that our results and strategies of selecting constraints iediately carry over to the ore recently developed Neuann-Neuann ethods with constraints, known as the BDDC algoriths; cf [3, 18, 19] This is so, since Mandel, Dohrann, and Tezaur [19] have shown that for any given set of constraints, the BDDC and FETI-DP ethods have alost all their eigenvalues in coon; see also Fragakis and Papadrakakis [10] for earlier experiental work We note that the results of [19] are obtained by algebra alone and that the authors do not address the question on how best to select the set of prial constraints The reainder of this article is organized as follows In Section 2, we introduce the equations of linear elasticity in three diensions and provide different Korn inequalities which are needed in our analysis In Section 3, we introduce our doain decoposition and soe geoetric notation as well as the associated finite eleent spaces In Section 4, we introduce our faily of dual-prial FETI ethods in an abstract setting and in Section 5, we establish certain conditions which will allow us to establish good bounds In Section 6, we discuss two different possibilities of ipleenting our FETI-DP algoriths, one using global optional Lagrange ultipliers and another where a change of basis is applied In Section 7, we collect soe auxiliary technical leas which are needed in our convergence analysis which is presented in Section 8 In a final subsection, we outline possible strategies of selecting a sufficiently rich set of constraints We note that soe of the results of this paper have previously been presented in a conference article; cf [14] 2 The equations of linear elasticity and Korn inequalities The equations of linear elasticity odel the displaceent of a linear elastic aterial under the action of external and internal forces The elastic body occupies a doain Ω IR 3, which is assued to be polyhedral and of diaeter one We denote its boundary by Ω and assue that one part of it, Ω D, is claped, ie, with hoogeneous Dirichlet boundary conditions, and that the rest, Ω N := Ω \ Ω D, is subject to a surface force g, ie, a natural boundary condition We can also introduce a body force f, eg, gravity With H 1 (Ω) := (H 1 (Ω)) 3, the appropriate space for a variational forulation is the Sobolev space H 1 0 (Ω, Ω D) := {v H 1 (Ω) : v = 0 on Ω D } The linear elasticity proble consists in finding the displaceent u H 1 0 (Ω, Ω D) of the elastic body Ω, such that (1) Ω G(x)ε(u) : ε(v)dx + G(x)β(x)div u divvdx = F,v v H 1 0 (Ω, Ω D) Ω Here G and β are aterial paraeters which depend on the Young s odulus E > 0 and the Poisson ratio ν (0, 1/2]; we have G = E/(1 + ν) and β = ν/(1 2ν) (The coefficients are also referred to as the Laé paraeters) In this article, we only consider the case of copressible elasticity, which eans that the Poisson ratio ν is bounded away fro 1/2 Furtherore, ε ij (u) := 1 2 ( ui x j + uj x i ) is the linearized strain

3 3 tensor, and 3 ε(u) : ε(v) = ε ij (u)ε ij (v), F,v := f T v dx + i,j=1 Ω Ω N g T v dσ For convenience, we also introduce the notation (ε(u), ε(v)) L2(Ω) := ε(u) : ε(v)dx The bilinear for associated with linear elasticity is then a(u,v) = (Gε(u), ε(v)) L2(Ω) + (Gβ divu, divv) L2(Ω) We will also use the standard Sobolev space nor 3 with u 2 L := 2(Ω) i=1 u H 1 (Ω) := Ω Ω ( ) 1/2 u 2 H 1 (Ω) + u 2 L 2(Ω) u i 2 dx, and u 2 H 1 (Ω) := 3 i=1 u i 2 L 2(Ω) We note that if we rescale our region by a dilation, the two ters of the full H 1 nor scale differently and we should introduce a factor 1/H 2 in front of the square of the L 2 nor if the diaeter of the region is on the order of H It is obvious that the bilinear for a(, ) is continuous with respect to H 1 (Ω), although the bound depends on the Laé paraeters Continuity follows fro the eleentary inequalities (2) (div (u), div (v)) L2(Ω i) u H1 (Ω i) v H1 (Ω i) u,v H 1 (Ω i ), (ε(u), ε(v)) L2(Ω i) u H 1 (Ω i) v H 1 (Ω i) u,v H 1 (Ω i ) However, proving ellipticity is less trivial but it can be established fro Korn s first inequality; see, eg, Ciarlet [2] Lea 1 (Korn s first inequality) Let Ω IR 3 be a Lipschitz doain Then, there exists a positive constant C = C(Ω, Ω D ) > 0, invariant under dilation, such that u 2 H 1 (Ω) C (ε(u), ε(u)) L 2(Ω) u H 1 0 (Ω, Ω D) The wellposedness of the linear syste (1) follows iediately fro the continuity and ellipticity of the bilinear for a(, ) It follows fro Korn s first inequality that ε(u) L2(Ω) is equivalent to u H 1 (Ω) on H 1 0 (Ω, Ω D) This result is not directly valid for the case of purely natural boundary conditions when we work with the entire space H 1 (Ω) This case is of interest when considering floating subregions, ie, those that do not touch Ω D However, a Gårding inequality is provided by Korn s second inequality This inequality will only be needed for our purposes on subdoains on which the Laé paraeters are assued to be hoogeneous, ie, do not vary greatly Lea 2 (Korn s second inequality) Let Ω IR 3 be a Lipschitz doain of diaeter one Then, there exists a positive constant C = C(Ω), such that u 2 H 1 (Ω) C ((ε(u), ε(u)) L 2(Ω) + u 2 L 2(Ω) ) u H1 (Ω)

