Extended variational formulation for heterogeneous partial differential equations

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1 To appear in Computational Methods in Applied Mathematics (CMAM) Extended variational formulation for heterogeneous partial differential equations Pablo Blanco, Paola Gervasio, Alfio Quarteroni Abstract We address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out. Mathematical Subject Classification: 35Q35, 65N55, 65N35, 65F8 Keywords: coupled elliptic-hyperbolic, advection-diffusion-reaction, domain decomposition, spectral elements, preconditioning Introduction Subdomain splitting is an interesting path towards multiphysics, i.e. the use of mathematical models based on different kinds of partial differential equations to address physical problems of heterogeneous nature in different subregions of a given computational domain. In this presentation we focus on the framework of advection-diffusion equations with boundary layer solutions. The complete elliptic advection-diffusion problem is solved only in a small subdomain embodying the layer, while the reduced hyperbolic model, that is obtained by neglecting the diffusion term, is used on the remainder of the computational domain. Gastaldi et al. (see [8, 7]) analysed this problem and they derived a suitable set of matching conditions at the interface between subdomains, which guarantee the well posedness of the heterogeneous problem. Such conditions express the continuity of the velocity field across the inflow part of the interface (i.e. the part of interface which is an inflow for the hyperbolic domain) and the continuity of the fluxes (i.e. the conormal derivatives associated to the differential operators) across the whole interface, If properly split, these conditions can be used to numerically solve the problem through a Dirichlet/Neumann-like algorithm. Different kinds of boundary conditions were derived in [, 5]. The set of interface conditions proposed in [8] can be formulated in two ways which are equivalent at continuous level and yield two equivalent Steklov-Poincaré formulations, but they differ one another in imposing the continuity of fluxes on the interface. The first set LNCC, Laboratório Nacional de Computação Científica and INCT-MACC, Instituto Nacional de Ciência e Tecnologia em Medicina Assistida por Computação Científica, Av. Getúlio Vargas 333, Quitandinha, Petrópolis, RJ, Brazil. pjblanco@lncc.br Department of Mathematics, University of Brescia, 533 Brescia (Italy). gervasio@ing.unibs.it MOX, Department of Mathematics F. Brioschi, Politecnico di Milano, 33 Milano (Italy) and SB-MATHICSE-CMCS École Polytechnique Fédérale de Lausanne, CH-5 Lausanne, (Switzerland). alfio.quarteroni@epfl.ch

2 of interface conditions (IC) enforces the continuity of fluxes on the whole interface, while the second set (IC) exploits the continuity of traces across the inflow interface, so that, there, the continuity of fluxes is achieved by imposing null normal derivative of the elliptic solution. A special attention is given in this work to the comparison of the two sets of interface conditions IC and IC. At discrete level, the formulation of flux continuity on the inflow interface is responsible for the efficiency of the corresponding approach, the interface conditions IC perform better than IC, mainly when the viscosity is small. More recently ([3, 4]), an extended variational approach has been proposed to solve heterogeneous problems, including those coupled problems featuring different geometrical dimensions. The starting point of this approach is the reduction of the geometrical dimension of the problem in a part of the computational domain, motivated by the need of reducing the computational cost in applications of practical interest. Such geometrical reduction entails different kinematic assumptions within the different subregions of the domain, so that a heterogeneous problem arises. This problem is then re-formulated globally (i.e. on the whole computational domain), by resorting to a saddle-point approach in which the constraint expresses the continuity of the solution across either the interface or a subset of it. The associated Lagrange multipliers are the fluxes across the interface, more precisely, the conormal derivatives associated to the differential operators defined in the different subdomains. Consequently, the matching conditions at the interface are identified by writing the Euler-Lagrange system associated to the saddle-point problem. In this paper we re-formulate the heterogeneous advection/advection-diffusion problem in terms of the extended variational formulation. How to choose the functional spaces and the bilinear forms is suggested by the well-posedness of the saddle-point problem. A first goal of this paper consists to find out which interface conditions, alternative to those proposed in [8], are admissible and lead to a well-posed extended variational formulation. In Section 3 we prove that the saddle-point problem whose constraint enforces the continuity of the solution only across the inflow (and not on the whole) interface is well-posed and the associated Euler-Lagrange equations provide the same interface conditions given in [8]. In Section 4 the extended variational problem is re-formulated as an interface problem in terms of Steklov-Poincaré operators. Four possible coupling strategies are analyzed depending on how the interface unknowns are chosen, as traces of order zero (Dirichlet s) or one (Neumann s). They are named DD, NN, DN and ND, where the first letter identifies the kind of trace (D=Dirichlet, N=Neumann) used for the hyperbolic solution, the second one that for the elliptic problem. In Section 5.3 we propose optimal preconditioners for the finite dimensional counterpart of the extended interface problems. Such preconditioners are built as inexact factorizations of the primal matrices in which the Schur complement matrix is replaced by its optimal preconditioner. No preconditioners for extended problems have been developed so far and it has been proved tat the proposed preconditioners have good properties. In all cases, the condition number of the preconditioned matrices are bounded from above independently of the discretization parameters (grid space and polynomial degree). The second aim of this work is to set up a systematic comparison between the heterogeneous approach ([8]) and XVF, from the computational point of view. We discretize the PDE s by conforming Spectral Elements and we solve the interface problems by the preconditioned Bi-CGStab method [4].

