Chemnitz Scientific Computing Preprints

S. Beuchler T. Eibner U. Langer

Primal and Dual Interface Concentrated Iterative Substructuring Methods

CSC/07-04

Chemnitz Scientiﬁc Computing Preprints Chemnitz Scientiﬁc Computing Preprints – ISSN 1864-0087 Impressum:

Chemnitz Scientific Computing Preprints — ISSN 1864-0087 (1995–2005: Preprintreihe des Chemnitzer SFB393) Herausgeber: Postanschrift: Professuren für TU Chemnitz, Fakultät für Mathematik Numerische und Angewandte Mathematik 09107 Chemnitz an der Fakultät für Mathematik Sitz: der Technischen Universität Chemnitz Reichenhainer Str. 41, 09126 Chemnitz http://www.tu-chemnitz.de/mathematik/csc/ Chemnitz Scientific Computing Engineering XVI, volume 55 of Lecture Notes in Computational Science and Engineering, pages 405–412. Springer-Verlag, 2007. Preprints [24] U. Langer and O. Steinbach. Boundary element tearing and interconnecting method. Computing, 71(3):205–228, 2003. [25] U. Langer and O. Steinbach. Coupled finite and boundary element domain decomposition methods. In In ”Boundary Element Analysis: Mathematical Aspects and Application”, ed. by M. Schanz and O. Steinbach, Lecture Notes in Applied and Computational Mechanic, Volume 29, S. Beuchler T. Eibner U. Langer pages 29–59, Berlin, 2007. Springer. [26] J. Mandel and C.R. Dohrmann. Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Lin. Alg. Appl., 10:639–659, 2003. [27] A. M. Matsokin and S. V. Nepomnyaschikh. The Schwarz alternation method in a subspace. Iz. VUZ Mat., 29(10):61–66, 1985. Primal and Dual Interface Concentrated [28] A. M. Matsokin and S. V. Nepomnyaschikh. The fictitious domain method and explicit continu- ation operators. Comput. Maths Math. Phys., 33(1):45–59, 1993. Iterative Substructuring Methods [29] J. M. Melenk, C.G. A. Pechstein, J. Schöberl, and S.C. Zaglmayr. Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. Technical Report Report RICAM 2005/11, Johann Radon Institute for Computational and Applied Mathematics, Linz, December 2005. accepted for publication in IMA J. Num. Anal. ´ [30] J. Neˇcas. Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Editeurs, CSC/07-04 Paris, 1967. [31] S.V. Nepomnyaschikh. Domain Decomposition and Schwarz Methods in a Subspace for the Ap- proximate Solution of Elliptic Boundary Value Problems. PhD thesis, Computing Center of Siberian Division of Russian Academy of Sciences, 1986. [32] S.V. Nepomnyaschikh. Method of splitting into subspaces for solving elliptic boundary value problems in compllex-form domains. Sov. J. Numer. Anal. Math. Modelling, 6:151–168, 1991. 2. [33] L. F. Pavarino. Additive schwarz methods for the p-version finite element method. Numer. Math., Abstract 66(4):493–515, 1994. This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed [34] Y. Saad. Iterative methods for sparse linear systems. Frontiers in Applied Mathematics. PWS, on the basis of a non-overlapping domain decomposition where a conforming Boston, 1992. boundary concentrated finite element approximation is used in every subdomain. Similar to data-sparse boundary element domain decomposition meth- [35] C.H. Tong, T.F. Chan, and C.-C.J. Kuo. A domain decomposition preconditioner based on a ods the total number of unknowns per subdomain behaves like O((H/h)(d 1)), change to a multilevel nodal basis. SIAM J. Sci. Statist. Comput., 12(6):1486–1495, 1991. − where H, h, and d denote the usual scaling parameter of the subdomains, the [36] A. Toselli and O. Widlund. Domain Decoposition Methods - Algorithms and Theory, volume 34 average discretization parameter of the subdomain boundaries, and the spatial of Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, 2004. dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost [37] H. Yserentant. Coarse grid spaces for domains with a complicated boundary. Numer. Algorithms, the same complexity as the number of unknowns. In particular, the so-called 21(1-4):387–392, 1999. All-Floating Finite Element Tearing and Interconnecting solvers are highly par- [38] W. Zulehner. Analysis of iterative methods for saddle point problems: A unified approach. allel and very robust with respect to large coefficient jumps. Mathematics of Computation, 71:479–505, 2002.

26 CSC/07-04 ISSN 1864-0087 April 2007 Sven Beuchler [4] S.C. Brenner. An additive Schwarz preconditioner for the FETI method. Numerische Mathematik, Johannes Kepler University Linz 94(1):1–31, 2003. Institute of Computational Mathematics [5] C.R. Dohrmann. A preconditioner for substructuring based on constrained energy minimization. Altenberger Straße 69, SIAM J. Sci. Comput., 25(1):246–258, 2003. A-4040 Linz, Austria [6] M. Dryja. A capacitance matrix method for Dirichlet problem on polygon region. Numer. Math., [email protected] 39(1):51–64, 1982. Tino Eibner [7] M. Dryja. A finite element-capacitance matrix method for elliptic problems on regions partitioned TU Chemnitz into substructures. Numer. Math., 44:153–168, 1984. Fakultät für Mathematik [8] T. Eibner and J. M. Melenk. Multilevel preconditioning for the boundary concentrated hp-fem. Reichenhainer Straße 39/41 Technical Report 05-13, SFB 393, TU Chemnitz, 2005. submitted. D-09107 Chemnitz, Germany [email protected] [9] T. Eibner and J.M. Melenk. A local error analysis of the boundary-concentrated hp-FEM. IMA J. Numer. Anal., 27(1):752–778, 2007. Ulrich Langer [10] C. Farhat, M. Lesoinne, and K. Pierson. A scalable dual-primal domain decomposition method. Johannes Kepler University Linz Numer. Linear Algebra Appl., 7:687–714, 2000. Institute of Computational Mathematics and [11] C. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int.J.Numer.Meth.Engng., 32:1205 – 1227, 1991. Johann Radon Institute for Compuational and Applied Mathematics Austrian Academy of Sciences [12] G. Haase. Hierarchical extension operators plus smoothing in domain decomposition precondi- Altenberger Straße 69,A-4040 Linz, Austria tioners. Applied Numerical Mathematics, 20:1–20, 1996. [email protected] [13] G. Haase, U. Langer, and A. Meyer. The approximate Dirichlet decomposition method. part I : An algebraic approach. Computing, 47:137–151, 1991. [14] G. Haase, U. Langer, and A. Meyer. The approximate Dirichlet domain decomposition method. II. Applications to 2nd-order elliptic BVPs. Computing, 47(2):153–167, 1991. [15] G. Haase, U. Langer, A. Meyer, and S.V. Nepomnyaschikh. Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners. East-West Journal of Numerical Mathematics, 2:173–193, 1994. [16] G. Haase and S.V. Nepomnyaschikh. Extension explicite operators on hierarchical grids. East- West Journal of Numerical Mathematics, 5:231–248, 1997. [17] W. Hackbusch, B.N. Khoromskij, and R. Kriemann. Direct Schur complement method by domain decomposition based on -matrix approximation. Comput. Vis. Sci., 8(3-4):179–188, 2005. H [18] B.N. Khoromskij and J.M. Melenk. Boundary concentrated finite element methods. SIAM J. Numer. Anal., 41(1):1–36, 2003. [19] A. Klawonn and O.B. Widlund. A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity. SIAM J. Sci. Comput., 22(4):1199–1219, 2000. [20] A. Klawonn and O.B. Widlund. FETI and Neumann–Neumann iterative substructuring methods: connections and new results. Comm. Pure Appl. Math., 54:57–90, 2001. [21] V.G. Korneev and U. Langer. Domain decomposition and preconditioning. In E. Stein, R. de Borst, and Th.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, volume 1 of Chapter 22. John Wiley & Sons, 2004. [22] U. Langer, G. Of, O. Steinbach, and W. Zulehner. Inexact data-sparse boundary element tearing and interconnecting methods. SIAM Journal on Scientific Computing, 29(1):290–314, 2007. [23] U. Langer, G. Of, O. Steinbach, and W. Zulehner. Inexact fast multipole beti methods. In Olof B. Widlund and David E. Keyes, editors, Domain Decomposition Methods in Science and

9 mat=[1 2 3 4] 1 Introduction mat=[1 10 100 1000] 8 mat=[1 1000 100 10] mat=[1 10 1000 100] 2 3 7 mat=[1 100 100 100 ] Boundary concentrated finite element methods (BC-FEM) introduced and investigated by Khoromskij and Melenk in [18] allow us to solve potential problems with smooth coefficients 6 as accurate as the corresponding standard finite element methods (FEM) but with a signifi- 5 cantly smaller number of unknowns. Indeed, the total number of unknowns in the BC-FEM (d 1) 4 is proportional to O(h− − ) whereas the total number of unknowns in the standard FEM maximal eigenvalue d behaves like O(h− ), where h and d denote the the average discretization parameter of the 3 boundary and the spatial dimension, respectively. Moreover, the stiffness matrices of the 2 BC-FEM are significantly better conditioned than the usual stiffness matrices. This property