4 4 There are several proofs; see, eg, Nitsche [22] We can now derive a Korn inequality on the space {u H 1 (Ω) : u ker (ε)} The null space ker (ε) is the space of rigid body otions and orthogonality is defined with respect to the L 2 inner product Thus, the linearized strain tensor of u and its divergence vanish only for the eleents of the space spanned by the three translations (3) r 1 := 1 0 0,r 2 := 0 1 0,r 3 := 0 0 1, and the three rotations (4) r 4 := 1 x 2 ˆx 2 x 1 + ˆx 1,r 5 := 1 x 3 + ˆx 3 0 H H 0 x 1 ˆx 1,r 6 := 1 H 0 x 3 ˆx 3 x 2 + ˆx 2 Here ˆx Ω and H denotes the diaeter of Ω The shift of the origin akes the basis for the space of rigid body odes well conditioned and the scaling and shift ake the L 2 (Ω) nors of these six functions scale in the sae way with H We will also use the notation r k = (r kl ) l=1,2,3, k = 1,,6, with r kl the l-th coponent of the k-th rigid body ode We now introduce two alternative inner products in H 1 (Ω), for a region Ω of diaeter one, (u,v) E1 := (ε(u), ε(v)) L2(Ω) + (u,v) L2(Ω) and (u,v) E2 := (ε(u), ε(v)) L2(Ω) + (u,r i ) L2(Σ)(v,r i ) L2(Σ), i=1 where (5) (u,r i ) L2(Σ) = u T r i dx Σ Here, Σ Ω is assued to have positive easure By Lea 2, E1, given by the inner product (, ) E1, is a nor and so, by construction, is E2 These nors are equivalent: Lea 3 Let Ω IR 3 be a Lipschitz doain of diaeter one and let Σ Ω be of positive easure Then, there exist constants 0 < c C <, such that c u E1 u E2 C u E1 u H 1 (Ω) Proof The proof of the right inequality follows iediately fro the Cauchy Schwarz inequality and a siple trace theore The left inequality is proven by contradiction and by using Rellich s theore as in a proof of generalized Poincaré Friedrichs inequalities, cf, eg, Nečas [21, Chap 27] For such an arguent, it is iportant that the linear functionals l i (u) = (u,r i ) L2(Σ) be bounded on H 1 (Ω); this is a consequence of a Cauchy-Schwarz inequality and the sae trace theore

5 Using analogous arguents, cobined with Lea 2, we obtain: Lea 4 Let Ω IR 3 be a Lipschitz doain of diaeter one and let Σ Ω be of positive easure Then, there exists a positive constant C > 0, such that ) u 2 H 1 (Ω) + u 2 L C 2(Σ) ((ε(u), ε(u)) L2(Ω) + u 2 L 2(Σ) u H 1 (Ω) Using (2) and Leas 2 and 3, in cobination with a scaling arguent, we obtain a Korn inequality on a subspace of H 1 (Ω) Lea 5 Let Ω IR 3 be a Lipschitz doain Then, there exists a constant c > 0, invariant under dilation, such that c u H 1 (Ω) ε(u) L2(Ω) u H 1 (Ω), where u {v H 1 (Ω) : (v,r) L2(Σ) = 0 r ker(ε)} In the following, we will ake extensive use of trace spaces equipped with trace nors We therefore recall soe definitions We will first consider scalar valued Sobolev spaces Let Σ, again, be a subset of Ω with positive easure Then, a seinor, which is equivalent to H 1/2 (Σ) on H 1/2 (Σ), can be defined for u H 1/2 (Σ) as inf v H 1 (Ω) v Σ =u v H 1 (Ω) Clearly, u 2 H 1/2 (Σ) := 3 i=1 u i 2 defines a seinor on the product trace space H 1/2 (Σ) H 1/2 (Σ) := (H 1/2 (Σ)) 3 Another useful seinor on H 1/2 (Σ) is given by u E(Σ) := inf v H 1 (Ω) v Σ =u ε(v) L2(Ω) We will denote by u har and u elast {v H 1 (Ω) : v Σ = u} the haronic and elastic extension of u, respectively, defined by u har H 1 (Ω) = u H 1/2 (Σ) and ε(u elast ) L2(Ω) = u E(Σ) Fro Lea 4, we iediately see that for u H 1/2 (Σ) 5 (6) u 2 H 1/2 (Σ) + u 2 L 2(Σ) u elast 2 H 1 (Ω) + u 2 L 2(Σ) ( ) 1/c ε(u elast ) 2 L + 2(Ω) u 2 L 2(Σ) ( ) = 1/c u 2 E(Σ) + u 2 L 2(Σ) Using these estiates and a standard scaling arguent, we also have a Korn inequality on the trace space H 1/2 (Σ) Lea 6 Let Ω IR 3 be a Lipschitz doain of diaeter H and Σ Ω be an open subset with positive surface easure Then, there exists a constant C > 0, invariant under dilation, such that u 2 H 1/2 (Σ) + 1 H u 2 L C 2(Σ) ( u 2 E(Σ) + 1 ) H u 2 L 2(Σ) where u H 1/2 (Σ) We also have the following Korn inequality