3 The non-conforming discretization of the hyperbolic-elliptic heterogeneous problem is a matter of current research ([5]). A paper that deals with a similar class of problems (anisotropic semidefinite diffusion problems with advection) is [], where the authors approximate the solution by Discontinuous Galerkin methods. We compare the Steklov-Poincaré approach with all the proposed extended variational forms for what concerns accuracy, boundedness of the condition number of the preconditioned matrix, and computational efficiency. In order to analyze the accuracy, we measure trace and flux jumps across interface between elliptic and hyperbolic solutions. The best accurate approaches are those based on the Steklov-Poincaré equation and the Dirichlet-Dirichlet version of the XVF, while both Dirichlet-Neumann and Neumann-Neumann forms are ill-posed for advection-dominated problems when interface conditions IC are considered. From the computational point of view, the most efficient approaches are those based on the Steklov- Poincaré equation, as they entail the lowest number of elliptic and hyperbolic subproblems at each preconditioned Bi-CGStab iteration. In conclusion, XVF is a valid alternative to the heterogeneous form proposed in [8] from the theoretical point of view. Moreover, it is interesting to see that XVF provides the same set of interface conditions derived in [8] and then the latter approach strengthens the validity of the former one. At discrete level, Extended Variational Formulation is not so efficient as the Steklov-Poincaré approach, as a matter of fact the computational complexity of the augmented linear system associated to XVF is larger than that of Steklov-Poincaré equation. An outline of the paper is as follows. In Section we introduce the problem setting and resume known theoretical results about the well-posedness of the heterogeneous advection/ advection-diffusion problem. In Section 3 we present and analyse the XVF of the heterogeneous problem, while in Section 4 we write the interface problem (XIP) associated to XVF. Section 5 is devoted to the discretization of the XIP, the development and analysis of suitable preconditioners for XIP and the numerical results comparing Steklov-Poincaré formulation and XIP. Problem setting We consider an open bounded domain Ω R d (d =, 3) with Lipschitz boundary Ω, split into two open subsets Ω and Ω such that Ω = Ω Ω, Ω Ω =. () Then, we denote by Γ = Ω Ω () the interface between the subdomains (see Fig. ) and we assume that Γ is of class C, ; Γ denotes the interior of Γ. Given two scalar functions f and b defined in Ω, a positive function ν defined in Ω Γ, a d dimensional vector valued function b defined in Ω satisfying the following inequalities: ν R : ν(x) ν >, x Ω Γ σ R : b (x) + divb(x) σ >, x Ω, (3) 3