2 3 4 was already recovered by Yserentant [37]. 10 10 10 Number of nodes Interface concentrated finite element methods (IC-FEM) are a combination of nonoverlapping domain decomposition (DD) methods with the BC-FEM that is used in the subdomains Ωk Figure 5: Maximal eigenvalue of the FETI preconditioner. in which the initial domain Ω is decomposed. Now we can admit piecewise smooth coefficients in the partial differential equation that we are going to solve. An efficient method for the solution of symmetric and positive definite systems of linear algebraic equations arising from proposed primal and dual domain decomposition preconditioners resulting in asymptotically the finite element discretization of elliptic boundary value problems is the preconditioned optimal or almost optimal solvers for the interface concentrated finite element equations. conjugate gradient method with DD preconditioners. The nonoverlapping Dirichlet-Dirichlet The first class of preconditioners are based on inexact Dirichlet-Dirichlet preconditioning DD preconditioning technique and the Finite-Element-Tearing and Interconnecting (FETI) techniques. The second class of preconditioners are constructed on the basis of the so-called method are typical representatives of primal and dual nonoverlapping DD approaches, respec- All-Floating Finite Element Tearing and Interconnecting technology. The latter class of tively. We refer the reader to the monograph [36] for an excellent overview over the different preconditioners is very robust with respect to jumps in the coefficients of the PDE. It is DD methods. clear that the AF-FETI technology can be replaced by the FETI-DP technique. The number The Dirichlet-Dirichlet DD-preconditioner requires preconditioners for the local Dirichlet of unknowns involved in the IC-FEM is comparable to the number of unknowns living on the problems on the subdomains, a Schur complement preconditioner related to the interfaces skeleton of the domain decomposition. Thus, the coupling of IC-FEM with the Boundary and approximate discrete harmonic extension operators, or more precisely, bounded discrete Element Method (BEM) seems to be very attractive for many applications. In contrast to extension operators acting from the subdomain boundaries into the interior, see [27], [31], the standard FEM the IC-FEM will not perturb the complexity of the BEM. We refer the [14], [15] and [21]. reader to the survey paper [25] for the standard symmetric BEM-FEM coupling. Nowadays, for preconditioning the local Dirichlet problems, a lot of very efficient techniques are available from preconditioning standard h, p or hp finite element equations, see, e.g., Acknowlegdement: This work was supported by the Johann Radon Institute for Compua- [21]. First preconditioners for the Schur complement were proposed by Dryja for some special tional and Applied Mathematics (RICAM) of the Austrian Academy of Sciences and by the cases in [6] and [7]. The BPS preconditioners proposed and developed by Bramble, Pasciak Austrian Science Fund ‘Fonds zur Förderung der wissenschaftlichen Forschung (FWF)’ under and Schatz in a series of papers can be seen as a generalization to more general decompo- the grant SFB F013 (subproject F1306). The work of the second author was supported by sitions and discretizations [2]. During the last 20 years many alternative Schur complement the SFB 393 “Parallele numerische Simulation für Physik und Kontinuumsmechanik” of DFG preconditioners have been proposed in the literature. Let us here only mention the Schur and the DFG-project “Software für parallele irreguläre Algorithmen”. complement BPX [35], techniques borrowed from the boundary element method [25] and - H matrix techniques [17]. In the case of uniform refined meshes, two basic techniques for the fast evaluation of the approximate discrete harmonic extension of a finite element function on References the boundary to the interior exist, namely, an averaging technique [31], [28] and hierarchical decomposition techniques [15], [12], [16]. [1] S. Beuchler. Extension operators on tensor product structures in two and three dimensions. SIAM J. Sci. Comput., 26(5):1775–1795, 2005. The FETI technology introduced by Farhat and Roux in [11] has been developed to a very efficient and robust parallel solver technique for large scale finite element equations. There are [2] J. Bramble, J. Pasciak, and A. Schatz. The constraction of preconditioners for elliptic problems a lot of modifications and new versions of the classical one-level FETI method like the FETI- by substructuring, I. Math. Comp., 47(175):103–134, 1986. DP (Dual-Primal) [10] and BDDC (Balanced Domain Decomposition by Constraints) [5], [26] [3] J. H. Bramble and J. E. Pasciak. A preconditioning technique for indefinite systems resulting from methods. The main building blocks of the FETI methods are solvers or preconditioners for mixed approximations of elliptic problems. Mathematics of Computation, 50(181):1–17, 1988. the local Dirichlet and the local Neumann problems in the subdomains and a component

24 1 managing the global information exchange. We again refer to the monograph [36] for more the constants in comparison to = . CD DL information on and references to FETI methods.

C =S ,C =D C =S ,C =D S B D L 18 S B D L 80 C =C ,C =R RT C =C ,C =R RT In this paper, we propose and investigate fast, robust and highly parallel DD solvers for S B,BPX D S B,BPX D 16 C =S ,C =K C =S ,C =K large-scale interface concentrated ﬁnite element equations. The polynomials of high order 70 S B D L S B D L C =C ,C =D C =C ,C =D S B,BPX D L 14 S B,BPX D L 60 are removed by an overlapping preconditioner with inexact subproblem solvers which was C =C ,C =C C =C ,C =C S B,BPX D D,BPX 12 S B,BPX D D,BPX suggested by Pavarino [33]. This preconditioner requires 50 10 40 a solver for the each patch corresponding to a node, and 8 30 • maximal eigenvalue 1/minimal eigenvalue 6 a coarse space solver for p = 1 on the computational domain. 20 • 4 10 The ﬁrst problem is treated by a direct solver, whereas the second problem is treated by 2

2 3 2 3 two different types of domain decomposition preconditioners. The first preconditioner is a 10 10 10 10 nodes nodes nonoverlapping domain decomposition preconditioner of Dirichlet-Dirichlet type with inexact subproblem solvers. The second type of preconditioners are All-Floating FETI preconditioners. We will prove that the design of both preconditioner leads to an almost optimal solver Figure 4: Eigenvalues of 1 for strongly jumping coefficients: minimal eigenvalue left, max- C− K for the corresponding system of algebraic finite element equations the complexity of which is imal eigenvalue right almost proportional to the number of unknowns corresponding to the skeleton of the domain i decomposition. Moreover, we show that the FETI preconditioners are robust with respect to In a second experiment, we have chosen ai = 10 , i = 1, 2, 3, 4. From the experiments, it jump of the coefficients. can seen that the primal DD-preconditioner is not robust with respect to the jumps of the diffusion coefficient, see Figure 4. The rest of the paper is organized as follows: In Section 2, we formulate the boundary value problem and describe its discretization with the help of the IC-FEM. In Section 3, we con- In a last experiment, we determine the numbers of iterations of the pcg-method for the sider the primal DD-preconditioners. In Section 4, we investigate the FETI-preconditioners. solution of u = b using the preconditioner (3.29) with S = B,BPX and D = D,BPX K 5 C C C C C Several numerical experiments are presented in Section 5. and a relative accuracy of 10− . Table 1 displays the numbers of iterations for the diffusion i coefficients ai = i and ai = 10 , = 1, 2, 3, 4, respectively. In both cases, the numbers of 2 Model Problem and Finite Element Discretization Level 0123456 7 8 9 N 41 129 337 785 1713 3601 7409 15057 30385 61073 ai = i 5 13 16 17 17 18 18 19 19 19 2.1 Model Problem i ai = 10 5 23 35 41 53 57 57 55 56 56 For a bounded Lipschitz domain Ω with Table 1: Numbers of pcg-iterations for the solution of x = b with the preconditioner . s K C 2 Ω= Ωi R , Ωi Ωj = with i = j, (2.1) pcg-iterations are bounded. ⊂ ∩ ∅ 6 i=1 [ In a last experiment, we investigate the FETI preconditioner (4.21) for (4.16). Again, 2,2 CF F a symmetric positive definite matrix function R which is piecewise constant on Ωj the finite element mesh of Figure 2 with p = 1 is used. We consider five different distributions 2 D ∈ and a right hand side f L (Ω) analytic on Ωj, we consider the following Dirichlet problem, 4 ∈ for the coefficients mat = [ai]i=1 in the subdomains Ωi, i = 1, 2, 3, 4. The maximal eigenvalue given in weak formulation: 1 of − is displayed in Figure 5, whereas the minimal eigenvalue is 1 in all experiments. Both Problem 2.1. (model problem) Find u H1(Ω) such that CF F ∈ 0 eigenvalues are robust with respect to material jumps. Moreover, a logarithmic growth of the condition number can be seen. a(u, v) := u(x) (x) v(x) dx = f(x)v(x) dx =: l(v) v H1(Ω). (2.2) ∇ · D ∇ ∀ ∈ 0 ZΩ ZΩ