6 6 Lea 7 Let Ω IR 3 be a Lipschitz doain of diaeter H and Σ Ω be an open subset with positive surface easure There exists a positive constant C, independent of H, such that inf r ker(ε) u r 2 L 2(Σ) C H u 2 E(Σ) u H 1/2 (Σ) Proof We first prove the lea for a doain Ω of unit diaeter Let u H 1/2 (Σ) be arbitrary but fixed and let r ker (ε) be the iniizing rigid body ode, for which (u r,r i ) L2(Σ) = 0, i = 1,,6 Then, by using a standard trace theore, and Leas 2 and 3, we obtain u r 2 L 2(Σ) C ( (u r) elast 2 H 1 (Ω) + (u r) elast 2 L 2(Ω) ) C ( u r 2 E(Σ) + (u r) elast 2 L 2(Ω) ) C ( u r 2 E(Σ) + ((u r,r i ) L2(Σ)) 2 ) = C u 2 E(Σ) i=1 We now obtain the result using a standard scaling arguent 3 Finite eleents and geoetry We will only consider copressible elastic aterials It then follows fro Lea 1 that the bilinear for a(, ) is uniforly elliptic and uniforly continuous It is therefore sufficient to discretize our elliptic proble (1) by low order, conforing finite eleents, eg, linear or trilinear eleents Let us assue that a triangulation τ h of Ω is given which is shape regular and has a typical diaeter of h We denote by W h := W h (Ω) H 1 0 (Ω, Ω D) the corresponding conforing finite eleent space of finite eleent functions The corresponding discrete proble is then (7) a(u h,v h ) = F,v h v h W h When there is no risk of confusion, we will drop the subscript h Let the doain Ω IR 3 be decoposed into nonoverlapping subdoains Ω i, i = 1,,N, each of which is the union of finite eleents with atching finite eleent nodes on the boundaries of neighboring subdoains across the interface Γ The interface Γ is the union of subdoain faces, edges, all of the regarded as open sets, and subdoain vertices Faces are sets which are shared by two subregions, edges norally by ore than two subregions, and vertices are endpoints of edges Vectors of interior variables will be equipped with the subscript I We denote the faces of Ω i by F ij, its edges by E ik, and its vertices by V il Subdoain vertices that lie on Ω N are part of Γ, while subdoain faces that are part of Ω N are not; the nodes on those faces will always be treated as interior If Γ intersects Ω N along an edge coon to the boundaries of only two subdoains, we will norally regard it as part of the face coon to this pair of subdoains; if there are ore than two subdoains, it will be regarded as an edge of Γ Siilarly, we will regard a subdoain vertex on Ω N part of an interior edge unless there are several such edges that end at the vertex In the latter case, we treat the vertex the sae way as a vertex in the interior of the doain We note that any subdoain the boundary of which does not intersect Ω D

7 is a floating subdoain, ie, a subdoain for which only natural boundary conditions are iposed These geoetrical entities can also be defined in ters of certain equivalence classes Let us denote the sets of nodes on Ω, Ω i, and Γ by Ω h, Ω i,h, and Γ h, respectively For any interface nodal point x Γ h, we define N x := {j {1,,N} : x Ω j }, ie, N x is the set of indices of all subdoains with x in the closure of the subdoain Associated with the nodes of the finite eleent esh, we have a graph, the nodal graph, which represents the node-to-node adjacency For a given node x Γ h, we denote by C con (x) the connected coponent of the nodal subgraph, defined by N x, to which x belongs For two interface points x, y Γ h, we introduce an equivalence relation by x y : N x = N y and y C con (x) We can now describe faces, edges, and vertices using their equivalence classes Here, G denotes the cardinality of the set G We find that x F : N x = 2 x E : N x 3 and y Γ h, y x, such that y x x V : N x 3 and y Γ h, such that x y In our theoretical analysis, we assue that each subregion Ω i is the union of a nuber of shape regular tetrahedral coarse eleents and that the nuber of such tetrahedra is uniforly bounded for each subdoain Therefore, the subregions are not very thin and we can also easily show that the diaeters of any pair of neighboring subdoains are coparable In such a case, our definition of faces, edges, and vertices confor with our basic geoetric intuition On the other hand, for subdoains generated by an autoatic esh partitioner, the situation can be quite coplicated We can, eg, have several edges with the sae index set N x or an edge and a vertex with the sae N x In practice, we can also have situations when there are not enough edges and potential edge constraints for soe subdoains Then, we have to use constraints on soe extra edges on Ω N, which otherwise would be regarded as part of a face; see above We denote the standard finite eleent space of continuous, piecewise linear functions on Ω i by W h (Ω i ); we always assue that these functions vanish on Ω D For siplicity, we assue that the triangulation of each subdoain is quasi unifor The diaeter of Ω i is H i, or generically, H We denote the corresponding finite eleent trace spaces by W (i) := W h ( Ω i Γ), i = 1,,N, and by W := N i=1 W(i) the associated product space We will often consider eleents of W, which are discontinuous across the interface For each subdoain Ω i, we define the local stiffness atrix, which we view as an operator on W h (Ω i ) On the product space N i=1 Wh (Ω i ), we define the operator K as the direct su of the local stiffness operators, ie, (8) K := N i=1 In an ipleentation, K corresponds to a block diagonal atrix since, so far, there is no coupling across the interface The finite eleent approxiation of the elliptic 7