4 Γ in n Γ Ω Γ Ω ( Ω \ Γ) in Figure : Example of a computational domain Ω R split into two disjoint subdomains. we look for two functions u and u (defined in Ω and Ω, respectively) such that u satisfies the advection-reaction equation A u div(bu ) + b u = f, in Ω, (4) while u satisfies the advection-diffusion-reaction equation A u div( ν u + bu ) + b u = f, in Ω. (5) For each subdomain, we distinguish between the external (or physical) boundary Ω Ω k = Ω k \ Γ (for k =, ) and the internal one (i.e. the interface) Γ. Let us denote by n k the outward normal unit vector to Ω k and by n Γ the normal unit vector to Γ oriented from Ω to Ω, so that n Γ (x) = n (x) = n (x), x Γ. Moreover, for any non-empty subset S Ω, we define: the inflow part of S : S in = {x S : b(x) n (x) < }, (6) the outflow part of S : S out = {x S : b(x) n (x) }. (7) Boundary conditions for problem (4) are assigned on the inflow boundary ( Ω ) in. Then, we set homogeneous Dirichlet boundary conditions on the external boundaries: u = on ( Ω \ Γ) in, u = on Ω \ Γ. (8) A crucial issue is the setting of the interface (or transmission) conditions on Γ. In [8], the heterogeneous problem (4), (5), (8) is closed with the following interface conditions (that are named IC) u = u on Γ in, b n Γ u = ν u n Γ b n Γ u on Γ. (9) They express the continuity of the velocity field across the inflow part of the interface and the continuity of the fluxes across the whole interface. The final formulation of the heterogeneous problem reads div(bu ) + b u = f in Ω, div( ν u + bu ) + b u = f in Ω, u = on ( Ω \ Γ) in u = on Ω \ Γ () u = u on Γ in b n Γ u + ν u b n Γ u = on Γ n Γ 4

5 and its solution will be named heterogeneous solution (or else solution of the heterogeneous problem). Note that the interface conditions (9) can be equivalently expressed as: u = u, ν u n Γ = on Γ in, b n Γ u = ν u n Γ b n Γ u on Γ out. () The last set of interface conditions is named IC. Let b [W, (Ω)] d and Ω be either Ω or Ω. Thanks to the assumption made on both Ω and Γ, Ω has a Lipschitz continuous boundary, piecewise C,. Given an open subset Σ Ω of class C,, whose outward normal unit vector is denoted by n Σ, we define the following Hilbert spaces (see [8, ]) endowed with their standard norms: L b(σ) = {v : Σ R : ( / b nσ v L (Σ)}, v L b (Σ) = b n Σ v dσ), () Σ H / (Σ) = {v : L (Σ) : ṽ H / ( Ω) : ṽ Σ = v, ṽ Ω e \ Σ = }, (3) v / H = inf v (Σ) v H ( Ω) e H ( Ω) e, L div,b ( Ω) = {v L ( Ω), div(bv) L ( Ω)}, v L div,b ( e Ω) = ( v L ( e Ω) + div(bv) L ( e Ω)) / ; and finally X b ( Ω) = {v L div,b( Ω) : v L b( Ω)}, v Xb ( e Ω) = ( v L div,b (e Ω) + v L b ( e Ω)) /. The following result has been proved in [8]: Theorem Assume the following regularity properties on the data: Ω and Ω are Lipschitz continuous, piecewise C, ; Γ is of class C, ; (6) (4) (5) ν L (Ω ), b [ W, (Ω) ] d, b L (Ω), f L (Ω). (7) Finally, assume that (3) holds. Then there is a unique solution (u, u ) L (Ω ) H (Ω ) of (), where: equations (), hold in the sense of distributions in Ω and Ω, respectively; boundary condition () 3 holds a.e. on ( Ω \Γ) in ; boundary condition () 4 holds in H / ( Ω ); interface condition () 5 holds a.e. on Γ in, interface condition () 6 holds in (H / (Γ)). Finally, the heterogeneous problem () can be regarded as the limit of a family of globally elliptic variational problems. Remark Other interface conditions have been proposed in the literature to close system (4), (5), (8). For instance, (see []) b n Γ u = ν u n Γ b n Γ u on Γ out, u = u, u n Γ = u n Γ on Γ in, (8) 5

