A BDDC ALGORITHM FOR RAVIART-THOMAS VECTOR FIELDS TR

Transcription

1 A BDDC ALGORITHM FOR RAVIART-THOMAS VECTOR FIELDS DUK-SOON OH, OLOF B WIDLUND, AND CLARK R DOHRMANN TR Abstract A BDDC preconditioner is defined by a coarse component, expressed in terms of primal constraints a weighted average across the interface between the subdomains, local components given in terms of Schur complements of local subdomain problems A BDDC method for vector field problems discretized with Raviart-Thomas finite elements is introduced Our method is based on a new type of weighted average developed to deal with more than one variable coefficient A bound on the condition number of the preconditioned linear system is also provided which is independent of the values jumps of the coefficients across the interface has a polylogarithmic condition number bound in terms of the number of degrees of freedom of the individual subdomains Numerical experiments for two three dimensional problems are also presented, which support the theory show the effectiveness of our algorithm even for certain problems not covered by our theory Key words domain decomposition, BDDC preconditioner, Raviart-Thomas finite elements AMS subject classifications 65F08, 65F10, 65N30, 65N55 1 Introduction Let Ω be a bounded polyhedral domain in R 3 We will work withthehilbertspacehdiv;ω, thesubspaceofvectorvaluedfunctionsu L Ω 3 with divu L Ω The space H 0 div;ω is the subspace of Hdiv;Ω with a vanishing normal component on the boundary Ω We will consider the following problem: Find u H 0 div;ω, such that au,v := αdivudivv +βu vdx = f vdx, v H 0 div;ω 11 Ω We will assume that the coefficient α is a nonnegative L Ω-function, that β is a strictly positive L Ω-function, that the right h side f L Ω 3 We note that the norm of u Hdiv;Ω for a domain with unit diameter is given by au,u 1/ with α 1 β 1 The bilinear form 11 arises from the following boundary value problem: Ω Lu := gradαdivu+βu = f in Ω, 1 u n = 0 on Ω Here, n is the outward unit normal vector of Ω The boundary value problem 1 is equivalent to a mixed formulation or a first order system least-squares problem as in [9] There are also other applications of Hdiv, eg, in iterative solvers for the Reissner-Mindlin plate the sequential regularization method for the Navier-Stokes equations For more details, see [1, 8] Center for Computation Technology, Louisiana State University, Baton Route, LA 70803, USA duksoon@cctlsuedu, Courant Institute of Mathematical Sciences, 51 Mercer Street, New York, NY 1001, USA widlund@cimsnyuedu, The work of this author was supported in part by the National Science Foundation Grant DMS in part by the US Department of Energy under contracts DE-FG0-06ER5718 Computational Solid Mechanics Structural Dynamics, Sia National Laboratories, Albuquerque, New Mexico, 87185, USA crdohrm@siagov Sia is a multiprogram laboratory operated by Sia Corporation, a Lockheed Martin Company, for the United States Department of Energy s National Nuclear Security Administration under Contract DE-AC04-94-AL

2 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann Domain decomposition methods of iterative substructuring type for solving large linear algebraic systems originating from elliptic partial differential equations have been studied extensively They are all preconditioned Krylov space methods; see [44] Among them, the balancing Neumann-Neumann BNN the finite element tearing interconnecting FETI algorithms have proven quite successful; see, eg, [14,16,17,4,9] The balancing domain decomposition by constraints BDDC methods, introduced in [10], are modified BNN methods with the global component of the preconditioner determined by a set of primal continuity constraints between the substructures For a pioneering analysis for scalar elliptic problems, see [30, 31] The BDDC methods are closely related to the dual-primal FETI FETI-DP methods; see [15,3] the same can be said about the earlier BNN one-level FETI methods Thus the spectra relevant to the performance of a BDDC a FETI-DP algorithm will be the same, except possibly for eigenvalues of 0 1, for the same set of primal constraints; see [7,7,31] Hence, we can use results for BDDC methods to obtain results for FETI-DP methods vice versa The main purpose of this paper is to construct analyze a BDDC preconditioner for vector field problems discretized with Raviart-Thomas finite elements Iterative substructuring methods for Raviart-Thomas problems were first considered in [5] we will use several auxiliary results from that study in the analysis of our method BNN, FETI, FETI-DP methods for these types of problems have been developed in [40, 4, 43] Overlapping Schwarz methods have also been introduced for vector field problems; see [, 19, 35, 36, 41] Other methods such as multigrid methods have been applied successfully in [3, 18, 3] BDDC methods have also been widely extended to other problems such as flow in porous media in [45, 46], incompressible Stokes equations in [6], Reissner-Mindlin plate models in [4, 5], advection-diffusion problems in [50] Multilevel BDDC methods were introduced in [10,47 49] other discretization methods, eg, spectral element methods discontinuous Galerkin methods, have been considered in [13,37, 38] Recently, there has also been pioneering work on isogeometric element problems; see [5] In the construction of a BDDC preconditioners, a set of primal constraints a weighted averaging technique have to be chosen these choices will very directly affect the rate of convergence Effective primal constraints are very simple for the Raviart-Thomas elements; we need only choose the average value of the normal component over the subdomain faces as primal variables However, the choice of averaging is much more intricate We will use a new type of weighted averaging technique introduced in [1] for three dimensional Hcurl problems In several previous studies on domain decomposition methods for vector field problems, see [40,4,43,5], the bound on the condition number of the preconditioned linear system depends on the ratio of the coefficients α β the diameters of the subdomains This defect has been removed in several recent studies Among them is a paper on an iterative substructuring method for two dimensional problems posed in Hcurl; see [11] In addition, a BDDC algorithm for three-dimensional problems in Hcurl has been considered in [1] An overlapping Schwarz method for threedimensional Hdiv problems has also been developed; see [36] However, we know of no previous full analysis of BDDC or FETI-DP type methods for three-dimensional Hdiv problems We will provide a BDDC method with an upper bound on the condition number which is indepedent of the values jumps of the coefficients across the interface provide a condition number bound which is polylogarithmic