2.2 Discretization - The Geometric Mesh, the Linear Degree Vector and the 6 Conclusions Fine Element Space The solution of potential problems with piecewise smooth coefficients can be approximated by We will restrict our considerations to γ-shape-regular triangulations τ of Ω consisting of affine the interface concentrated finite element method with the same asymptotical accuracy as the triangles, i.e., each element K τ is the image F (Kˆ ) of the reference triangle Kˆ , where the standard finite element method but with a considerably smaller number of unknowns. We have ∈ K

2 23 mapping FK satisﬁes the inequality

1 1 h− F ′ + hK F ′ − γ K τ. a1 a2 K k K kL∞(K) k K kL∞(K) ≤ ∀ ∈ Here h denotes the diameter of the element K. Moreover, we assume τ = s τ (j) with K ∪j=1 τ (j) := K τ supp K Ω to be a geometric mesh with boundary mesh size h (see { ∈ | { } ⊂ j} also Figure 2). The precise description is given in the following definition. Definition 2.2. (geometric mesh) There exist c , c > 0 such that for all K τ (j): 1 2 ∈ 1. if K ∂Ωj = , then h hK c2h, a3 a4 ∩ 6 ∅ ≤ ≤ 2. if K ∂Ωj = , then c1 inf dist(x, ∂Ωj ) hK c2 sup dist(x, ∂Ωj ). ∩ ∅ x K ≤ ≤ x K ∈ ∈ In order to define hp-FEM spaces on the mesh τ, we associate a polynomial degree pK N ∈ with each element K τ and collect these pK in the polynomial degree vector p := (pK)K τ . ∈ ∈ Figure 2: Coefficient matrix (left) and IC-concentrated mesh (right) Furthermore we associate a polynomial degree

In all experiments, we use the preconditioner (3.29), where is the matrix representation pe := min pK e is an edge of element K (2.3) C EDB { | } of the extension operator of Algorithm 3.8. and the preconditioners S = B, B,BPX T C {S C } with each edge e of the triangulation. We denote the vector containing the polynomial and D = L, L, D,BPX , RR for the Schur complement and the block of the interior C K D C distribution of the triangle K τ with edges ei i = 1, 2, 3 by functions, where ∈ { | } and denote the BPX-preconditioner for and , respectively, p(K) := (pe1,pe2,pe3,pK ). (2.4) • CB,BPX CD,BPX SB KD R denotes the Cholesky factor of the incomplete Cholesky factorization for , i.e. • KD In conjunction with geometric meshes a particularly useful polynomial degree distribution is = RRT , and Cichol the linear degree vector: s (j) L denotes the diagonal part of L. Definition 2.3 (linear degree vector). Let Ω = j=1Ωj. For all j = 1,...,s, let τ be a • D K ∪ s (j) geometric mesh on Ωj with boundary mesh size h and let τ = j=1τ be a mesh on Ω. A Since p = 1 on all elements, the block H does not exist. Experiments for the overlapping ∪ K polynomial degree vector p = (pK)K τ is said to be a linear degree vector with slope α> 0 if preconditioner be defined via (3.16) have already been presented in [9], [29]. ∈ CD h h K K 14 C =S ,C =D C =S ,C =D 1+ αc1 log pK 1+ αc2 log (2.5) S B D L 3.5 S B D L C =C ,C =R RT C =C ,C =R RT h ≤ ≤ h S B,BPX D S B,BPX D 12 C =S ,C =K C =S ,C =K S B D L 3 S B D L C =C ,C =D C =C ,C =D for some c1, c2 > 0. 10 S B,BPX D L S B,BPX D L C =C ,C =C C =C ,C =C S B,BPX D D,BPX S B,BPX D D,BPX 2.5 Now we are in the position to define our hp-FEM spaces: 8 Definition 2.4 (FEM spaces). Let Ω= s Ω . For all j = 1,...,s, let τ (j) be a geometric ∪j=1 j l 6 2 s (j)

maximal eigenvalue mesh on Ω with boundary mesh size h and let τ = τ be a mesh on Ω. Let p be a 1/minimal eigenvalue j l l ∪j=1 l 4 linear degree vector. Furthermore, for all edges e let p be given by (2.3) and for all K τ 1.5 e ∈ 2 let p(K) be given by (2.4). Then we set

2 3 2 3 10 10 10 10 p (j) (j)

nodes nodes p 1 ˆ Ë Ë = (Ω ,τ ) := u H (Ω ) u F (K) K τ , l,j j l { ∈ j | ◦ K ∈ Pp(K) ∀ ∈ l }

p p (j) p 1

Ë Ë Ël,j,0 = 0 (Ωj,τl ) := l,j H0 (Ωj), 1 ∩

Figure 3: Eigenvalues of − for moderate jumping coeﬃcients: minimal eigenvalue left, p p 1 p

Î Ë C K Îl = (Ω,τl) := u H (Ωj) u Ωj l,j j = 1,...,s , maximal eigenvalue right { ∈ | | ∈ ∀ }

p p 1 Î Îl,0 := l H0 (Ω), 1 ∩ Figure 3 displays the maximal and minimal eigenvalue of − for several variants of S and C K C where D in the case of moderate jumping coeﬃcients ai = i, i = 1, 2, 3, 4. The optimality of the C T (Kˆ ) := u p (Kˆ ) u e p , i = 1, 2, 3 . condition number can be seen in all cases. The usage of = C or = RR reduces p(K) K i ei CD D,BPX CD P { ∈ P | | ∈ P }

22 3 The FE-discretization of Problem 2.1 then reads: which can be done in parallel and which can be performed by means of a sparse direct method p

Problem 2.5 (FE-approximation). Find u Î such that with the matrix factorization in a preprocessing step, or with the help of the PCG method h ∈ l,0 with the Dirichlet preconditioner D,i proposed by relation (3.16) in Subsection 3.2. p C a(u , v )= l(v ) v Î . h h h ∀ h ∈ l,0 4.2.3 AF-FETI3 Solvers Considering the approximation properties of boundary concentrated hp-FEM, we have the following theorem 2 The AF-FETI3 system (4.20) is again an one-fold saddle point problem which can be solved Theorem 2.6. Let Ωi R and τi a geometric mesh with boundary mesh size h on Ωi. Let ⊂ p be a linear degree vector with slope α sufficiently large. Then for u H1+δ(Ω ), δ (0, 1) by the solvers mentioned above. In any case, we need ∈ i ∈ we have an efficient preconditioners i for the regularized local Neumann matrices (4.19) and δ • C inf u vh 1 Ch . p H (Ωi) vh Ë (Ωi,τi) k − k ≤ a FETI preconditioner , ∈ • CF Proof. See [18]. which are available from Section 3 and (4.21)–(4.23), respectively. Theorem 4.3. Assume that the AF-FETI3 system (4.20) is solved by means of the Bramble Pasciak PCG method with the scaled inexact Dirichlet FETI preconditioner (4.22) and an ap- 2.3 The System of Finite Element Equations propriately scaled local Neumann preconditioners (3.27) composed of the optimal components 1 D,i, S,i and DB,i given in Section 3. Then not more than I(ε)= O((1 + log(H/h)) log ε− ) p C C E 1 To solve Problem 2.5 numerically, we equip the space Îl,0 with a basis iterations and ops(ε) = O((H/h)(1 + log(H/h)) log ε− ) arithmetical operations are required in order to reduce the initial error by the factor ε (0, 1) in a parallel regime, where Φ= Φ , Φ , Φ = φ i = 1,...,N (2.6) ∈ { B L H } { i | } H/h = max Hi/hi. The number of iterations I(ε) is robust with respect to the jumps in the coefficients. Moreover, at most O(H/h) storage units are needed per processor. in the following way:

The functions φ1,...,φN are the usual nodal hat functions with supp φi Ωj for all j = Proof. The estimate for the number I(ε) of iterations is a direct consequence of results given B 6⊂ 1,...,s, i = 1,...,NB, i.e. the support is not contained in one subdomain Ωj only. The in [3] (see [38] for improved estimates) and of the spectral estimates (3.14) and (4.25). The ∀ functions φN +1,...,φN are the usual nodal hat functions with supp φi Ωj for some j, complexity estimates for the arithmetical costs and the memory demand follow from the B L ⊂ i = NB + 1,...,NL. The remaining functions φN +1,...,φN are all polynomials of degree corresponding complexity estimates for the block matrices involved in the matrix-vector mul- ∀ L H p> 1. Since hK = h for all elements with K ∂Ωj = , we have p = 1 for all these boundary tiplications and in the preconditioning. ∩ 6 ∅ elements by (2.5). With this definition, the basis functions φi can be divided into three groups, We emphasize that we can solve the interface concentrated finite element equations with the same complexity (even with a slightly better complexity) as data-sparse boundary element the boundary functions ΦB = φi i = 1,...,NB , • { | } DD equations (cf. [23] and [22]). In contrary to boundary element technology we can easily the interior functions of low order ΦL = φi i = NB + 1,...,NL , include source terms f( ) and we can even admit non-constant, but smooth coefficients a ( ) • { | } · i · the interior functions of high order Φ = φ i = N + 1,...,N . in the subdomains Ωi. • H { i | L H } In the same way, the basis Φi on the subdomains Ωi, i = 1,...,s, is introduced. Analogously, this basis can be partitioned into ΦB,i, ΦL,i and ΦH,i, i.e. boundary functions (B), interior 5 Numerical Experiments functions of low order (L) and interior functions of high order (H). More precisely, let

Φi =Φ Ωi Ri and ΦB,i =ΦB Ωi RB,i (2.7) In this section, we first present some numerical experiments for the primal DD-solver proposed | | in Section 3. Finally, we investigate the behavior of the FETI preconditioner studied in with the corresponding restriction matrices Ri and RB,i. Now, the solution of Problem 2.5 is Section 4. equivalent to solve the system of linear equations We consider problem (2.2) in Ω = ( 1, 1)2 using a piecewise constant coefficient matrix − u = f, (2.8) (x,y) with coefficients ai, i = 1, 2, 3, 4 and a right hand side f(x,y) = 1, see Figure 2. The D K problem is discretized by an IC-FEM with p = 1 in all elements. The interfaces are at the where lines x = 0 and y = 0, whereas the boundaries are situated at x = 1 and y = 1. Here, we ± ± = [a(φ , φ ]N , f = [f(φ )]N , u = [u ]N . choose Dirichlet boundary conditions, see Figure 2. K j k k,j=1 j j=1 k k=1

4 21 spectral inequalities (4.25) were proved for FEM (see [20], [4], or [36]), they are also valid if Then, u = N u φ provides the solution of (2.5). Moreover, let be the stiffness matrix SB,i h k=1 k k Ki we replace FEM in and by the matrices , + , or . This completes the restricted to Φi, i = 1,...,s, i.e. SB,i F CF SB,i SB,i TB,i WB,i P proof of the theorem. e = [a(φ , φ ]Ni . (2.9) e e e f Ki j,i k,i k,j=1 Now we can solve the SPD system (4.16) via the PCG method that can efficiently be per- Then, a consequence of (2.7)-(2.9) is the following formula formed on the subspace Λ0 with one of the preconditioners F proposed above. Theorem 4.2 C 1 immediately implies that we need at most I(ε)= O((1+log(H/h)) log ε− ) iterations in order s to reduce the initial error by the factor ε (0, 1) in a parallel regime. The matrix-vector = Ri iRi⊤ (2.10) ∈ K K product λ, or λ , requires the (parallel) solution of the systems i=1 F F 0 X with the restriction matrix introduced in (2.7). In the following, we are interested in con- b B,iw = r , (4.26) S B,i B,i structing fast solver for the systems of linear algebraic equations (2.8). which are equivalent to the solution ofe the regularized finite element Neumann problems

w r 3 Primal Iterative Substructuring Methods B,i BD,i B,i = B,1 , i = 1,...,s. (4.27) K K w 0 KDB,i KD,i D,i D,1 e In this section, we will derive a primal DD preconditioner for the matrix (2.8) with in- K The discrete Neumann problem (4.27) can now be solved either by means of a sparse direct exact subproblem solvers. In Subsection 3.1, we expose the ingredients of a nonoverlapping method with the matrix factorization in a preprocessing step, or with the help of the PCG DD-preconditioner. In Subsection 3.2, we develop the preconditioner on the subdomains. method with the Neumann preconditioner (3.27) proposed in Subsection 3.4. The precondi- 1 Subsection 3.2 is devoted to the construction of the preconditioner on the subdomains. Using tioning step − ρ requires 1/2 CF this extension, we show the equivalence of the Schur complement norm to the H (Γ)- the (parallel) solution of local discrete Dirichlet problems with the system matrices D,i, seminorm. In Subsection 3.4, we prove the independence of the condition number for the • K preconditioned system from the discretization parameter. In Subsection 3.5, we show that 1 the (parallel) matrix-vector multiplications ( B,i + B,i)wB,i costing O((Hi/hi)) arith- the preconditioning operation r can be performed in optimal arithmetical complexity. • S T − metical operations, or C e the (parallel) multiplications w costing O((H /h )(ln(H /h ))2) arithmetical op- • WB,i B,i i i i i 3.1 Block Partitioning of the Stiﬀness Matrix erations f for the preconditioners (4.21), (4.22), or (4.23), respectively. Thus, the scaled inexact Dirichlet We partition the stiﬀness matrix into 3 times 3 blocks, which corresponds to the partition of FETI preconditioner (4.22) is the cheapest FETI preconditioner. the shape functions into ΦB, ΦL and ΦH , see (2.6),

B B,L B,H K K K B B,D 4.2.2 AF-FETI2 Solvers = L,B L L,H = K K K  K K K  D,B D K K KH,B KH,L KH The saddle point problem (4.14) can be solved by the Bramble-Pasciak PCG [3] or by any   with D = L H. The two times two block matrix can be factorized into other suitable Krylov subspace iterative solver [34], e.g. the preconditioned conjugate residual ∪ method [19]. In any case, we need 1 I B,D D− B 0 I 0 = K K S 1 (3.1) eﬃcient preconditioners for the regularized local Schur complements and K 0 I 0 D D− D,B I • CB,i SB,i K K K a FETI preconditioner F with the Schur-complement • C e 1 B = B B,D − D,B. (3.2) which are available from Section 3, cf. (3.27), and (4.21)–(4.23), respectively. It follows from S K − K KD K Theorem 4.2, Theorem 3.7 and the results of [3] and [38] that the number I(ε) of iterations In this section, we will investigate a preconditioner of the type 1 of the Bramble-Pasciak PCG behaves like O((1 + log(H/h)) log ε− ). I 0 I 0 = −EDB⊤ CS , (3.3) The matrix-vector multiplication B,ivB,i requires the solution of the local Dirichlet systems C 0 I 0 D DB I S C −E