6 which are based on absorbing boundary condition theory, or else (see [4, 5]) u = u on Γ, u n Γ = u n Γ on Γ in. (9) However, the coupled problem with either one of these sets of conditions ((8), (9)) cannot be regarded as a limit of the same variational problem as ν in Ω. For a survey on this subject we refer to [6,, 6] for D problems, to [9] for D problems with convection limited to only one coordinate direction, and to [, 4, 5] for D problems. In the next sections we will consider another possible approach to close the heterogeneous problem (4), (5), (8) based on a saddle-point formulation of the coupled problem ([3, 4]). The following results will be useful later. (We refer to [8] and [] for their proof.) Theorem Under the same assumptions on the data as in Theorem, if λ H / ( Ω ) such that λ Ω in L b ( Ωin ), the first-order problem admits a unique solution u X b (Ω ). A u = f in Ω, u = λ on ( Ω ) in () If the Dirichlet condition () is replaced by a flux condition b n u = φ on ( Ω ) in, () with φ (H / ( Ωin )), the first-order problem (), () is still well-posed. Theorem 3 Under the same assumptions of Theorem,. if λ H / (Γ), the second-order problem A u = f in Ω, u = on Ω \ Γ, u = λ on Γ () admits a unique solution in H (Ω );. if µ (H / (Γ)), with µ Γ out L b (Γout ), the second-order problem A u = f in Ω, u = on Ω \ Γ, ν u = on Γ in, ν u (3) b n u = µ on Γ out n n admits a unique solution in H (Ω ); 3. if µ (H / (Γ)), and b L (Γ) ε, ε min{ν, σ } C (where C is the constant of the trace inequality v L ( Ω ) C v H (Ω ), v H (Ω )), the second-order problem A u = f in Ω, u = on Ω \ Γ, ν u n b n u = µ on Γ (5) admits a unique solution in H (Ω ). (4) 6

7 3 Extended Variational Formulation (XVF) In this section we reformulate the coupled advection/advection-diffusion problem by regarding the continuity across Γ in as a constraint, yielding a saddle-point problem. Let the regularity assumptions (3) and (6)-(7) hold on the data. Let us consider the Hilbert spaces L b (Γin ) and X b (Ω ) defined in () and (5), respectively. Moreover we define Λ = L b (Γin ), Λ = H / (Γ), V = X b (Ω ), V = H (Ω ), V = V V. (6) The space V, endowed with the graph norm v V = Hilbert space, then we set ( v X b (Ω ) + v H (Ω )) /, is a V = {v X b (Ω ), v = a.e. in ( Ω \ Γ) in } V, (7) V = {v H (Ω ) : v ( Ω\Γ) = } V (8) and V = V V. We introduce the bilinear form: a : V V R : a(u, v) = div(bu )v dω + b u v dω b n Γ u α v α dγ Ω Ω Γ in + ν u v dω u b v dω + b u v dω Ω Ω Ω b n Γ u v dγ Γ out where u α = αu + ( α)u, v α = αv + ( α)v, α can be either or, and the linear continuous functional F : V R : F(v) = fv dω + Ω fv dω. Ω (3) As we will see in the proof of the next Theorem, the choice of the parameter α is responsible for the setting of interface conditions across the inflow interface, more precisely, α = (α =, resp.) will provide interface conditions (9) ((), resp.). The bilinear form a is continuous. Let us bound ourselves to check only the integrals on the interface. Since u, v V, then u, v L b (Γin ) and b n Γ u v dγ is therefore bounded. Γ in For the regularity assumptions (7) on b, it holds H / (Γ) L b (Γ), so that also the boundedness of the mixed integral b n Γ u v dγ is guaranteed. Finally the integral Γ out b n Γ u v dγ can be interpreted as a duality pair between H / (Γ) and its dual space. Γ in We denote by s an element in Λ. Note that the dual space of Λ is (9) Λ = L /b (Γin ) = {s : Γ in R : ( b n Γ ) / s L (Γ in )} (3) 7

8 and the duality between Λ and Λ can be written as Λ s, φ Λ = sφ dγ. (3) Γ in Finally, let us define the following bilinear continuous form: b : V Λ R : b(v, s) = Λ s, (v v ) Γ in Λ (33) Note that (v v ) Γ in is well defined and it belongs to L b (Γin ). As a matter of fact, v Γ H / (Γ) and, since regularity assumptions (6) (7) hold, its restriction to Γin belongs to L b (Γin ). Then we set: Z = {v V : b(v, s) =, s Λ }, since b is continuous on V, then Z is a closed subspace of V and then it is a Hilbert space with respect to the norm of V. We are now able to define the saddle-point problem: seek u V, t Λ : { a(u, v) + b(v, t) = F(v) v V b(u, s) = s Λ. (34) 3. Well-posedness and Euler-Lagrange equations Our aim is twofold. From one hand we want to prove the well-posedness of the saddle-point problem (34). On the other hand we want to characterize the multiplier t Λ and recover the interface conditions on Γ that are hidden in this formulation. We begin by defining the following linear and continuous operators: A : V (V ) V Au, v V = a(u, v) u, v V, B : V Λ Λ s, Bv Λ = b(v, s) v V, s Λ. By definition (33), we have Bv = (v v ) Γ in, and Z = ker(b) V, i.e., Z = {v V : v = v a.e. on Γ in }. We now introduce the orthogonal of Z: Z = {v V : (v, z) = z Z}, where (, ) denotes the inner product on V, and the so called polar set of Z : (Z ) = {f (V ) : V f, v V = v Z }. The dual space Z of Z can be identified with (Z ) (see [7, ]), the latter being a closed subspace of (V ). Finally, we define the linear and continuous operator πa : Z Z : V πau, v V = V Au, v V, u, v Z, where π : (V ) Z is the orthogonal projection from (V ) onto Z. The following theorem states the well-posedness of the saddle-point problem (34). Theorem 4 If regularity assumptions (3) and (6) (7) hold on the data and b satisfies the smallness assumption (4), then there exists a unique solution (u, t) V Λ of (34) and the solution u = (u, u ) satisfies the interface conditions (9) ((), resp.) when α = (α =, resp.). 8