3 BDDC FOR RAVIART-THOMAS VECTOR FIELDS in the number of degrees of freedom of the individual subdomains The rest of this paper is organized as follows In section, we introduce some stard Sobolev spaces, a finite element approximation based on Raviart-Thomas elements, decompositions of the interface spaces We introduce our BDDC algorithms for an interface problem define various operators used to describe the algorithms in section 3 We next provide some auxiliary results a proof of our main result in section 4 Finally, section 5 contains results of numerical experiments, which support our findings Function finite element spaces 1 Continuous spaces We will use the Sobolev spaces H 1 Ω its trace space H 1/ Ω their norms seminorms for bounded domains Let H be the diameter of Ω Then, u 1;Ω := u 1;Ω + 1 H u 0;Ω, u 1/; Ω := u 1/; Ω + 1 H u 0; Ω, where the L -norm 0;Ω the seminorms 1;Ω 1/; Ω are defined by u 0;Ω := Ω u dx, u 1;Ω := Ω u dx u 1/; Ω := Ω Ω ux uy x y 3 dxdy, respectively The weights for the L terms result from the stard definitions of the norms for a domain of diameter 1 a dilation We can also easily extend these definitions to vector-valued cases The space Hdiv;Ω is defined by with the scaled graph norm: Hdiv;Ω := {u L Ω 3 divu L Ω} u div;ω := divu 0;Ω + 1 H u 0;Ω The normal component of any u Hdiv;Ω belongs to H 1/ Ω; see [8,33] The norm for the space H 1/ Ω is given by u n 1/; Ω := u n,φ sup φ H 1/ Ω,φ 0 φ 1/; Ω The angle brackets st for the duality product of H 1/ Ω H 1/ Ω We have the following trace theorem Lemma 1 There exists a constant C, which is independent of the diameter of Ω, such that, for all u Hdiv;Ω, u n 1/; Ω CH divu 0;Ω + u 0;Ω Proof This follows directly from Green s identity on a domain with unit diameter by applying a dilation; see [5, Lemma 1] 3

4 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann by In developing our theory, we also need to work with the Hcurl;Ω space defined with the scaled graph norm: Hcurl;Ω := {u L Ω 3 curlu L Ω 3 } u curl;ω := curlu 0;Ω + 1 H u 0;Ω We finally introduce H 1 0 Ω, H 0div;Ω, H 0 curl;ω as the subspaces of H 1 Ω, Hdiv;Ω, Hcurl;Ω with a vanishing boundary value, a vanishing normal component, a vanishing tangential component on Ω, respectively Remark 1 The curl operator for two dimensions is just a simple rotation of the divergence operator In two dimensions, we therefore can use results for Hdiv; Ω to obtain results for Hcurl;Ω vice versa Finite element spaces In this paper, we will only develop our theory for tetrahedral elements but we note that our results are equally valid for hexahedral elements We first introduce a triangulation T h of Ω of tetrahedral elements We then decompose the domain Ω into N nonoverlapping subdomains Ω i of diameter H i We assume that each subdomain Ω i is a union of elements of the triangulation T h that each Ω i is simply connected has a connected boundary Later, when we develop our theory, we will introduce additional assumptions on the subdomains We also assume that the triangulation T h is shape regular with nodes matching across the interface between the subdomains The smallest diameter of the elements of Ω i is denoted by h i We will use the fraction H/h in our estimates, to denote H/h := max 1 i N {H i/h i } We also define the interface by N := Ω i \ Ω the local interfaces i by i=0 i := Ω i We will consider the lowest order Raviart-Thomas Nédélec elements on the mesh T h ; see [8, Chapter 3] [34] The Raviart-Thomas elements are conforming in Hdiv;Ω those of lowest order are defined by W := {u u K RT K,K T h u Hdiv;Ω}, where the shape function RT K is given by four scalar parameters RT K := a 1 a +b x y a 3 z for a tetrahedral element The degrees of freedom for an element K in T h are given by λ f u := 1 u n ds, f K, f f 4