e D,iw = r , (4.28) where K D,i D,i

20 5 is a preconditioner for , 4.2 AF-FETI Solvers and Preconditioners • CD KD S is a preconditioner for the Schur-complement B or the inexact Schur-complement • C S Throughout this subsection, we assume that (x) = αi on Ωi. We are interested in the 1 D I B + B := B + ( ⊤ ⊤ ) D( DB DB) with DB = − DB, (3.4) robustness of our FETI preconditioners with respect to large jumps of these coefficients across S T S EDB −HDB K E −H H KD K the interfaces. We note that all results remain valid if the spectral condition number κ( )= is the matrix representation of the extension operators E acting from Γ = s ∂Ω ∂Ω D • EDB j ∪i=1 i\ λmax( )/λmin( ) of the diffusion matrix is small on Ωi. into the interior of the domains Ωj. D D The following result is the key in order to analyze the preconditioner , see e.g. [13], [32]. C 4.2.1 AF-FETI1 Solvers Lemma 3.1. Let and be spd preconditioners for and , i.e. CS CD SB KD c ( v, v) ( v, v) v, (3.5) Since the system matrix ˆ of the AF-FETI1 system (4.16) is symmetric and positive definite S CS ≤ SB ∀ F cI ( Dv, v) ( Dv, v) CI ( Dv, v) v. (3.6) (SPD), it can efficiently be solved with the help of the PCG method with an appropriate FETI C ≤ K ≤ C ∀ preconditioner ˆ . It is clear that we only have to construct preconditioner for on the CF CF F Moreover, let subspace Λ = ker = (range ) . Candidates for are the following preconditioners: I I 0 G⊤ G ⊥ CF g, g c2 g, g g. (3.7) K ≤ E CS ∀ scaled exact Dirichlet FETI preconditioner (see, e.g., [36]) EDB EDB • Then, the inequalities 1 − = ⊤, = diag( ), (4.21) c ( v, v) ( v, v) C ( v, v) v CF ASBA SB SB,i C ≤ K ≤ C ∀ cI cS 2 hold with c = 2 min 1, c +c2 c and C = 2max cE,CI . e { I E− S } { } scaled inexact Dirichlet FETI preconditioner • Proof. We start the proof with the following equation 1 − = ( + ) ⊤, + = diag( + ) (4.22) CF A SB TB A SB TB SB,i TB,i I ⊤ I = B + ( DB⊤ DB⊤ ) D( DB DB) (3.8) with the disturbed Schur complements B , see (3.4), DB K DB S E −H K E −H e E E T 1 scaled data-sparse hypersingular BETI preconditioner with the discrete harmonic extension = − . HDB −KD KDB • 1 Next, we prove the upper estimate. First, we transform the inequality C into an − = ⊤, = diag( ), (4.23) K ≤ C CF AWA W WB,i equivalent formulation. Using (3.1) and (3.3), we obtain with the matrices B,i arising from an appropriate data-sparsef approximation of the I ⊤ B 0 I 0 I ⊤ S 0 I 0 W −HDB S C −EDB C hypersingular operator on ∂Ω [24]. 0 I 0 I ≤ 0 I 0 I i KD −HDB CD −EDB f which is equivalent to The matrix 1 1 1 = ( ˜− ⊤)− ˜− (4.24) I DB⊤ DB⊤ B 0 I 0 S 0 A BD B BD E −H S C C ˜ 0 I 0 D DB DB I ≤ 0 D is defined by the interconnecting matrix = diag( B,i) and by the scaling matrix = B B D K E −H C diag( ˜ ) with appropriate diagonal matrices ˜ . The diagonal entries depend on the and DB,i DB,i coefficients αi, see [20], [4], [36] for further details. B + ( DB⊤ DB⊤ ) D( DB DB) ( DB⊤ DB⊤ ) D S 0 S E −H K E −H E −H K C C . (3.9) Theorem 4.2. Let F be one of the FETI preconditioners defined by (4.21), (4.22), or (4.23). D( DB DB) D ≤ 0 D C K E −H K C Then, there exist positive constants cF and cF such that the spectral inequalities Both matrices are postive definite. Using the Cauchy-Schwarz inequality, (3.8), (3.7), (3.6), 2 we can estimate c ( F λ, λ) ( λ, λ) cF (1 + log(H/h)) ( F λ, λ) (4.25) F C ≤ F ≤ C B + ( DB⊤ DB⊤ ) D( DB DB) ( DB⊤ DB⊤ ) D S E −H K E −H E −H K hold for all λ Λ0 = ker ⊤, where the constant cF and cF are independent of hi, Hi, and D( DB DB) D ∈ G K E −H K the αi’s (coefficient jumps), H/h = max Hi/hi. + ( ) ( ) 0 2 SB EDB⊤ −HDB⊤ KD EDB −HDB ≤ 0 KD Proof. By Theorem 3.13, the matrices B,i, B,i + B,i and B,i are spectrally equivalent to FEM S S T W I ⊤ I an auxiliary Schur complement B,i arising from a standard finite element discretization 0 2 S 0 S e e f = 2 DB K DB 2max cE ,CI C . with linear triangular element on an auxiliary quasi-uniform triangular mesh generated from  E E  ≤ { } 0 D 0 D C the given quasi-uniform boundarye mesh on ∂Ωi with the average mesh size h [24]. Since the  K 

6 19 Remark 4.1. We mention that the iteration updates for λ completely life in the subspace Using (3.9), we can conclude that C with C = 2max c2 ,C . K≤ C { E I } Λ if the initial guess is chosen as λ . Therefore, a basis of Λ , forming the columns of , 0 e 0 L0 Finally, we prove the lower estimate. Using similar arguments as for the proof of the upper is not explicitly required in the computation. We further mention that in the case of large estimate, we can show that the assertion c is equivalent to jumps in the coefficients of the PDE the scalar product in Λ has to be changed according to C ≤ K the proposal made in [20], see also [4] and [36]. Of course, the change of the scalar product + ( ) ( ) ( ) 0 c CS EDB⊤ −HDB⊤ CD EDB −HDB − EDB⊤ −HDB⊤ CD SB , changes the orthoprojection , too. D( DB DB) D ≤ 0 D P −C E −H C K Eliminating the unknowns vB,1,...,vB,s from the AF-FETI2 system (4.14), we get the FETI see (3.9). The positive definitness of the above matrices implies Schur complement problem ˆ ˆ S + ( DB⊤ DB⊤ ) D( DB DB) ( DB⊤ DB⊤ ) D λ0 = d0 (4.16) C E −H C E −H − E −H C F D( DB DB) D −C E −H C with the FETI Schur complement + ( ) ( ) 0 2 CS EDB⊤ −HDB⊤ CD EDB −HDB . ≤ 0 D s C 1 ˆ = ⊤ ⊤ 0 with = B,i − ⊤ (4.17) Using (3.6), (3.8), (3.7) and (3.5), we can estimate F L0 PFP L F B SB,iBB,i Xi=1 e cI S + ( DB⊤ DB⊤ ) D( DB DB cI S + ( DB⊤ DB⊤ ) D( DB DB) and the corresponding right-hand side C E −H C E −H ≤ C E −H K E −H I ⊤ I s = cI S + B 1 C DB K DB − S dˆ = ⊤ d ⊤ − d . (4.18) E E 0 L0 P 0 −L0 P BB,iSB,i B,i 2 i=1 cI S + cE S B X ≤ C 2 C − S e cI + cE cS − B. The symmetric and positive definite system (4.16) is called AF-FETI1 system. ≤ cS S On the other hand, we can again unfold the AF-FETI2 system (4.14) arriving at a saddle Due to the postive definitness of +( ) ( ), we have c +c2 c > 0. CS EDB⊤ −HDB⊤ CD EDB −HDB I E − S point system that is similar to the saddle point system (4.3) but now with regular diagonal cI cS Together with cI D D, we can conclude that c with c = 2 min 1, c +c2 c . blocks C ≤ K C ≤ K I E− S n o B,i BD,i Remark 3.2. A direct consequence of (3.7) is the spectrally equivalence of the Schur-complement i = K K , (4.19) K B and the disturbed Schur-complement B + B, see [13]. KDB,i KD,i S S T e e Summarizing, we have to find the preconditioners , and the extension E in order with the regularized matrices = + α 1 1 . Moreover, let CD CS ↔EBD KB,i KB,i i B,i B,i⊤ to derive a good preconditioner . C e f λ For an inexact FETI preconditioner, a preconditioner i for i (cf. (2.9)), i.e. for the vB,i B,i B,⊤ 1 e C K i = ⊤ B,i 0 , u = , and f = −B . Neumann problem on domain Ω , is required. Due to (2.7), the block partitioning for in B L0 PB i u i f i D,i D,i ! K (3.1), can be done for the subdomain stiffness matrices, too. A simple computation shows e e e 1 I B,D,i D,i− B,i 0 I 0 Then, the resulting system called AF-FETI3 system can be written in the form: i = K K S 1 (3.10) K 0 I 0 D,i − D,B,i I K KD,iK 1 0 . . . 0 ⊤ u f with the local Schur-complement K B1 1 1 . . .  0 2 . 2⊤   .   .  1 e K Be e B,i = B,i B,D,i D,i− D,B,i. (3.11) . .. . e. = . . (4.20) S K − K K K  . . 0 .   .   .   e e      Then, the global Schur complement can be computed via the local Schur-complements, i.e.  0 . . . 0 s s⊤   us   f   K B     s  s  1 . . . s 0   λ0   0   B B      = R R⊤ (3.12)  e e   e   e  SB B,iSB,i B,i Let us mention that all AF-FETIe systemse derived in this subsection are equivalent. In the Xi=1 following subsection we propose and analyze fast and robust solvers for the AF-FETI1–AF- with the assembling matrices R (2.7). Using the DD-approach applied to instead of B,i Ki K FETI3 systems (4.16), (4.14), and (4.20). this preconditioner will be developed in Section 3.

18 7 3.2 An Overlapping Preconditioner for the Dirichlet Subproblems on H1(Ω ) H1(Ω ) with a positive regularization parameter α , e.g. α = h2. Taking into i × i i i i account the solvability conditions for local Neumann problems In this subsection, we present the preconditioner for the matrix . Here, we use an overlap- KD ping preconditioner which has been developed in [29], see also [8], [33]. The preconditioner is (g ⊤ λ, 1 ) = 0 (4.9) B,i −BB,i B,i based on the additive Schwarz framework (see, e.g., [36]). We decompose the hp-FEM space

p K

Î Î Ë and the representation of the vectors u as (Ωi,τi)= i=0 i for subspaces j that will be specified below. This splitting defines an B,i ASM-preconditioner which eliminates all high order degrees of freedom and, moreover, the P properties of this preconditioner can be expressed by the two numbers C0 and ρ(ǫ) defined be- uB,i = vB,i + γi1B,i with (vB,i, 1B,i) = 0, (4.10) low. As a well-known fact from the ASM-theory the condition number of the preconditioned 2 we get the two–fold saddle point problem system is bounded by C0 (1 + ρ(ǫ)).