9 Proof. We have seen above that V and Λ are Hilbert spaces, the bilinear forms a and b are continuous and F (V ). Thus, existence and uniqueness of solution (u, t) of (34) are ensured if (see [7]): i) πa is an isomorphism from Z onto Z, ii) β > such that b(v, s) inf sup β. (35) s Λ v V v V s Λ Proof of i). Thanks to the Banach-Nečas-Babuska theorem (see, e.g. [3]), πa is an isomorphism from Z onto Z iff C > : πau V C u V u Z (36) w Z (a(w, v) = v Z) w =. (37) Condition (36) is often referred to by saying that πa is bounding ([3, Thm 4.5]) and it is equivalent to prove that πa is injective and its range is closed in Z, while condition (37) means that the adjoint operator (πa) of (πa) is injective. Let us start by proving that πa is bounding. For any v Z it holds v α = v = v and V πau, u V = a(u, u) ( ) = Ω divb + b u dω + b n u dγ b n Γ u dγ Ω Γ ( ) in + ν u dω + Ω Ω divb + b u dω b n u dγ Γ b n Γ u u dγ Γ out σ u L (Ω + min{ν ), σ } u H (Ω + ) b n u dγ ( Ω \Γ) out + b n Γ u dγ b n Γ u Γ dγ b n u out Γ dγ in Γ in b n u dγ b n Γ u u dγ Γ out Γ out = σ u L (Ω + min{ν ), σ } u H (Ω ) + b n u + b n Γ (u u ) dγ ( Ω \Γ) out Γ out σ u L (Ω + min{ν ), σ } u H (Ω ) + ) ( u Lb( Ω) 4 b n Γ u dγ, Γ where we have used the fact that u = u on Γ in and (u u ) ( ε)u + ( /ε)u with ε = /. By the trace inequality u L (Γ) C u H (Ω ), it holds u L b (Γ) C b L (Γ) u H (Ω ) 9

10 and, under assumption (4) we define the positive constant C = min{ν, σ } C b L (Γ), so that V πau, u V σ u L (Ω + C ) u H (Ω + ) 4 u L ( Ω). (38) b For any v = (v, v ) V we define the norm v L = ( v L (Ω ) + v H (Ω ) + v L b ( Ω) ) /. (39) It is straightforward to prove that M > such that a(u, v) M u V v L for any u, v V, while (38) says that C > : a(u, u) C u L u Z. (4) We set ã(u, v) = a(u, v) (div(bu ), v ) L (Ω ), therefore there exists M > such that ã(u, v) M u L v L for any u, v V. Since v L v V for any v V, it holds a(u, v) a(u, v) (div(bu ), v ) L sup sup sup (Ω ) v Z v V v Z v L v Z v M u L L (div(bu ), v ) sup L (Ω) M a(u, v) sup v V v V C v Z v L so that ( + M C + C ) sup v Z a(u, v) v V div(bu ) L (Ω) + u L C 3 u V, where C 3 >. (36) follows with C = C C 3 /(C + M + ). In order to prove (37), we start by noting that a(w, v) = for any v Z, then we take v = w. By applying (4) we conclude that w = a.e. in Ω. Proof of ii). As a consequence of the Closed Range theorem ([5]) together with the Open Mapping Theorem, the inf-sup condition (35) is satisfied iff the operator B is surjective from V onto Λ. For the regularity assumption (6) (7), it holds that H / (Γin ) L b (Γin ). Let λ L b (Γin ), for any λ H / (Γin ) we can define λ L b (Γin ) such that λ = λ + λ, solve the differential problems A v = in Ω v = a.e. on ( Ω \ Γ) in v = λ a.e. on Γ in A v = in Ω v = on Ω \ Γ in (4) v = λ on Γ in and define v = (v, v ). By Theorems, 3 it holds v V and Bv = (v v ) Γ in = λ λ = λ, i.e. B is surjective and the saddle-point problem (34) is well-posed.