5 BDDC FOR RAVIART-THOMAS VECTOR FIELDS ie, the average values of the normal components over the faces of the elements These four values determine a 1,a,a 3, b We note that the direction of the normal of a subdomain face can be fixed arbitrarily but that we should choose the same direction for all element faces of any subdomain face The basis functions of the lowest order Raviart-Thomas element space are supported in two elements of T h, with the normal component equal to 1 for one of the two elements 1 for the other on a specified face while vanishing on all others The l norm of the vector of the coefficients λ f u can be used to estimate the L norm of u; the proof of the following lemma is elementary a simple modification of [39, Proposition 631] Lemma Let K T h Then, there exist strictly positive constants, c C, depending only on the aspect ratio of K, such that for all u W, c h 3 f λ f u u 0;K C h 3 f λ f u 1 f K f K divu 0;K C f K h f λ f u, where h f is the diameter of f The following lemma follows directly from Lemma Lemma 3 inverse inequality Let K T h Then, there exist a constant C, depending only on the aspect ratio of K, such that for all u W, h K divu 0;K C u 0;K, 3 where h K is the diameter of K We also need Ŵ0, the finite element counterpart of H 0 div;ω: Ŵ 0 Ω := WΩ H 0 div;ω We will now consider the variational problem11 We obtain the stiffness matrix A by restricting this problem to Ŵ0; A is symmetric positive definite When developing our theory, we will need several additional spaces Let S be the space of continuous, piecewise linear functions on the tetrahedral elements, let S 0 be the subspace of elements of S which vanish on Ω Let Q be the space of piecewise constant functions on the same elements Finally, let X be the space of the lowest order Nédélec elements We recall that the Lagrange P 1, Raviart-Thomas, Nédélec spaces are conforming finite element spaces in H 1, Hdiv, Hcurl, respectively LetV h F h bethesetofverticesfacesoft h,respectively Theinterpolation operators I h, RT h for sufficiently smooth functions u H 1 v Hdiv onto S W, respectively, are defined as follows: I h u := upφ P p RT h v := λ f vφ RT f, p V h f F h where φ P p φrt f are the basis functions of P 1 Raviart-Thomas finite elements associated with the node p the element face f, respectively We also denote by h the projection operator from L onto Q 5

6 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann We finally recall the following error estimate for the Raviart-Thomas interpolation operator a commuting property Lemma 4 For any u H 1 Ω 3, we have u RT h u 0;Ω Ch u 1;Ω 4 Proof See [6, Lemma 55] Lemma 5 Let u be sufficiently regular Then, the following commuting property holds: div RT h u = h divu 5 Proof See [6, Property 53] We note that the commuting property 5 is a part of the discrete de Rham diagram described, eg, in [33, section 57] 3 The discrete problem The description of the BDDC algorithm its analysis require the introduction of a number of spaces Let W i be the space of the lowest order Raviart-Thomas finite elements on Ω i with a zero normal component on Ω Ω i We decompose W i into two subspaces, an interior space W i I an interface space W i The interface space Wi is then decomposed into a primal space W i a dual space Wi Hence, we have the following decompositions: W i := W i I W i := Wi I W i Wi We will also use the following product spaces: We then have N N W 0 := W i, W I := W i I, W := i=1 W := N i=1 i=1 W i, W := N i=1 W i N i=1 W 0 = W I W = W I W W W i, In general, the functions in W have discontinuous normal components across the interface while those of the finite element solutions are continuous We denote the subspace with continuous normal components by Ŵ We also consider a space W, for which all the primal constraints are enforced We can then decompose Ŵ W into Ŵ Ŵ W Ŵ, respectively, where Ŵ is the continuous dual variable subspace Ŵ is the continuous primal variable subspace We can now obtain the local stiffness matrix A i by restricting the bilinear form to Ω i replacing Hdiv;Ω i by the finite element space W i But before we do so, it is convenient to make a change of variables by introducing a basis for the primal degrees of freedom a complementary basis for the dual subspace W i Here we 6

7 BDDC FOR RAVIART-THOMAS VECTOR FIELDS can follow the recipes of [7, subsection 33] closely For our problem, the only primal variables will be the averages of the normal component over the subdomain faces, after our change of variables, the complementary dual subspace will be represented by elements for which the same averages vanish We note that there is also evidence that such a change of variables enhances the numerical stability of BDDC FETI-DP algorithms; see [] After this change of variable, the restriction of our problem to the subdomain Ω i can be written in terms of a local stiffness matrix A i as A i u i I u i u i = A i II A i I A i I A i I Ai I A i Ai A i Ai u i I u i u i = f i I f i f i, 6 where u i I W i I, u i Wi, ui Wi We can then obtain the global linear system of algebraic equations by assembling the local subdomain problems: A u I u u = A II A I A I A I A A A I A A where u I W I, u Ŵ, u Ŵ 3 The BDDC methods u I u u = f I f f, 7 31 Some useful operators We will now define several operators which perform restrictions, extensions, scalings, averaging between different spaces They will be used to present our algorithm in our proofs We first consider the restriction operators R i maps the space Ŵ to the subdomain subspace W i Similarly, we can define R i : W W i Moreover, Ri : W W i Ri : Ŵ W i mapglobalinterfacevectorsdefinedontotheircomponentson i R R are the restrictionoperatorsfrom the intermediatespace W to W Ŵ, respectively Similarly, we can define the restriction operator R i from Wi to Wi R R are the direct sums of the R i is the direct sum of R the Ri, respectively Furthermore, R : Ŵ W R i, where R represents the restriction from Ŵ to i Ŵ R maps the space Ŵ into W i We next introduce scaling matrices, D i, acting on the degrees of freedom associated with the i They are combined into a block diagonal matrix should provide a discrete partition of unity, ie, R T D 1 D D N R = I 31 We can now define a scaled operator R i D, := Di R i scaling matrix D i Another locally scaled operator by pre-multiplying Ri R i D, by the is defined by Ri Ri D, We next consider a globally scaled operator R D, defined by the direct sum of R 7