Since D = blockdiag [ D,i] it is possible to restrict ourselves to the case of one subdomain 0 v g K K i SB,1 BB,⊤ 1 B,1 B,1 Ωi. . . . .  .. .  .  .  Lemma 3.3 (Clement interpolation). Let τ be a γ-shape-regular aﬃne triangulation. For e   v = g (4.11) K τ deﬁne the patch  B,s B,s⊤ 0  B,s  B,s   S B      ∈  B,1 . . . B,s 0   λ   0  ωK := K′ τ K K′ = .  B B G      ∪{ ∈ | ∩ 6 ∅}  0 . . . e0 ⊤ 0   γ   e  C 1 1 G  

Ë     Then there exists C > 0 depending solely on γ and a linear operator I : H (Ω) (τ)       7→ s such that with the notations γ = (γi)i=1,...,s R , = ( B,11 ,..., B,s1 ) and e = (ei)i=1,...,s = ∈ G B B,1 B B,s ( (g , 1 ) s. The solution u can be recovered via formula (4.10). c c B,i B,1 i=1,...,s R B,i u I u 2 Ch u 2 and I u 2 C u 2 . − ∈ k − kL (K) ≤ K k∇ kL (ωK ) k∇ kL (K) ≤ k∇ kL (ωK ) Applying the orthogonal projection Proof. The proof is given in [9]. 1 = ( ⊤ )− ⊤ (4.12) Lemma 3.4. Let Kˆ be the reference triangle and p(K) = (p ,p ,p ,p ) with p p , P I − G G G G e1 e2 e3 K ei ≤ K i = 1,..., 3, a polynomial degree distribution. Let ip be the one-dimensional Gauss-Lobatto L from the space Λ := R onto the subspace Λ0 = ker ⊤ = (range )⊥ with respect to the interpolation operator. Then there exists a constant C > 0 and a linear operator G G scalar product ( , ) = ( , ) = ( , ) L to the last but one block equation of system (4.11), we · · · · Λ · · R can exclude γ from the first s + 1 block equations of (4.11). Moreover, let us represent λ in I : (Kˆ ) (Kˆ ) (3.13) p(K) P2pK 7→ Pp the form λ = λ + λ (4.13) such that L0 0 e 1 Ipu = u for all u p(Kˆ ), with known λe = ( ⊤ )− e (ker ⊤)⊥ = range , fulfilling the constraints ⊤λe = e, and • ∈ P G G G ∈ G G L0 G unknown 0λ0 ker ⊤, i.e. ⊤ 0λ0 = 0, where λ0 R and L0 = dim Λ0. The columns I u C u for all u (Kˆ ), L ∈ G G L ∈ p H1(Kˆ ) H1(Kˆ ) 2pK of span a basis of Λ . Now we can define v ,...,v and λ from the one–fold saddle • k k ≤ k k ∈ P L0 0 B,1 B,s 0 point problem (Ipu) Γ = ip ,Γ u for i = 1,..., 3. • | i ei i Proof. The proof is given in [9, Theorem 4.4]. B,1 B,⊤ 1 ⊤ 0 vB,1 dB,1 S . B P. L ...... Definition 3.5. Let τ be a γ-shape-regular triangulation with polynomial degree distribution  e    =   , (4.14) B,s ⊤ ⊤ 0 vB,s dB,s p = (pK )K τ . For each K τ let the operator Ip(K) be given by (3.13). Then we define  S BB,sP L      ∈ ∈    λ   0   0⊤ B,1 . . . 0⊤ B,s 0   0   

2p p 1  L PB L ePB      Ë Ip : Ë (τ) (τ) by (Ipu) K = (Ip(u K FK )) FK− . 7→ | | ◦ ◦ where d = g λ . Once the vectors v ,...,v and λ are defined from (4.14), B,i B,i −B1⊤,B e B,1 B,s 0 we get λ from (4.13), γ from the last but one block equation of system (4.11), i.e. For a finite element mesh τ, we denote the set of its vertices by V (τ) and the polynomial degree vector by p = (pK )K τ . In order to define our ASM-preconditioner we associate with 1 ∈ γ = ( ⊤ )− ⊤( B,1vB,1 + + B,svB,s), (4.15) each vertex vj a patch − G G G B · · · B ωj := K τ vj is a vertex of K ∪{ ∈ | } and, finally, uB,1,...,uB,s from (4.10). We will call the one–fold saddle point problem (4.14) p and decompose the space Ë (τ) into the following subspaces: AF-FETI2 system.

8 17

1 Ë interconnecting the local potential vectors across the subdomain boundaries and the nodal 1. The space of piecewise linear polynomials Î0 := (τ).

parameters on the boundary ΓD with the corresponding Dirichlet data which are zero for p Ë 2. Local higher dimensional spaces Îj := u (τ) supp u ωj . our model problem. Each row of the matrix = ( 1,..., s) = (( 1,B, 0),..., ( s,B, 0)) is { ∈ | ⊂ } B B B B B p connected with a pair of matching nodes across the subdomain boundaries or with a Dirichlet To analyze the properties of the decomposition of Ë (τ), we introduce C0 via: node. The entries of the former rows are 1 and 1 for the indices corresponding to the − #V (τ) #V (τ) matching nodes on the interface (coupling boundaries) ΓC =ΓS ΓD and 0 otherwise, whereas 2 2 2 \ min uj u = u0 + uj, uj Îj C u , (3.14) a entry corresponding to a Dirichlet node on ΓD is 1 and again 0 otherwise. Therefore, (4.2)  k kA ∈  ≤ 0 k kA j=0 j=1 implies that the corresponding ﬁnite element functions ui,h are continuous across the interface  X X 

ΓC , i.e. ui,h = uj,h on Γi Γj = , and are vanishing on Γi ΓD. We assume here that the (#V (τ)) (#V (τ)) ∩ 6 ∅ ∩ and deﬁne a symmetric matrix ǫ R × with entries ǫik = 0 if ωi and ωk are number of constraints at some matching node is equal to the number of matching subdomains ∈ disjoint and ǫ = 1 otherwise. Thus minus one. This method of a minimal number of constraints respectively multipliers is called ik

non–redundant, see, e.g., [20] for the use of redundant constraints. Î a(ui, uk) ǫik ui A vk A ui Îi, uk k. (3.15) By introducing Lagrange multipliers λ RL, the linear system (2.8) is equivalent to the | |≤ k k k k ∀ ∈ ∀ ∈ ∈ following extended system Theorem 3.6. Let τ be a γ-shape regular triangulation and p a corresponding polynomial degree distribution. Let f 1 1⊤ u1 1 #V (τ)

K . B. . . p

Î Î . . . . Ë (τ)= +  . .   .  =  .  , (4.3) 0 j u f j=1  Ks Bs⊤   s   s  X  1 . . . s 0   λ   0  and let a( , ) : H1 H1 R symmetric positive deﬁnite. Then there exists a constant C > 0  B B      · · × 7→       depending solely on the bilinear form a( , ) and γ, such that where now all matrices i are corresponding to local Neumann problems given by the Neu- · · K 1 mann bilinear form u vdΩ on H (Ωi). Due to the choice of the matrices, all matrices Ωi C C, ρ(ǫ) C ∇ ·D∇ 0 ≤ ≤ ⊤ i are singular withR kernels spanned by the vector 1i,B⊤ , 1i,L⊤ , 0i,H⊤ as was already mentioned K where C0 and ǫ are from (3.14), (3.15), respectively. in Subsection 3.4. h i Proof. Since there exists M > 0 depending only on γ, such that Eliminating all interior unknowns ui,D, i = 1,...,s, from system (4.3), we arrive at the equivalent reduced system # v′ V ωv ωv′ = M v V g { ∈ | ∩ 6 ∅} ≤ ∀ ∈ B,1 B,⊤ 1 uB,1 B,1 S . B . . . we have ρ(ǫ) ǫ M. In order to bound C0, we exploit the properties of the operators .. . .  .  C ≤ k k∞ ≤ p     = , (4.4) I and Ip and decompose u Ë (τ) as follows u g ∈ B,s B,s⊤ B,s  B,s   S B      #V (τ)  B,1 . . . B,s 0   λ   0   B B      C C C C       u = I u + Ip(u I u) = I u + Ip φj(u I u) with the singular Schur complements −  −  Xj=1 1 #V (τ)   B,i = B,i BD,i D,i− DB,i, i = 1,...,s, (4.5) S K − K K K = u0 + uj and the corresponding right-hand sides Xj=1 g = f 1 f , i = 1,...,s. (4.6) B,i B,i BD,i D,i− D,i with