8 8 proble is continuous across Γ and we denote the corresponding subspace of W by Ŵ We note that while the stiffness atrix K and its Schur copleent, obtained after eliinating the variables interior to the subregions, which corresponds to the product space W, are singular if there are any floating subdoains, those of Ŵ are not In the present study, as in others of FETI DP ethods, we also work with subspaces W W for which sufficiently any constraints are enforced so that the resulting leading diagonal block atrix of the FETI saddle point proble, to be introduced in (20), though no longer block diagonal, is strictly positive definite These are called prial constraints and in our discussion they usually consist of certain edge averages and first order oents, which have coon values across the interface of neighboring subdoains, and possibly of constraints at well chosen subdoain vertices (or other nodes), for which a partial subassebly is carried out One of the benefits of working in W, rather than in W, will be that certain related Schur copleents, S ε and S ε, are strictly positive definite; cf (10) and (12) We further introduce two subspaces, Ŵ Π Ŵ and W, corresponding to a prial and a dual part of the space W These subspaces play an iportant role in the description and analysis of our iterative ethod We note that the dual subspace W will be directly associated with jups across the interface and with the Lagrange ultipliers that are introduced to eliinate these jups The direct su of these spaces equals W, ie, (9) W = ŴΠ W W (i) The second subspace, W, is the direct su of local subspaces of W, where (i) each subdoain Ω i contributes a subspace W ; only its i-th coponent in the sense of the product space W is non trivial We now define certain Schur copleent operators by using a variational forulation; for a atrix representation, see Section 6 Here,, will denote the l 2 inner product We first define Schur copleent operators S ε (i), i = 1,,N, operating on W (i), by (10) S (i) ε w (i),w (i) = in v (i),v (i) w (i) W (i), where we take the iniu over all v (i) W h (Ω i ) with v (i) Γ = w (i) We can now define the Schur copleent S ε operating on W as the direct su of the local Schur copleents (11) S ε := N i=1 S (i) ε Next, we introduce a positive definite Schur copleent S ε, operating on W, by a variational proble: for all w W, (12) S ε w,w = in w Π Ŵ Π S ε (w + w Π ),w + w Π We note that any Schur copleent of a positive definite, syetric atrix is always associated with such a variational proble We also obtain, analogously, a reduced right hand side f, fro the load vectors associated with the individual subdoains

9 We now consider the relation between the Schur copleent of the elasticity stiffness atrix, S ε, and S the one arising fro discretizing a vector valued Laplace operator scaled by the values of G Obviously, S can also be defined as the direct su of local Schur copleents (13) S := N S (i), i=1 where the S (i) are again given by a variational arguent as in (10), using the discrete, scaled, vector valued Laplace operator instead of We furtherore introduce bilinear fors which represent the contributions of the individual subdoains to the bilinear fro a(u,v): with G i := Ei 1+ν i, β i := νi 1 2ν i, and a(u,v) := N a i (u,v) i=1 a i (u,v) := G i ( (ε(u), ε(v))l2(ω i) + β i (div (u), div (v) L2(Ω i)) We will assue that G i and β i are constant on the subdoain Ω i We obviously have for u W h (14) u 2 S ε ax (1 + β i) u 2 S i=1,,n In order to define certain scaling operators, we need to define weighted counting functions δ i for each subdoain Ω i These are functions in the scalar finite eleent trace space on Ω i They are defined, for γ [1/2, ), by j N δ i (x) := x G γ j (15) (x) G γ i (x), x Ω i,h Γ h Here, as before, N x is the set of indices of the subregions which have x on its boundary This forula can also be used for aterial coefficients G i which vary over the boundary of the subdoains but in our theory, we only consider the case when the coefficients are constant in each subdoain We note that any node of Γ h belongs either to a face coon to two subdoains, to an edge coon to at least three subregions, or is a vertex of several substructures The pseudo inverses δ i are defined as (16) δ i (x) = δ 1 i (x), x Ω i,h Γ h We further introduce an extension operators Ri T : W (i) Ŵ, such that the continuous global function Ri Tw i Ŵ shares the nodal values with w i on Ω i,h Γ h and vanishes at all other nodes of Γ h We note that these functions provide a partition of unity: (17) Ri T (δ i (x)1) 1 x Γ h, i where 1 Ŵ is the vector valued function with coponents equal to one at every point of Γ h For γ 1/2, we can easily show that 9 (18) G i (δ k )2 in(g i, G k )

10 10 4 The dual-prial FETI ethod We reforulate the original finite eleent proble, reduced to the degrees of freedo of the second subspace W, as a iniization proble with constraints given by the requireent of continuity across all of Γ h : find u W, such that (19) J(u ) := 1 2 S ε u,u f,u in B u = 0 The jup operator B operates on W and enforces pointwise continuity of the dual displaceent degrees of freedo At possible prial vertices, continuity is already enforced by subassebly and a jup operator applied to a function fro W would autoatically be zero at these special degrees of freedo By introducing a set of Lagrange ultipliers λ V := range (B ), to enforce the constraints B u = 0, we obtain a saddle point forulation of (19): (20) [ Sε B T O B ] [ u λ ] = [ f 0 We note that we can add any eleent fro ker (B T ) to λ without changing the displaceent solution u Since S ε is invertible, we can eliinate u and obtain the following syste for the Lagrange ultiplier variables: ] } (21) Fλ = d Here, our new syste atrix F is defined by (22) F := B S 1 ε B T and the new right hand side by d := B S 1 f ε Algorithically, S ε is only needed 1 in ters of S ε ties a vector w W and such an operation can be excecuted relatively inexpensively; see Section 6 The operator F will obviously depend on the choice of the subspaces ŴΠ and W To define the FETI-DP Dirichlet preconditioner, we need to introduce scaled jup operators B D, := [B (1) D,,, B(N) D, ] Here, the B (i) D, are defined as follows: each row of B(i) with a nonzero entry corresponds to a Lagrange ultiplier connecting the subdoain Ω i with a neighboring subdoain Ω j at a point x Ω i,h Ω j,h Multiplying each such row of B (i) with δ j (x) gives us B(i) D, As in Klawonn and Widlund [13, Section 5], we solve the dual syste (21) using the preconditioned conjugate gradient algorith with the preconditioner (23) M 1 := PB D, S ε B T D, P T, where P is the l 2 -orthogonal projection fro range (B D, ) onto V = range (B ), ie, P reoves the coponent fro ker(b T ) of an eleent in range (B D, ) We note that P and P T are only needed for the theoretical analysis to guarantee that the