8 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann the i R D, We note that R T R D, = R T D, R = I 3 Finally, we introduce an averaging operator E D : W Ŵ by E D := R RT D, 33 This operator, which is a projection, provides a weighted average across the interface We will provide details on our choice of scaling matrices in subsection 33 3 Schur complements a reduced interface problem Before we introduce the BDDC algorithm, we eliminate all interior unknowns locally by using direct solvers After this step, we obtain the local Schur complements: S i := Ai Ai I Ai 1 II A i I, 34 where A i := [ A i Ai A i Ai ] We will also consider the global Schur complement S, the direct sum of the local Schur complements S i By using the local Schur complements, we can build a reduced global interface problem The global problem is given by where Ŝ := N i=1 Ŝ u = g, 35 R it S i Ri := RT S R 36 g := N i=1 R it { [ ] [ f f A i I A i I ] A i 1 II f i I } 37 We note that once u i has been computed, we can find the interior values ui I by solving the following equation: [ ] A i II ui I = f i I A i I Ai I u i We will construct a preconditioner for the interface problem The BDDC algorithm We now define a different Schur complement S := R T S R 8

9 BDDC FOR RAVIART-THOMAS VECTOR FIELDS After eliminating the interior residuals, we obtain the following linear system which also defines S : u 1 0 I u 1 R à 1 R S u u N = 0 I u N R N R S u u R S u We note that à is the partially subassembled stiffness matrix S is the partially assembled Schur complement Hence, we need to further assemble it to obtain the fully assembled Schur complement Ŝ By using restriction extension operators, we find that Ŝ = R T S R We can then rewrite the interface problem 35 as The BDDC preconditioner has the following form: R T S R u = g 38 M 1 = R T D, S 1 R D, 39 Here, S 1 can be obtained by using a block Cholesky factorization of à as in [7, section 4]: with := R N [ T S 1 where S := i=1 Φ := R T R T N i=1 R it 0 R it N [ i=1 A i [ ] [ A i II A i I 0 R it A i I A i A i I A i ] [ A i II A i I ] [ A i II ] 1[ 0 R i A i I A i I A i A i I A i ] 1 [ ] R +ΦS 1 ΦT ] 1 [ A it I A it ] A it I A it R i 310 ] R i The first term of 310 is related to the local Schur complements the second term to the coarse-level problem associated with the primal space In order to specify the algorithm completely, we need to define the weighted averaging operator D i Conventional weighted averaging techniques, known as stiffness ρ scalings, are described in [10, 31] However, these methods are designed for constant coefficients or for one variable coefficient For more than one variable coefficient, we need a different approach we will use the new weighted averaging technique introduced in [1] for Hcurl problems We first consider the Schur complements related to, the common face of two adjacent subdomains Ω i Ω j Two local stiffness matrices associated with are given, for k = i j, by [ ] k A II A k := A k I 9 A k I A k 311

10 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann The two Schur complements associated with are given by S k := A k A k 1 I Ak k II A I 31 for k = i j We will use the scaling matrices D i j := S i +S j 1S i We 1 note that we can apply the operator S i +S j by solving a Dirichlet problem on Ω i Ω j with zero Dirichlet boundary conditions The scaling operator D i is then given by D i := D i j 1 D i j D i j k, 313 where j 1,j,,j k N i N i is the set of indices of the Ω j s i j which share a subdomain face with Ω i With this scaling operator D i the operators of section 31, the BDDC preconditioner 39 is completely determined We remark that there are two scaling matrices for each subdomain face that it is easy to show that the partition of unity condition 31 is satisfied, ie, that N i=1 We obtain the following preconditioned linear system: R i T D i R i = I 314 M 1 Ŝ u = R T D, S 1 R D, RT S R u = R T D, S 1 R D, g = M 1 g 315 We use the preconditioned conjugate gradient method to solve the linear system315 We next define norms related to the Schur complements The S -normis givenby u S := u T S u for all u W Similarly, the local norms associated with S i T i S u i S i are defined by u i := u i T i u S i S ui i := u i S i The norms Ŝ are defined by û S Ŝ := û T Ŝû ũ S := ũ T S ũ for û Ŵ ũ W, respectively Using the definition of R R, we can easily see that û Ŝ = R û S ũ S = R ũ S In developing FETI-DP BDDC theory, we need to estimate jump averaging operators the traditional variants require bounds for each subdomain face involve norms associated with the both subdomains that share a face, eg, [45, Lemma 53] Such results usually require relatively strong assumptions such as a shape regular coarse triangulation a quasi-uniform fine triangulation The following lemma for our averaging technique, introduced earlier in this subsection, will greatly simplify the analysis will reduce our work to estimates for individual subdomains We also no longer need to use of a finite element extension theorem Lemma 31 For any u i W i, let u ij be the restriction of u i to a subdomain 10

11 BDDC FOR RAVIART-THOMAS VECTOR FIELDS face = Ω i Ω j We then have the following estimates: 1S S i +S j i S u ij u ij i i S 316 1S S i +S j i S u ij u ij i j S 317 Proof These bounds are equivalent to the inequalities 1S S i S i +S j i 1S S i +S j i S i 1S S i S i +S j j 1S S i +S j i S i These inequalities follow easily by exping any function u ij in the eigenvectors of the generalized eigenvalue problem defined by the two Schur complements using the fact that all its eigenvalues are positive 4 Technical tools the main result From now on, we will assume that the coefficients α β are constant in each subdomain that they thus only can have jumps across the interface We will also assume that all subdomains are convex polyhedra We can then write the bilinear form of 11 in the following way: au,v := N a i u,v, where the local energy bilinear form for each subdomain Ω i are defined as follows: a i u,u := α i divudivudx+β i u udx Ω i Ω i i=1 41 Technical tools To have access to all the technical tools that we need, we from now on have to assume that our subdomains are polyhedra that each of them is the union of a few shape-regular large tetrahedra, which define a coarse finite element mesh T H In our proofs, we also need some stard tools for the space SΩ i, which we can borrow from [44, subsection 46] They are related to the subdomain faces the wire basket W i, which is the union of the edges vertices of Ω i Lemma 41 There are functions ϑ W i SΩ i ϑ F j SΩ i such that ϑ W i + Ω i ϑ Fij = 1, 41 where ϑ Fij 0 on Ω i \ Moreover, for any u SΩ i, there exists a constant independent of h i H i, such that I h ϑ W iu 1;Ω i C1+logH i /h i u 1;Ω i 4 11