− K K c C Î u0 := I (u) Î0, uj := Ip(φj(u I u)) j. The kernels of the matrices B,i are spanned by the vector 1B,i. We now replace the singular ∈ − ∈ S 2 #V (τ) 2 Neumann Schur complements B,i by the regularized matrices Now we want to bound u + u . By Lemma 3.3 and since 1 we S k 0kA j=1 k jkA k · kA ∼k·kH have B,i = B,i + αi1B,i1B,i⊤ , i = 1,...,s. (4.7) P 2 c 2 2 S S u0 = I (u) C u . k kA k k ≤ k kA This corresponds to the regularized Neumann bilinear form e Next we consider a ﬁxed j 1, ..., #V (τ) and a ﬁxed K ω : ∈ ⊂ j u(x) (x) v(x) dx + u(x) ds v(x) ds (4.8) 2 2 C 2 ∇ · D ∇ uj A C uj H1(K) = Ip(φj(u I u)) 1 ZΩi ZΓi ZΓi k k ≤ k k − H (K)

16 9 Denotingu ˜ := (u IC u),u ˆ :=u ˜ F , φˆ := φ F and applying Lemmas 3.4, 3.3 we 3.5 Computational Aspects − |K ◦ K j|K ◦ K get: 1 In the previous subsection, we have proved the optimality of the condition number of − . 2 2 ˆ 2 ˆ 2 1 C K uj H1(K) C hK Ip(φuˆ) 2 ˆ + Ip(φuˆ) 2 ˆ For the design of an optimal solver, the operation r should be performed in optimal k k ≤ k kL (K) k∇ kL (K) C− arithmetical complexity. This is the purpose of this subsection. C I (φûˆ) 2 ≤ k p kH1(Kˆ ) ˆ 2 2 The preconditioner (3.29) consists of three ingredients, namely the preconditioner D for C φuˆ 1 ˆ C uˆ 1 ˆ C C ≤ k kH (K) ≤ k kH (K) D, the Schur-complement preconditioner S and the extension operator DB. K C E 2 C 2 C 2 1 C hK− u I u L2(K) + (u I u) L2(K) For the preconditioning operation − r (3.16), we have to solve subproblems onto the high- ≤ k − k k∇ − k CD D 2 order subspaces Vj, j = 1,...,n. Those problems can be treated by direct solvers since C u L2(ω ). ≤ k∇ k j hK 3 dim(Vj) (1 + log h ) , see (2.5). On the subspace V0, we have to multiply with a diagonal 1 Thus, the relations matrix. Hence, the cost for D− rD is (N). 2 2 C O uj H1(ω ) C u L2(ω ) k k j ≤ k∇ k j Furthermore, we need a good preconditioner CS for B. Due to Theorem 3.13, the norm K ωj S X⊂ induced by the Schur complement is equivalent to the H1/2-norm on the skeleton. Hence, and any preconditioner which is known from the h-version of the FEM using uniform refined #V (τ) 2 2 2 meshes can be used as preconditioner for the Schur complements B,i or the assembled Schur uj H1(ω ) C u L2(Ω) C u A S k k j ≤ k∇ k ≤ k k complement B (see, e.g. [21] or [36]). Thus, nowadays many preconditioners S,i for B,i j=1 S 1 1 C S X are available such that κ( − 2 − 2 ) = (1) and the solution of a system w = r CS,i SB,iCS,i O CS,i follow. requires (N ) floating point operations. Similar results are valid for the assembled Schur O i complement . In our numerical experiments presented in Section 5 we use the so-called Now we are able to define the following preconditioner for . Let SB KD,i Schur complement BPX preconditioner proposed in [35]. #V (τ) The last ingredient is the matrix representation of the extension operator . Due to Al- 1 1 1 EDB D− = Qj⊤ P,j− Qj + Q0⊤( L)− Q0, (3.16) gorithm 3.8, the operations w and T w involve only BPX-like basis transformations C K D EDB B EDB D Xj=1 at the boundary or skeleton and on the interior, which are proportionally to the numbers of unknowns at the skeleton or the interior. Therefore both operations require (N) floating where O point operations. P,j denotes the stiffness matrix restricted to Vj, j = 1,..., #V (τ), 1 • K Summarizing, the total cost for − r is (N) flops. Q , j = 0,..., #V (τ) is the corresponding finite element restriction matrix onto V , C O • j j is the diagonal part of the matrix . • DL KL Theorem 3.7. Let D be defined via (3.16). Then, we have D D. 4 All-Floating Interface Concentrated Finite Element Tearing and C C ∼ K Interconnecting Methods Proof. Due to [37] and [18], the condition number of does not depend on the discretization KL parameter. This gives . Now, the result is a consequence of Theorem 3.7. DL ∼ KL 4.1 AF-FETI Formulations

3.3 Schur complement preconditioner and extension operator In order to avoid assembled matrices and vectors, we tear the global potential vector u on the subdomain boundaries Γi by introducing the individual local unknowns Next, we consider the preconditioner S for B (3.2). We will prove that the norm induced C S ui = Riu. (4.1) by the Schur complement (3.2) is equivalent to a suitable seminorm. Therefore, the SB extension operators E for the nearly discrete harmonic extension of a function g H1/2(∂Ω)˜ In contrast to the preceding sections the vector u of nodal parameters now contains also those ∈ to a function u = Eg H1(Ω)˜ has to be investigated. We follow the approach of Haase and nodal parameters belonging to the Dirichlet boundary ΓD that coincides with Γ = ∂Ω for ∈ Nepomnyaschikh [16]. our model problem (2.1). The global continuity of the potentials and the Dirichlet boundary conditions are now enforced by the constraints In a first step, we describe the extension of Haase and Nepomnyaschikh. Then, we prove that s this type of extension can be used for boundary concentrated meshes with p = 1 or p > 1 iu = 0 (4.2) defined via Definition 2.3, too. Finally, the equivalence of the Schur complement is shown. B i Xi=1

10 15 3.4 Final condition number estimates

Now, we are in the position to summarize the results of Subsections 3.2 and 3.3. In a first step, we propose the primal DD preconditioner for (3.10). Let Ki I DB,i⊤ S,i 0 I 0 i = −E C , (3.27) C 0 I 0 I Figure 1: Refinement of each element of the mesh. CD,i −EDB,i where The technique of Haase and Nepomnyaschikh has originally been derived for the h-version of is the preconditioner (3.16) for , • CD,i KD,i the finite element method with p = 1 using uniform refined meshes and uses the hierarchy is a preconditioner for with , of the nested finite element spaces. Let τ0 τ1 τ2 . . . τL be a sequence of nested • CS,i SB,i CS,i ∼ SB,i ⊂ ⊂ ⊂ ⊂ (l) triangular (d = 2) or tetrahedral (d = 3) finite element meshes with nodes xi , i = 1,...,Nl, DB,i is the matrix representation of the extension operator EL,i : ∂Ωi Ωi in Algo- ˜ • E 7→ l = 0,...,L, on a domain Ω. Here τ0 is the coarse mesh and τl is obtained from τl 1 be rithm 3.8. d − subdividing each element of τl 1 into 2 new congruent elements, cf. Figure 1 for d = 2. − Note that the matrix i is positive definite if ∂Ω ∂Ωi = . If ∂Ω ∂Ωi = , this matrix is 1 C ∩ 6 ∅ ∩ ∅ Due to Definition 2.4, Ël denotes the space of all functions which are piecewise linear, linear ⊤ 1 ˜ positive semidefinite with ker i = ker i = [uB, uL, uH ]⊤ = 1i,B⊤ , 1i,L⊤ , 0i,H⊤ . The choice of on each element of τl and belong to H (Ω). The corresponding trace space is denoted by Bl,

C K 1 Ë the preconditioners S,i is specified in the next subsection. h i i.e. Bl = l ∂Ω˜ . This space is spanned by nodal basis functions ψi,l, i.e. C | Theorem 3.14. Let i be defined via (3.27). Then, the spectral inequalities Nl C Bl = span ψi,l = spanΨl. (3.17) c ( v, v) ( v, v) c ( v, v) (3.28) { }i=1 8 Ci ≤ Ki ≤ 9 Ci Ni (i) are valid for all v R , where the spectral constants do not depend on Ni. Moreover, let Ï = span ψi,l be the one-dimensional subspace corresponding to the node ∈ l { } (l) 1/2 ˜ xi . Let g Bl H (∂Ω). In order to define a function u = Eg such that Proof. We show that the assumptions of Lemma 3.1 are satisfied. Due to Theorem 3.7, we ∈ ⊂ have D,i D,i. Our assumptions imply the estimate S,i B,i. Using Theorem 3.13, u H1(Ω)˜ c g H1/2(∂Ω)˜ , u ∂Ω˜ = g, (3.18) C ∼ K 2 C1 ∼ S k k ≤ k k | Theorem 3.11 and u 1 ( iu, u) for all u =Φiu Ë , the relation | |H (Ωi) ∼ K ∈ L,i Haase/Nepomnyaschikh used a BPX-like decomposition of the function g. Let Q : L (∂Ω)˜ I I i,l 2 7→