11 preconditioned residuals will belong to V; cf reark after (20) The projections can be dropped in the ipleentation; cf the arguent at the end of this section This definition of M 1 clearly depends on the choice of the subspaces ŴΠ and W The FETI-DP ethod is the standard preconditioned conjugate gradient algorith for solving the preconditioned syste M 1 Fλ = M 1 d Algorith 1 (i) Initialization: r 0 := d Fλ 0 (ii) Iterate for k = 1, 2,, until convergence, z k 1 := M 1 r k 1 β k := zk 1,r k 1 z k 2,r k 2 [β 1 := 0] p k := z k 1 + β k p k 1 [p 1 := z 0 ] α k := zk 1,r k 1 p k, Fp k λ k r k := λ k 1 + α k p k := r k 1 α k Fp k We note that in the ipleentation of our preconditioner M 1, we can drop the projection operator P and its transpose as can be seen by the following arguent Applying B D, S ε BD, T to an eleent fro V results in a vector µ which can be written as a su µ = µ 0 +µ 1 of coponents µ 0 ker (B T ) and µ 1 V = range (B ) When F is applied to µ, the coponent Fµ 0 vanishes and we also have Fµ V Exaining Algorith 1, we can also easily see that dropping P and P T only affects the coputed Lagrange ultiplier solution but not the coputed displaceents The residuals r k are always in V and it is easy to show that the α k and β k are not affected 5 Selection of constraints In order to control the rigid body odes of a subregion, we need at least six constraints To get an understanding of the type of prial constraints that are required to ake our preconditioner effective, it is useful to exaine two special cases In the first, we assue that we have two subdoains ade of the sae aterial, which have a face in coon and are surrounded by subdoains ade of a aterial with uch saller Young s odulus E Such a proble will clearly have six low energy odes corresponding to the rigid body odes of the union of the two special subdoains Any preconditioner that has less than six linearly independent prial constraints across that face will have at least seven low energy odes and will be far fro spectrally equivalent to the original finite eleent odel In the second case, we again consider two subdoains surrounded by subdoains with uch saller stiffnesses, ie, Young s oduli We now assue that the two special subdoains share only an edge In this case, there are seven low energy odes of the finite eleent odel corresponding to the sae rigid body odes as before and an additional one The new ode corresponds to a relative rotation of the two subdoains around their coon edge We conclude that in such a case, we should introduce five linearly independent prial constraints related to the special edge 11

12 12 In the convergence theory presented in Section 8, we will first assue that the requireent of the first special case is et for each face, ie, there are at least six linearly independent edge constraints for each face of the interface Any such face will be called fully prial, cf Definition 1 We note that any such edge constraint will serve as a constraint for every face adjacent to the edge in question We do not have to ake every face fully prial but, for every face, we have to have an acceptable face path, cf Definition 3 and Section 83 We also note that using only constraints based on averages over faces ight not always lead to a robust algorith with respect to jups in the stiffnesses of different aterials; see [16, 12] For coefficient distributions with only odest jups across the interface Γ and for soe special decopositions, we are able to work exclusively with edge averages; cf Section 81 To be able to treat general coefficient distributions with arbitrarily large jups, we also need first order oents, in addition to the averages, on certain edges such as those in the second special case discussed above All these constraints will be written in ters of inner products of rigid body odes and the displaceent over individual edges There will be only five linearly independent constraints of this type since, restricted to an edge, one rotational rigid body ode is always linearly dependent of the others This can be seen easily by a direct coputation or by a change of coordinates such that the chosen edge coincides with the x 1 axis of the Cartesian coordinate syste; then the third rotation r 6 will vanish and the relevant first order oents are with respect to the second and third displaceent coponents Such an edge with three edge average constraints and two first order oent constraints will be called fully prial, cf Definition 2 An edge will be called prial if there is at least one constraint, expressed in ters of an average, for at least one of its displaceent coponents As with the fully prial faces, we do not have to ake every edge fully prial Instead, we can ake sure that there is an acceptable face path; cf Definition 3 Finally, to be able to treat the ost general distribution of coefficients, it can be necessary to ake soe vertices prial and we need the concept of an acceptable vertex path; cf Definition 4 Definition 1 (Fully prial face) Let F ij be a face A set f, = 1,,6, of linearly independent linear functionals on W (i) is called a set of prial constraint functionals on that face if it has the following properties: (i) f (w (i) ) 2 C H 1 i (1 + log(h i /h i )){ w (i) 2 H 1/2 (F ij ) + 1 H i w (i) 2 L 2(F ij ) } (ii) f (r l ) = δ l, l = 1,,6, r l ker (ε) Such a face is called a fully prial face We will soeties write f Fij instead of f As an exaple of functionals f, as considered in Definition 1, we can use appropriately chosen linear cobinations of certain edge averages, g n, of coponents of the displaceent, g n (w (i) ) := E ik w (i) l dx, n = 1,, 6, 1dx E ik for a function w (i) = (w (i) 1, w(i) 2, w(i) 3 ) W(i) and appropriately chosen edges E ik which belong to the boundary of the face F ij We can show that in order to obtain six linearly independent linear functionals associated with a rectangular face F ij, we have to work with at least three different edges E ik The functionals g 1,, g 6, provide a basis of the dual space (ker (ε)) There also exists a dual basis of (ker (ε)), which we denote by f 1,, f 6, defined by f (r l ) = δ l,, l = 1,,6; thus, there exist β lk IR, l, k = 1,,6, such that for w W (i),