12 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann Ih ϑfij u 1;Ω i C1+logH i /h i u 1;Ω i 43 In addition, the following estimate holds: u 0; C1+logH i /h i u 1;Ω i 44 Proof See [44, subsection 46] Unlike for the gradient operator, it is quite complicated to classify the kernel the range of the curl divergence operators The discrete regular decompositions given in [1] provide useful tools to analyze problems posed in Hcurl Hdiv We can then apply techniques developed for H 1 -functions by using Lemma 4 Hiptmair-Xu decomposition Let Ω i be a convex polyhedron Then, for all v h WΩ i, there exist Ψ h SΩ i, q h XΩ i, ṽ h WΩ i such that v h = ṽ h + RT h Ψ h +curlq h 45 h 1 i ṽ h + Ψ h 1;Ω i C divv h, 46 curlq h + Ψ h C v h 47 Proof See [1, Lemma 51 5] We note that this important paper was preceeded by [0], which concerns another application of the same decomposition We next introduce a stable operator which provides a divergence-free extension Lemma 43 divergence free extension There exists an extension operator H i from the normal trace space of W i, such that, for all u Wi, Hi µ n = µ, div H i µ = 0, where µ := u n Moreover, H i µ C µ 1/; Ωi 0;Ωi Proof See [5, Lemma 43] or [51, Lemma 6] We then have the following estimate for the discrete harmonic extensions which have the minimal energy property for a given normal trace For more detail, see [44, section 10] [36, section 31] Corollary 44 discrete harmonic extension Let H i be the energy minimizing discrete harmonic extension For all u W i, we have Furthermore, H i µ n = µ := u n α i divh i µ +β i H i µ Cβ i µ 1/; Ω i 1

13 BDDC FOR RAVIART-THOMAS VECTOR FIELDS Proof H i is the minimal-energy extension operator for a given subdomain interface Therefore, we have α i divh i µ +β i H i µ div α i Hi µ +β i Hi µ But div H i µ = 0 thus by using Lemma 43, α i divh i µ +β i H i µ Cβ i µ 1/; Ω i We next considerthe coarseinterpolationoperator RT H ontothe Raviart-Thomas space of the coarse mesh T H Lemma 45 stability of the coarse interpolant For all u i W i, we have the following estimates: div RT H u i divu i 48 RT H u i C 1+logH i /h i u i 0;Ωi +H i divu i 0;Ωi, 49 where the constant C depends neither on h i nor on H i Proof See [51, Lemma 4] or [5, Lemma 41] Lemma 46 Let u Fij W i with λ f ufij = 0, f Ωi \ Let u H := H i RT H u n assume that α i β i Hi We then have a i u H,u H Ca i ufij,u Fij, 410 where C is indepedent of H i, h i, α i, β i Proof We will modify the proof of [36, Lemma 56] Let Ω di i, Ω i be the set of all points which are within a distance d i of We first introduce a piecewise linear scalar cut-off function χ Fij which has the value 1 on vanishes in Ω i \Ω di i, for some h i d i H i Moreover, χfij 1 χfij C/d i We next consider the coarse basis function related to the discrete harmonic extension The basis function φ RT is obtained from the stard basis function φ RT, ie, φ RT = H i φ RT n We note that φ RT CHi 3 divφ RT F j CH i i We can then write the function u H as follows: u H = λ Fij ufij φrt 411 In order to estimate the energy of u H, we estimate the degree of freedom λ Fij ufij the energy of φ RT separately 13

14 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann We first estimate the degree of freedom By the divergence theorem, the fact that χ Fij u Fij n vanishes on Ω d i i, \, λ Fij ufij = Fij λ Fij χfij u Fij = χfij u Fij nds = = div χ Fij u Fij dx χfij u Fij nds i,f Ω d i ij i,f \ ij Ω d i Ω d i i, div χ Fij u Fij dx 41 By using the Cauchy-Schwarz inequality the shape-regularity, we obtain λfij ufij C d i H i C d i H i C d i H i div χfij u Fij 0;Ω d i i, χfij divufij + χfij ufij divufij +C 1 Hi d ufij 0;Ω i i 413 We now estimate the basis function From the minimal energy property, we find Hence, we have div α φrt i α i div RT h +β φrt i χ Fij φ RT +β i RT h χ Fij φ RT Cα i di +H i/d i +Cβi H id i Cα i H i/d i +Cβ i H id i 414 a i u H,u H λfij RT λfij = α i ufij div φ φrt +β i ufij F j i C α i +β i di divufij +C α i /d i +β i ufij 415 Let d i = max{ α i /β i,h i } We note that h i d i H i By using 415 Lemma 3, we obtain a i u H,u H Ca i ufij,u Fij 416 We next introduce a partition of unity associated with the faces of an individual subdomain Ω i as in [44, Chapter 101], where F Ω i ζ F 1,ae on Ω i \ Ω, { 1, x F ζ F x = 0, x Ω i \F