11 It remains to prove that the solution u = (u, u ) of (34) satisfies the interface conditions (9). To this aim, we integrate by parts the first equation of (34). It holds: a(u, v) + b(v, t) = (A u )v dω b n Γ u α v α dγ Ω Γ in ( + (A u )v dω ν u ) b n Γ u v dγ Ω Γ n Γ (4) b n Γ u v dγ + Λ t, (v v ) Γ in Λ Γ out = fv dω + fv dω v V Ω Ω By taking v C (Ω ), v = first, then v =, v C (Ω ), it holds A u = f a.e. in Ω, A u = f a.e. in Ω. (43) Let us take now v = and s Λ, from the second equation of (34) it holds u = u a.e. on Γ in, (44) that is the interface condition (9). Finally, starting from (4), for any v V, thanks to (43), it holds ( b n Γ u α v α dγ ν u ) b n Γ u v dγ Γ in Γ n Γ b n Γ u v dγ + Λ t, (v v ) Γ in Λ = v V. Γ out and, by exploiting the integral on Γ as the sum of integrals on Γ in and Γ out, it holds ( b n Γ u α v α dγ ν u ) b n Γ u v dγ Γ in Γ n in ( Γ + Λ t, (v v ) Γ in Λ ν u ) b n Γ u + b n Γ u v dγ = v V. n Γ Γ out The interface condition on Γ out easily reads as (45) b n Γ u + ν u n Γ b n Γ u = in (H / (Γout )), (46) while the characterization of t Λ depends on α as well as the interface condition on Γin. When α = the multiplier t is t = b n Γ u = ν u n Γ + b n Γ u and the corresponding interface condition on Γ in reads ν u n Γ + b n Γ u b n Γ u = in (H / (Γin )),

12 while when α = the multiplier t is t = = ν u n Γ and the corresponding interface condition on Γ in reads ν u n Γ = in (H / (Γin )). Therefore, when α = we obtain the interface conditions IC (9), while when α = we recover interface conditions IC (). In view of (44) and the characterization of t, it holds u Γ in H / (Γ in ) and t (H / (Γ)). From the previous theorem we deduce that the saddle-point problem (34) is equivalent to the heterogeneous problem (). Remark It is important to note that the interface condition u = u a.e. on Γ in, cannot be extended to the whole interface Γ since it would induce the definition of a linear operator B: B : V L b(γ) : Bv = (v v ) Γ (47) which is not surjective. As a matter of fact, if Γ out, it is not guaranteed that, given a function λ L b (Γ), there exists a function v V such that its restriction to Γ coincides with λ. 4 The interface problem In Sections -3 we have shown the equivalence at continuous level between the heterogeneous formulation () of Gastaldi et al. [8] and the extended variational formulation (34), by proving that both formulations provide the same set of interface conditions. In Sect. 4. we recall the Steklov-Poincaré equation associated to (), see [8], while in Sect. 4., starting from the saddle-point formulation (34), we derive the associated interface equation by setting the decomposition of u and u and their variations, introducing the interface variables and unveiling the interface balance equations. 4. Steklov-Poincaré equation for the heterogeneous problem () For k =,, we write each u k as the sum of two functions, the former u λ k k depending on the unknown trace λ k of u k at the interface Γ, the latter u f k depending on the forcing term f, i.e. u = u λ + uf, u = u λ + uf. (48) Thanks to the linearity of the differential problem, the corresponding test functions v k depend only on the trace µ k on Γ (on the external force f) i.e., v k = v µ k k (for k =, ).