13 13 we have (24) f (w) = β n g n (w), = 1,,6 n=1 Using a Cauchy-Schwarz inequality, we obtain g (w (i) ) 2 C H 1 i w (i) 2 L 2(E ik ) We can then show, by using Lea 11, that w (i) 2 L 2(E ik ) C (1 + log(h i/h i ))( w (i) 2 H 1/2 (F ij ) + 1 H i w (i) 2 L 2(F ij ) ) Thus, the first requireent of Definition 1 is satisfied for the functionals f It is also possible to construct five of the functionals f in Definition 1 using five constraints, three averages and two first order oents, on one single edge Before we discuss this possibility, we introduce the definition of a fully prial edge Definition 2 (Fully prial edge) Let F ij be a face and E ik an edge which belongs to the boundary of F ij A set f, = 1,, 5, of linearly independent linear functionals on W (i) is called a set of prial constraint functionals on the edge E ik if it has the following properties: (i) f (w (i) ) 2 C H 1 i (1 + log(h i /h i )){ w (i) 2 H 1/2 (F ij ) + 1 H i w (i) 2 L 2(F ij ) } (ii) f (r l ) = δ l, l = 1,,5, r l ker (ε) Such an edge is called a fully prial edge We will soeties write f Eik instead of f We recall that the rigid body odes r 1,,r 6, restricted to an edge provide only five linearly independent vectors, since one rotation is always linearly dependent of other rigid body odes In our arguents, we will assue that we have used an appropriate change of coordinates such that the edge under consideration coincides with the x 1 -axis; the special rotation is then r 6 As an exaple of functionals f, as required in Definition 2, we can use appropriate linear cobinations of certain edge averages and first order oents of the coponents of the displaceent given by (25) g n (w (i) ) := (w(i),r n ) L2(E ik ), n {1,,5} (r n,r n ) L2(E ik ) Using a Cauchy-Schwarz inequality, we obtain g n (w (i) ) 2 w(i) 2 L 2(E ik ) r n 2 L 2(E ik ) We can now proceed exactly as in the case of the fully prial faces and construct the functionals f in ters of a dual basis of (ker (ε)) Then, the second requireent of Definition 2 is again iediately satisfied by construction and the first requireent follows again fro Lea 11 We do not have to require that every face and edge be fully prial but we need the concept of an acceptable face path for those that are not Definition 3 (Acceptable face path) Consider a pair of subdoains (Ω i, Ω k ) which have a face or an edge in coon An acceptable face path {Ω i,ω j1,,ω jn, Ω k }

14 14 for this pair is a path fro Ω i to Ω k, via a uniforly bounded nuber of other subdoains Ω jq, q = 1,,n, such that the coefficients G jq of Ω jq satisfy the condition (26) TOL G jq in(g i, G k ) q = 1,,n, for soe tolerance TOL The path can pass fro one subdoain to another only through a fully prial face; cf Figure 1 Ð fully prial face ½ ¾ Ð fully prial face ½ fully prial face ¾ fully prial face dual face fully prial face fully prial face Ð fully prial face Ð Ð Ð dual edge fully prial face fully prial face Ð Fig 1 Exaples of acceptable face paths (planar cut): dual face (left) and dual edge (right) We note that any fully prial face has a trivial acceptable face path An edge should either be fully prial or have an acceptable face path Finally, we have to consider vertices, where we need to control only translational rigid body odes A vertex is prial if the three displaceent coponents are continuous These variables are then global and this is reflected in the subassebly of the stiffness atrix of the preconditioner We do not have to ake a vertex prial if for every pair of subregions, which have the vertex in coon, there is an acceptable vertex path Definition 4 (Acceptable vertex path) Consider a pair of subdoains (Ω i, Ω k ), which have a vertex in coon An acceptable vertex path {Ω i, Ω j1,, Ω jn, Ω k } fro Ω i to Ω k, is a path via a uniforly bounded nuber of other subdoains Ω jq, q = 1,,n, such that the coefficients G jq of Ω jq satisfy the condition (27) TOL G jq h i H i in(g i, G k ) q = 1,,n, for soe tolerance TOL We can only pass fro one subdoain to another through a fully prial face We will assue that for each pair (Ω i, Ω k ), which has a face, an edge or a vertex, in coon, there either exists an acceptable path as in Definitions 3 and 4, respectively, with a odest tolerance TOL and that the path does not exceed a prescribed length, or that the face or edge are fully prial or the vertex prial (In Subsection 84, we will look in ore detail at the consequences of long paths) If T OL becoes too large for a certain face, edge, or vertex or if the length of the acceptable path exceeds a given unifor bound, we should ake the face or edge fully prial, or the vertex prial; cf Figure 2 for an exaple where certain vertices should be ade prial We note that the bounds for the prial constraint functionals in Definitions 1 and 2 will allow us to prove alost unifor bounds for the condition nuber of our algoriths If point constraints were to replace the edge constraints, this would not be possible We note that while we will work with functionals which are not