15 BDDC FOR RAVIART-THOMAS VECTOR FIELDS We then have the following estimates for the subdomain face components; we recall that for u i W i, u i nds = 0 for each subdomain face Ω i Lemma 47 For any u i W i for any uh i W i, let µ i := u i n, µ Fij = ζ Fij µ i, µ H i := u H i n Then, µ Fij 1/; Ω i C1+logH i /h i 1+logH i /h i µ i +µ H i + µ 1/; Ω i i 1/; Ω i where C is independent of µ H i, H i, h i Proof See [5, Lemma 44] or [51, Lemma 7] Lemma 48 For any u i W i, there exist v i,fij v H i, W i such that { λf vi,fij = λf u i if f ; 418 λ f vi,fij = 0 if f Ωi \ λ f vi,f H ij = λ f RT H u i if f ; λ f vi,f H ij = 0 if f Ω i \ 419 Furthermore, a i v i,fij v H i,,v i,fij v H i, C1+logH i /h i a i u i,u i, 40 where C is independent of α i, β i, H i, h i Proof We will only consider the case where α i β i Hi since the proof is straightforward by using Corollary 44 Lemmas 47, 1, 45 if β i Hi α i; for more details, see [51, 5] By using Lemma 4, we can find ũ i, Ψ i, q i such that u i = ũ i + RT h Ψ i +curlq i 41 Wenotethatdivcurlq i = 0that4647provideboundsforthedifferent terms We first consider ũ i We define ũ i,fij = f λ f ũ i φ RT f, where φ RT f is the Raviart-Thomas basis function associated with the face f By using Lemmas, 3, 4, we have ũi,fij C h 3 i λ f ũ i C ũ i f Ch i divu i C u i 4 divũ i,fij C f h i λ f ũ i C h 1 i ũ i C divu i 43 Hence, from 4 43, a i ũi,fij,ũ i,fij Cai u i,u i 44 15

16 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann We alsodefine ũ H i, = H i RT H ũi, n Byusing Lemma46 44, weobtain a i ũ H i,,ũ H i, Ca i ũ i,fij,ũ i,fij = Ca i u i,u i 45 We next consider the second term RT h Ψ i of 41 Let Ψ i,fij := I h ϑfij Ψ i By using Lemmas 41 4, we obtain Ψ i,fij C Ψ i C u i divψ i,fij C Ψ i,fij 1;Ω i C1+logH i /h i Ψ i 1;Ω i C1+logH i /h i divu i Moreover, by using Lemma 4, an inverse estimate, Lemma 5, we obtain RT Ψi,Fij h Ψ,Fij + Ψ i,fij RT i h Ψi,Fij +Ch i Ψi,Fij Ψi,Fij 1;Ω i C Ψi,Fij, div RT Ψi,Fij h = h divψi,fij divψi,fij Therefore, RT Ψi,Fij h C u i 46 div RT Ψi,Fij h C1+logH i /h i divu i 47 Hence, from 46 47, we obtain a i RT h Let Ψ H i, = H i RT H Ψi,Fij, RT h Ψi,Fij C1+logHi /h i a i u i,u i 48 RT h Ψ i, n By using Lemma 46 48, we have a i Ψ H i,,ψ H i, Ca i RT h Ψi,Fij, RT Ψi,Fij h C1+logH i /h i a i u i,u i 49 Let Ψ i,wi = RT h I h ϑ W iψ i Ψ i, Fij = f λ f Ψ i,wi φ RT f By using Lemmas, 4, 41, 4, an inverse estimate, an estimate for the P 1 basis functions of SΩ i, we obtain Ψi, Fij C f h 3 iλ f Ψ i,wi C Ψ i,wi C I h ϑ W iψ i C Ψ i C u i

17 BDDC FOR RAVIART-THOMAS VECTOR FIELDS divψi, Fij C f h i λ f Ψ i,wi C 1 h i Ψ i,wi C 1 h i I h ϑ W iψ i C Ψ i 0; C1+logH i /h i Ψ i 1;Ω i Hence, combining , we obtain C1+logH i /h i divu i 431 a i Ψi, Fij,Ψ i, Fij C1+logHi /h i a i u i,u i 43 Let Ψ H i, := H i RT H Ψ i, n From Lemma 46 43, we obtain a i Ψ H i,,ψ H i, Ca i Ψi, Fij,Ψ i, Fij C1+logH i /h i a i u i,u i 433 We finally consider the curlq i -term of 41 Let qi H := RT H curlq i, q i := curlq i n, qi H := qi H n Moreover, let qi,f H ij := H i ζfij qi H qi,fij = H i ζfij qi qi H From Corollary 44 Lemma 47, we obtain a i qi,fij,q i,fij Cβ ζfij i qi qi H 1/; Ω i Cβ i 1+logH i /h i 1+logH i /h i q i 1/; Ω i + qi qi H 1/; Ω i Cβ i 1+logH i /h i 1+logH i /h i q i 1/; Ω i + q H i 1/; Ω i Cβ i 1+logH i /h i q i 1/; Ω i +Cβ i 1+logH i /h i q H i 1/; Ω i 434 We note that divqi H 0;Ωi divcurlq i 0;Ωi = 0 from Lemma 45 Hence, by using Lemmas 1 45, we obtain q H i 1/; Ω i C Hi divq H i 0;Ω + q H i = C 0;Ω q H i 0;Ω C 1+logH i /h i curlq i 0;Ω +H i divcurlq i 0;Ω q i 1/; Ω i C = C1+logH i /h i curlq i 0;Ω 435 Hi divcurlq i 0;Ω + curlq i 0;Ω = C curlq i 0;Ω 436 Therefore, by combining 434, 435, 436, Lemma 4, we obtain a i qi,fij,q i,fij C1+logHi /h i β i curlq i 0;Ω C1+logH i /h i β i u i 0;Ω C1+logH i /h i a i u i,u i