2 (i) ˜ (i) Ï i g, g cE S,ig, g g Ï be the L2-like projection from L2(∂Ω) onto , i.e. K BD,i BD,i ≤ C ∀ l l E E holds with a constant, which is independent of Ni, for the matrix representation of the Qi,lg = di,l(g)ψi,l g Bl, extension operator in Algorithm 3.8. ∀ ∈ where (g, ψi,l) ˜ (g, 1) Now, we will present the solver for the global stiffness matrix . Let L2(∂Ω) L2(supp ψi,l) K di,l = , or, di,l = . (ψi,l, 1) ˜ (1, 1) I 0 I 0 L2(∂Ω) L2(supp ψi,l) = −EDB⊤ CS , (3.29) C 0 I 0 D DB I Moreover, let C −E N where l Ql = Qi,l, l = 0,...,L, and Q 1 = 0. − is the block diagonal matrix of the preconditioner (3.16) for , i = 1,...,s, i=1 • CD KD,i X S is a preconditioner for B, which satisfies S B, ˜ Now, the extension of a function g B into Ω can be defined. • C S C ∼ S ∈ L is the matrix representation of the extension operators E : ∂Ω Ω in Algorithm Algorithm 3.8. 1. Let ψl = (Ql Ql 1)g Bl, l = 0,...,L. • EDB L,i i 7→ i − − ∈ 3.8. 1 2. Define the function u Ë via Theorem 3.15. Let be defined via (3.29). Then, ∈ l C ˜ c10 ( v, v) ( v, v) c11 ( v, v) v. ψl(xi,0) if xi,0 ∂Ω C ≤ K ≤ C ∀ ul(xi,0) = ∈ , (3.19) g if xi,0 / ∂Ω˜ The constants do not depend on N. ∈ ψl(xi,l) if xi,l ∂Ω˜ ul(xi,l) = ∈ ˜ . (3.20) Proof. The proof is similar to the proof of Theorem 3.14. 0 if xi,l / ∂Ω ∈

14 11 (i) (i) 3. Set u = Elg = u0 + u1 + . . . + uL. Theorem 3.13. Let τ . . . τ be a family of geometrically reﬁned meshes on the

0 ⊂ ⊂ L

B Ë Î Remark 3.9. Note that the function ul in (3.19), (3.20) is uniquely deﬁned by setting the domains Ωi, i = 1,...,s, and let BL,i and L,Γ be the trace spaces of L,i and L,0 onto (l) s function values in the nodes x of the mesh τ . For g, a coarse grid problem on the mesh τ ∂Ωi and Γ = ∂Ωi ∂Ω, respectively. Let ΦB,i and ΦB be deﬁned via (2.6) and (2.7), i l 0 ∪i=1 \ has to be solved. Alternatively, the mean value of g over ∂Ω˜ can be taken. respectively.

Now, the following result has been shown. Moreover, let B,i be deﬁned via (3.11). If ∂Ω ∂Ωi = , we have 1 S ∩ 6 ∅ Theorem 3.10. Let τ0 . . . τL be a family of uniformly reﬁned meshes and let Ël , 2 2 c g 1/2 ( g , g ) c g 1/2 g =Φ g B , g = 0. (3.22)

⊂ ⊂ 1 4 i H (∂Ωi Γ) B,i i i 5 i H (∂Ωi Γ) i B,i i L,i ∂Ω Ë l = 0,...,L be the space of the piecewise linear nodal basis functions on τ . Let E : B | | ≤ S ≤ | | ∀ ∈ | l L L 7→ L ∩ ∩ be the extension of Algorithm 3.8. Then, we have E g = g. Moreover, there is a constant L |∂Ω˜ If ∂Ω ∂Ωi = , we have c2 which is independent of L such that ∩ ∅ 2 2

c g 1/2 ( g , g ) c g 1/2 g =Φ g B . (3.23) 4 i H (∂Ωi Γ) B,i i i 5 i H (∂Ωi Γ) i B,i i L,i | | ∩ ≤ S ≤ | | ∩ ∀ ∈

E g c g g B . k L kH1(Ω)˜ ≤ 2 k kH1/2(∂Ω)˜ ∀ ∈ L Moreover, the relation

2 2 Proof. The proof was given in [16]. c6 g 1/2 ( Bg, g) c7 g 1/2 g =ΦBg BL,Γ (3.24) k kH (Γ)≤ S ≤ k kH (Γ) ∀ ∈

holds. The constants c4, c5, c6 and c7 are independent on L and N. Next, we consider a family of boundary concentrated finite element meshes and the corresponding finite element spaces Ël. Proof. The proof of (3.22) is adapted from [36], see also [1]. By the trace theorem and the Theorem 3.11. Let τ . . . τ be a family of meshes on a domain Ω˜, which are geo- 0 ⊂ ⊂ L Poincare Friedrichs inequality metrically refined to the boundary. Let Ël, l = 0,...,L, be defined via Definition 2.4. Then,

2 2 2 2

B B Ë ËL ˜ = L. Moreover, let EL : L L be the extension operator of Algorithm 3.8. Then, c14c− gi 1/2 c14 u 1 c15 u 1 |∂Ω 7→ T | |H (∂Ωi) ≤ k kH (Ωi) ≤ | |H (Ωi) there exists a constant c2 which is independent of L (and N) such that min( iu, u) ≤ uD K

ELg 1 ˜ c2 g 1/2 ˜ g BL. (3.21) g k kH (Ω)≤ k kH (∂Ω) ∀ ∈ = ( g , g ), u = i , u =Φ u. SB,i i i u i D

˜ B Proof. Due to (2.5), the polynomial degree on the elements on ∂Ω is 1. Thus, ËL ∂Ω˜ = L. 1 | Due to Theorem 3.11 and Remark 3.12, there exists a function u H (Ωi), u ∂Ωi = g such This proves the ﬁrst assertion. ∈ | that u 1 c2 g 1/2 . Thus, we can conclude that | |H (Ωi) ≤ | |H (∂Ωi) Due to (3.19) and (3.20), the image of extension operator described in Algorithm 3.8 for 2 2 2 1 ( B,ig , g ) ( iu, u) c12 u 1 c12c2 gi 1/2 . Ë H (Ω ) uniformly reﬁned meshes belongs to the space L. Theorem 3.10 implies the estimate S i i ≤ K ≤ | | i ≤ | |H (∂Ωi)

E g c g 1/2 g B To prove (3.23), we have k L kH1(Ω)˜ ≤ 2 k kH (∂Ω)˜ ∀ ∈ L

1 p 2 2

B Ë Ë c g 1/2 ( g , g ) c g 1/2 g =Φ g , g = 0 (3.25) with the constant of Theorem 3.10. Since L L, the second assertion follows. 4 i H (∂Ωi Γ) B,i i i 5 i H (∂Ωi Γ) i B,i i L,i i ⊂ | | ∩ ≤ S ≤ | | ∩ ∀ ∈ Z∂Ωi Remark 3.12. If g = 0, or g = 0 for Γ˜ ∂Ω˜ with meas(Γ)˜ > 0, relation (3.21) remains ∂Ω˜ |Γ˜ ⊂ by the same arguments. valid, if the seminorms are used instead of the seminorms in (3.21). This is a consequence of R the Poincare-Friedrichs type inequalities, see [36, Lemma A17], or, [30]. Let gi BL,i with g = gi. Thus, one obtains gi = g 1 + gi,0 with gi,0 = 0. Since ∈ ∂Ω · ∂Ω gi H1/2(∂Ω ) = gi,0 H1/2(∂Ω ), and ( B,ig , g ) = ( B,ig , g ) for gi,0 =ΦB,ig , relation (3.23) Hence, we have shown that the extension of Haase/Nepomnyaschikh for uniformly reﬁned | | i | | R i S i i S 0,i 0,i R i 1 is a consequence of (3.25). meshes can be used for geometrically reﬁned meshes with grading factor 2 , too. To prove (3.24), we have In a next theorem, we show that the norm induced by the Schur complement B,i (3.11) is 1/2 S s equivalent to the H (∂Ωi) norm and that the norm induced by the Schur complement B 2 2 S g 1/2 = g 1/2 . (3.26) (3.2) is equivalent to the following norm: H (Γ) H (∂Ωi Γ) k k k k ∩ Xi=1 2 2 2 2 2 1/2 1/2 g H1/2(Γ):= g L2(Γ) + g H1/2(Γ), where g H1/2(Γ) = inf u H1(Ω). Since meas(∂Ω) > 0, the H (Γ) seminorm is equivalent to the H (Γ) norm. This is k k k k | | | | u = g | | |Γ implied by the Poincare-Friedrichs inequality. Using (3.26), (3.12), (3.23) and (3.22), the u = 0 |∂Ω assertion follows.

12 13