15 Fig 2 Exaple of a decoposition where no acceptable vertex path exists for the vertices which connect the black subdoains, which have uch larger coefficients than those of the white These vertices should be ade prial uniforly bounded, the growth of these bounds is quite odest when H/h grows These growth factors will appear in the ain theore as is custoary for any doain decoposition ethods We also note that the logarithic factors cannot be eliinated if we wish to obtain a result which is unifor with respect to arbitrary variations of the Laé paraeters Finally, we define the spaces ŴΠ and W = N (i) i=1 W For the definition of these spaces, we will use certain standard scalar finite eleent cutoff functions θ F ij, θ E ik, and θ V il The first two equal one on F ij and E ik, respectively, and vanish elsewhere on Γ h ; the cutoff function θ V il equals one at the vertex and vanishes elsewhere on the interface Additionally, for an edge, we denote by E ik a linear function which equals 1 at one end of the edge and 1 at the other The first space, Ŵ Π, is spanned coponent by coponent by the nodal finite eleent functions θ V il which are associated with vertices V il that have been chosen to be prial, by the cutoff functions θ E ik for the prial edges, and for fully prial edges by θ E ik and I h ( E ikθ E ik) Each such prial constraint is associated with a basis eleent of ŴΠ; all these functions are continuous across the interface Γ For each subdoain Ω i, we (i) then define a subspace W by those functions in W(i) which vanish at the prial variables, ie, these functions vanish at prial vertices and have zero averages over prial edges and additionally certain zero first order oents over fully prial edges More details will be provided especially in Section 8 6 Linear algebra aspects of the FETI-DP ethod In this section, we first introduce atrix representations of the operators used in the description of our FETI- DP algorith and given in Section 4 We then describe two ways of ipleenting these algoriths The atrix representing the jup operator B is constructed fro {0, 1, 1}, in such a way that the values of the solution u, associated with ore than one subdoain, coincide when B u = 0 These constraints are very siple and just express that the nodal values coincide across the interface; in coparison with the one-level FETI ethod, cf [13], we can drop soe of the constraints, in particular those associated with the prial vertices However, we will otherwise use all possible constraints and thus work with a fully redundant set of Lagrange ultipliers as in [13, Section 5]; cf also [23] Thus, for an edge node coon to four subdoains, we will use six constraints rather than choosing just three To define the FETI-DP Dirichlet preconditioner, we also need to introduce a atrix representation of the scaled jup operator B D, ; this is done by scaling the contributions of B fro individual subdoains Additionally, we add a zero colun to B D, for each prial

16 16 vertex variable Let us now consider the atrix representation of our preconditioner M 1 We need the atrix for of S ε ; this is a Schur copleent atrix, which is obtained fro the block diagonal atrix K, (8), by eliinating the interior variables The associated block diagonal atrix is given by S ε := diag N i=1 (S(i) ε ) Each local stiffness atrix can be written as [ = II ΓI T ΓI ΓΓ ] Then, each local Schur copleent atrix S (i) ε S (i) ε can be written as = ΓΓ K(i) ΓI (K(i) II ) 1 T ΓI Thus, we can copute S ε ties a vector w W by solving local Dirichlet probles and foring soe sparse atrix-vector products Our preconditioner is then given in atrix for by M 1 = PB D, S ε B T D, P T Finally, we have to consider the syste atrix F = B S 1 ε B T We have to describe 1 how the edge (and face) constraints can be ipleented and how S ε ties a vector can be coputed efficiently There are two different approaches to the edge and face constraints, one using optional Lagrange ultipliers, which will for a part of the global, coarse proble, and the other which uses a change of basis; cf Subsections 61 and 62, respectively The second approach generally leads to saller and coputationally ore efficient coarse probles; with this approach, the invertibility of the local probles and the positive definiteness of the entire proble can be guaranteed without any vertex constraints In fact, vertex constraints are only needed for probles with very challenging distributions of the aterial coefficients We will outline two different ways of ipleenting the change of basis; we can either explicitly carry out the basis transforation on the prial and fully prial edges for both the prial and dual displaceent variables, or one can apply the transforation explicitly only for the prial displaceent variables and use local Lagrange ultipliers to enforce zero edge averages and first order oents for the dual displaceents The latter approach has the advantage of retaining ore of the original sparsity of the stiffness atrices 61 An ipleentation using global optional Lagrange ultipliers We first briefly review the approach taken by Farhat, Lesoinne, and Pierson in [9] They assue that a sufficient nuber of vertices have been chosen as prial variables such that the stiffness atrix which results fro K by partial assebly at those vertices is invertible even without any additional prial constraints In two diensions, such a set of vertex constraints is sufficient to obtain a fast and scalable algorith but in three diensions, to be copetitive, we have to choose a prial space, which also ensures that certain face and/or edge averages and first order oents are the sae across the interface This approach can be ipleented by introducing additional

17 17 optional Lagrange ultipliers originating fro constraints of the for (28) Q B u = N i=1 Q B (i) u(i) = 0 Here, Q is a rectangular atrix which has as any coluns as there are Lagrange ultipliers It has one row for each prial constraint, which is not related to a vertex and the atrix Q is constructed such that (28) guarantees that certain linear cobinations of the rows of B u vanish Thus, these linear cobinations are directly related to the constraints of the prial edges and faces; (28) forces appropriate edge averages and first order oents at fully prial edges or faces to have coon values across the interface We note that this approach could also be used for coon face averages at selected faces across the interface; here we will work exclusively with edge averages and oents Let us now order the unknowns such that the interior and dual variables coe first, grouped together in blocks by the subdoains and denoted with the subscript r, and that the prial vertices, with the subscript c, are ordered last We note that the atrix K is partially assebled with respect to the prial vertices Thus, we have K := K rr KT cr Q T r K cr Kcc O Q r O O, where K rr := K (1) rr O O K (N) rr, K(i) rr := [ II I T I ], K T cr := K cr (1)T R c (1)T K cr (N)T R c (N)T, K cc = N i=1 R (i) c cc R c (i) T, Qr := [Q (1) r Q (N) r ], and Q (i) r := [O Q B (i) ] Here, we denote by R c (i) the atrix which perfors the partial assebly at the relevant prial vertices and K cc is the subatrix which is assebled at the prial vertices The resulting leading two by two block of K is thus non singular Using the notation B r := [B r (1) B r (N) ] with B r (i) := [O B (i) ], for a atrix with sae structure as Q r and introducing Lagrange ultipliers λ range (B r ), we can reforulate the original finite eleent proble as follows K rr KT cr Q T r Br T K cr Kcc O O Q r O O O B r O O O u r u c µ λ = We can now derive the unpreconditioned FETI-DP linear syste by eliinating the variables u r,u c, and µ to obtain Fλ = d In order to exploit the sparsity of K, the Schur copleent S ε is never built explicitly In fact, we only need to be able to copute the action of S 1 ε f r f c 0 0 on a vector f This can be done in a coputationally