18 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann We can now define v i,fij as follows: v H i, by v i,fij := ũ i,fij + RT h Ψi,Fij +Ψi, Fij +q H i, +q i, 438 v H i, := ũ H i, +Ψ H i, +Ψ H i, +q H i, 439 Then, v i,fij v H i, satisfy the conditions , respectively Furthermore, we obtain the following estimate by using 44, 45, 48, 43, 49, 433, 437: a i v i,fij v H i,,v i,fij v H i, C1+logH i /h i a i u i,u i 440 We conclude this section by introducing an extension lemma which extend the face components to the entire interface Lemma 49 face extension lemma For all v i W i, we can find vh i W i which satisfies vi v H F i nds = 0 for each subdomain face F of Ωi Let v ij vij H be the restriction of v i vi H to the subdomain face, respectively We then have the following estimate: vij vij H C1+logH S i i /h i v i, 441 S i where C is independent of h i H i Proof Letu i := H i v i n Then, bylemma48,thereexistv i,fij v H i, such that v ij vij H n,hi ζfij vi vi H n a i v i,fij vi,f H ij,v i,fij vi,f H ij C1+logH i /h i a i u i,u i S i = a i Hi ζfij vi v H i = C1+logH i /h i v i S i 4 A stability estimate The averaging operator E D, defined in 33, satisfies the following estimate: Lemma 410 There is a constant C, which is independent of H i, h i, α i, β i, such that, for all u W, E D u S C1+logH/h u S 44 Proof We have that E D u S u S + u E D u S = u S + R u E D u S N = u S + R i u E D u i=1 18 S i 443

19 BDDC FOR RAVIART-THOMAS VECTOR FIELDS We will use the scaling operators introduced in subsection 33 Let u i := R i u We then have u E D u = D j i u i u j on := Ω i Ω j Let u ij u ji be the restrictions of u i u j to, respectively We then obtain R i u E D u S i = i j, Ω i D i j u ij u ji S i 444 Letu H i W i definedbyh i RT H H iu i n n Wethenhave F ui u H i nds = 0 for each subdomain face F of Ω i We define u H j similarly Let u H ij uh ji be the restrictions of u H i u H j to By our choice of primal constraints, u H ij = uh ji on From Lemmas 31 49, we obtain D i j u ij u ji = D i S i j D i j uij u H ij uji u H ji S i uij u H ij uij u H ij S i C +D i S i j + uji u H ji 1+logH i /h i u i S i By summing over all faces subdomains, we obtain uji u H ji S j S i +1+logH j /h j u j S j 445 E D u S C1+logH/h u S Main result We consider the preconditioned linear system M 1 Ŝ u = M 1 g Theorem 411 condition number estimate The condition number of the preconditioned linear system M 1 Ŝ u = M 1 g satisfies κ M 1 Ŝ C1+logH/h 447 Proof We need upper lower bounds on the eigenvalues of the generalized eigenvalue problem Ŝu = λmu We note that the lower bound is always greater than or equal to 1 the upper bound is given by the norm of the averagingoperator 33; see [31, Theorem 5], [6, Theorem 1], [46, Theorem 61] Therefore, 447 follows from Lemma Numerical results 51 The two-dimensional case We have applied the BDDC algorithm to our model problem 1 For algorithmic details, we follow [7] section 33 We set Ω = 0,1 decompose the unit square into N subdomains Each subdomain has a side length H = 1/N Moreover, we assume that the coefficients α β have jumps across the interface between the subdomains with a checkerboard pattern as in Fig 51 We discretize the model problem 1 by using the lowest order Raviart- Thomas finite elements for triangles use the preconditioned conjugate gradient 19

20 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann Fig 51 Checkerboard distribution of the coefficients D case method to solve the discretized problem The iteration is stopped when the l norm of the residual has been reduced by a factor of 10 6 We have three different sets of experiments We first fix the value of β vary α Second, we fix the value of α vary β Tables 51 5 show the first two sets of results For the final set of experiments, we use a different distribution, instead of the checkerboard distribution, We first generate N rom numbers {r αi } i=1,,n {r βi } i=1,,n in [ 3,3] with a uniform distribution We then assign 10 rα i 10 r β i for α i β i, respectively The last results can be found in Table 53 In Fig 5, we see that the condition number grows quadratically with the logarithm of H/h; it is insensitive to the jumps of coefficients Table 51 Condition numbers Cond iteration counts Iters: D case α i = α w = 1 for the white subregions α i = α b for the black subregions as indicated in a checkerboard pattern as in Fig 51, β i 1, N = 4 H/h = 4 H/h = 8 H/h = 16 H/h = 3 H/h = 64 Cond Iters Cond Iters Cond Iters Cond Iters Cond Iters α i = α i = α i = α i = α i = Table 5 Condition numbers Cond iteration counts Iters: D case β i = β w = 1 for the white subregions β i = β b for the black subregions as indicated in a checkerboard pattern as in Fig 51, α i 1, N = 4 H/h = 4 H/h = 8 H/h = 16 H/h = 3 H/h = 64 Cond Iters Cond Iters Cond Iters Cond Iters Cond Iters β i = β i = β i = β i = β i =