18 18 efficient way as described at the end of Section 62 We note that although the eliination of the interior and dual variables u r leads to an indefinite syste with respect to the prial variables u c and the optional global Lagrange ultipliers µ, it can still be solved without pivoting for stability Instead, we can syetrically reorder a large leading principal inor of our atrices so as to aintain sparsity Since the arguent is the sae as for a variant described in the next section, we refer to the discussion at the end of Section An ipleentation using a change of basis As a second approach, we present a ethod which uses a change of basis to force certain edge averages and first order oents to vanish This change of basis will introduce the edge averages and oents as new prial variables and explicitly separate the dual fro the prial variables As a consequence, this will allow us to write our syste atrix in a block structured for with respect to the interior, dual, and prial variables Two variants are discussed, one where the transforation is carried out for the prial and selected dual displaceent variables and a second one, where local Lagrange ultipliers are used to enforce certain zero edge averages and oents instead of explicitly perforing the change of basis for the associated dual displaceent variables Both approaches generally lead to saller and coputationally ore efficient coarse probles Such an ipleentation also works for face constraints instead of or in addition to edge constraints but since in this article we only consider edge based algoriths, we restrict ourselves to the case of edges only 621 First approach We first describe the approach using an explicit change of basis As a result, the dual displaceent vectors should have zero edge averages over prial edges and the selected displaceent coponents and additionally zero first order edge oents over fully prial edges In addition, we introduce these averages and oents as prial variables in ŴΠ We first describe how the transforation atrix for such a change of basis can be built We first consider the unknowns u E on a fully prial edge E It is sufficient to describe the transforation of a single coponent of u E = (u T 1,E, ut 2,E, ut 3,E )T subject to two constraints; we note that the edge average constraints are constructed for each of the three coponents whereas the constraints of first order oents are only used for two appropriately chosen coponents; cf Section 5 For siplicity, we drop the subscripts indicating the coponent and consider only a scalar vector of unknowns u E = (u 1 u N ) T We define a transforation atrix T E which perfors the desired change of basis In the new basis, we introduce the edge average ū a E and the first order edge oent ū E as new variables Additionally, the representation of the dual part of u E in the new basis should have a zero edge average and a zero first order edge oent Our transforation atrix T E perfors the change of basis fro the new basis to the original nodal basis If we denote the edge unknowns in the new basis by û E, we will have where u E = T E û E, T E = [t 1 t N 2 t N 1 t N ] We consider an edge coponent with two constraints and can define T E in ters of second-order differences The first N 2 vectors t j are defined by first setting the j-th vector t j to zero at all but the j-th and its next two coponents We set the j-th

19 entry to one and define the next two entries such that t j has a zero average and a zero first order oent The vector t N 1 is defined as being one at each esh point of that edge and the last vector, t N, is obtained by evaluating, at the esh points, the linear function which is orthogonal to t N 1 in the L 2 -inner product; cf the definition of E ik at the end of Section 5 A siilar construction can be carried out for a prial edge Then, only edge averages are introduced as new variables and the reaining new variables should have zero edge average In this case the first N 1 coluns of T E are defined by setting all except for the j-th and (j+1)-th coponent to zero and aking its average zero The last colun t N is then obtained by setting all entries to one Such a transforation can be constructed separately for each coponent of u E = (u T 1,E, ut 2,E, ut 3,E )T and for each edge with prial edge constraints We denote the resulting transforation, which operates on all relevant coponents of u E and all relevant edges, by T (i) E The transforation for all variables of one subdoain Ω i is then of the for T (i) = I O O O I O O O T (i) E where we assue that the variables are ordered interior variables first, interface variables not related to the (fully) prial edges second, and the variables on the (fully) prial edges last, ie, a typical vector of nodal unknowns is of the for [u (i)t I,u (i)t,u (i)t Γ E ]T We note that T (i) E is a direct su of the relevant transforation atrices associated with the prial and fully prial edges of that subdoain; T (i) E, 19 is a block-diagonal atrix where each block represents the transforation for a coponent of a prial or fully prial edge Decoposing the subdoain stiffness atrices in the sae anner, we obtain = II ΓI EI IΓ ΓΓ EΓ IE ΓE EE Using the transforation u (i) = T (i) û (i), we obtain T (i)t T (i) = II ΓI T (i)t E K(i) EI IΓ ΓΓ T (i)t E K(i) EΓ IE T (i) E T (i) ΓE E T (i)t E K(i) EE T (i) E The averages and oents are now new prial variables We note that there ight be additional prial variables, eg, selected prial vertices The prial variables belonging to Ω i are denoted by û (i) Π and the reaining, dual displaceent variables by û (i) By construction, the new dual displaceent variables û(i) satisfy the zero edge average and oent properties Using this decoposition of the unknowns into interior, dual, and prial displaceent variables, the transforation atrix T (i) E can be written as [T (i) E T (i) Π E ] Here, the indices E and Π E indicate the dual and