21 BDDC FOR RAVIART-THOMAS VECTOR FIELDS Table 53 Condition numbers Cond iteration counts Iters with rom coefficients: D case N = 4 H/h = 4 H/h = 8 H/h = 16 H/h = 3 H/h = 64 Cond Iters Cond Iters Cond Iters Cond Iters Cond Iters Set Set Set Set Set Fig 5 Estimated condition numbers in Table 53 least-squares fit to a degree polynomial in logh/h, versus H/h D case 5 The three-dimensional case For the three-dimensional case, we use the unit cube 0,1 3 for Ω In a way similar to the two-dimensional case, we decompose the domain into N 3 subdomains with the side length H = 1/N We use the lowest order hexahedral Raviart-Thomas elements for this case a similar checkerboard distribution of the coefficients as in the two-dimensional case; see Fig 53 We use the stopping criteria of reducing the l norm of the residual by a factor of 10 6 for the preconditioned conjugate gradient method Other general settings are also similar to those of the two-dimensional case We find that the results for the three-dimensional case are quite similar to those of the two-dimensional case; see Tables 54, 55, 56, Fig 54 The condition numbers depend quadratically on the value of log H/h are independent of the jumps of coefficients across the interface 53 Jumps inside subdomains We report on numerical experiments for the case where coefficients have jumps inside the subdomains We follow the general settings in section 5 for these experiments but use different coefficient distributions For each subdomain Ω i, we let Ω o i = {x,y,z 1/4 x o,y o,z o 1/,wherex o = x/h x/h,y o = y/h y/h,z o = z/h z/h } Here, x = max{m Z m x}, where Z is the set of integers We use the α i β i specified in section 5 ascoefficientsfor Ω i \Ω o i For Ωo i, weassign100α i 100β i ascoefficients in the black subregions α i β i as coefficients in the white subregions Table show the results We note that our theory does not cover these cases However, we see that our method works well even though we have discontinuities inside the 1

22 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann Fig 53 Checkerboard distribution of the coefficients 3D case Table 54 Condition numbers Cond iteration counts Iters: 3D case α i = α w = 1 for the white subregions α i = α b for the black subregions as indicated in a checkerboard pattern as in Fig 53, β i 1, N = 4 H/h = H/h = 4 H/h = 8 H/h = 16 Cond Iters Cond Iters Cond Iters Cond Iters α b = α b = α b = α b = α b = Table 55 Condition numbers Cond iteration counts Iters: 3D case β i = β w = 1 for the white subregions β i = β b for the black subregions as indicated in a checkerboard pattern as in Fig 53, α i 1, N = 4 H/h = H/h = 4 H/h = 8 H/h = 16 Cond Iters Cond Iters Cond Iters Cond Iters β b = β b = β b = β b = β b = Table 56 Condition numbers Cond iteration counts Iters with rom coefficients: 3D case N = 4 H/h = H/h = 4 H/h = 8 H/h = 16 Cond Iters Cond Iters Cond Iters Cond Iters Set Set Set Set Set subdomains 54 The effect of using a conventional weighted averaging technique In this section, for a comparison, we report on some numerical experiments using conventional techniques We have performed three different types of experiments with

23 BDDC FOR RAVIART-THOMAS VECTOR FIELDS Fig 54 Estimated condition numbers in Table 56 least-squares fit to a degree polynomial in logh/h, versus H/h 3D case Table 57 Condition numbers Cond iteration counts Iters Specified values as indicated in Table 54 with jumps inside subdomains N = 4 H/h = 4 H/h = 8 H/h = 16 Cond Iters Cond Iters Cond Iters α i = α i = α i = α i = α i = Table 58 Condition numbers Cond iteration counts Iters Specified values as indicated in Table 55 with jumps inside subdomains N = 4 H/h = 4 H/h = 8 H/h = 16 Cond Iters Cond Iters Cond Iters β i = β i = β i = β i = β i = the same set of coefficients distribution The first set of experiments, named sc, is based on the weighted averaging techniques described in 313 In the second, diag, we use the conventional methods described in [10,31] In this case, the scaling is based on the diagonal entries of each subdomain matrix We use the cardinality in the last set, card For Raviart-Thomas elements, only two subdomains share a subdomain face in common Hence, we use 1/ as scaling factors For other general settings, we follow section 5 As we see in Table 59, our weighted averaging technique works well while the others are sensitive to the discontinuities across the interface REFERENCES 3

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26 Duk-Soon Oh, Olof B Widlund, Clark R Dohrmann Trans Numer Anal, 0 005, pp [46], A BDDC algorithm for flow in porous media with a hybrid finite element discretization, Electron Trans Numer Anal, 6 007, pp [47], Three-level BDDC in three dimensions, SIAM J Sci Comput, 9 007, pp [48], Three-level BDDC in two dimensions, Internat J Numer Methods Engrg, , pp [49], A three-level BDDC algorithm for a saddle point problem, Numer Math, , pp [50] Xuemin Tu Jing Li, A balancing domain decomposition method by constraints for advection-diffusion problems, Commun Appl Math Comput Sci, 3 008, pp 5 60 [51] Barbara I Wohlmuth, Discretization methods iterative solvers based on domain decomposition, vol 17 of Lecture Notes in Computational Science Engineering, Springer- Verlag, Berlin, 001 [5] Barbara I Wohlmuth, Andrea Toselli, Olof B Widlund, An iterative substructuring method for Raviart-Thomas vector fields in three dimensions, SIAM J Numer Anal, , pp