13 More precisely, u f and uf are the solutions of problems { A u f = f in Ω u f = on ( Ω ) in { A u f = f in Ω u f = on Ω, (49) while u λ and u λ are the solutions of A u λ = in Ω u λ = on ( Ω \ Γ) in u λ = λ on Γ in, A u λ = in Ω u λ = on Ω \ Γ u λ = λ on Γ. For any λ L (Γ), let λ in = λ Γ in denote the restriction of λ to Γ in. Similarly, v µ and v µ are the solutions of problems (5) with data µ and µ instead of λ and λ on Γ, respectively. We define the Steklov-Poincaré operators on the interface: S : L b (Γ) (H/ (Γ)) : S λ, µ Γ = (actually S depends only on λ in S : H / (Γ) (H/ (Γ)) : ) S λ, µ Γ = a (u λ, vµ ) = Γ Γ (5) b n u λ µ dγ µ H / (Γ), (5) ( ν uλ b n u λ n ) µ dγ, (5) where for any subset Σ Γ,, Γ denotes the duality between H / (Γ) and (H/ (Γ)). System () can be equivalently written as: seek λ Λ = H / (Γ) such that where Sλ, µ Γ = χ, µ Γ µ Λ, (53) Sλ, µ Γ = S λ in, µ Γ + S λ, µ Γ, χ, µ Γ = χ, µ Γ + χ, µ Γ, (54) λ in = λ Γ in, while χ and χ are the fluxes on Γ associated to u f and uf, respectively, i.e. χ = b n u f, χ = ν uf n + b n u f = ν uf n. (55) Note that χ Γ in =. We set the interface operators S : L b (Γ) (H/ (Γ)) and S : H/ (Γ) (H/ (Γ)) such that { on Γ in ν uλ on Γ in Sλ = Sλ = n (56) b n u λ on Γ out, ν uλ b n u λ on Γ out, n 3

14 (actually S λ depends only on λin ) and S = S + S on Λ, such that S λ, µ Γ = S λ, µ Γ + S λ, µ Γ. (57) Since u λ = uλ = λ on Γin, the Steklov-Poincaré equation (53) can also be written as: find λ Λ : S λ, µ Γ = χ, µ Γ µ Λ. (58) Remark 3 In view of Theorem 3, it is straightforward to prove that the operators S, S and S are coercive on H / (Γ), whereas S is coercive only if smallness assumption for b are assumed, as required in (4). (See [8, ] for a proof.) Remark 4 The Steklov-Poincaré equations (53) and (58) realize the interface conditions IC (9) and IC (), respectively, and they are equivalent one another at continuous level. Nevertheless they might not coincide at discrete level, when nonconforming discretization across the interface Γ is used (see [5]). Moreover, their discrete counterparts feature a different computational performance, in particular for what concerns preconditioner efficiency. 4. The interface problem associated with XVF Let us start by writing the saddle-point problem (34) in terms of u, u and t. It reads: seek u V, u V, t Λ such that a (u, v ) + Λ t, v Λ = fv dω v V Ω a (u, v ) + d (u, v ) Λ t, v Λ = fv dω v V Ω Λ s, u u Λ = s Λ, (59) where a : V V R : a (u, v ) = [div(bu ) + b u ]v dω α Ω b n Γ u v dγ, Γ in a : V a (u, v ) = ν u v dω Ω u b v dω+ Ω b u v dω ( α) Ω b n Γ u v dγ, Γ in with α =, and d : V V R : d (u, v ) = b n u v dγ. Γ out 4

15 By decomposing both u and u as done in (48), problem (59) reads: seek λ Λ, λ Λ, t Λ such that a (u λ, vµ ) + Λ t, vµ Λ = fv µ dω a (u f, vµ ) µ Λ Ω a (u λ, vµ ) + d (u λ, vµ ) Λ t, vµ Λ (6) = fv µ dω a (u f, vµ ) d (u f, vµ ) µ Λ Ω Λ s, uλ uλ Λ = s Λ. We consider the Steklov-Poincaré operators previously introduced. Moreover, we denote by S in and S out the restrictions of the operator S to Γ in and Γ out, respectively. Therefore, we note that a (u λ, vµ ) = α b n λ µ dγ = α S in λ, µ Γ in µ Λ, Γ in d (u λ, vµ ) = b n u λ µ dγ = S out λ, µ Γ out µ Λ, Γ out where, Γ in is an abridged notation for Λ, Λ, while a (u λ, vµ ) = α S λ, µ Γ + ( α) S λ, µ Γ and we set S α = αs + ( α)s, for α =,. The interface problem equivalent to (6) takes the following form: seek λ Λ, λ Λ, t Λ such that α S in λ, µ Γ in + t, µ Γ in = µ Λ S α λ, µ Γ + S out λ, µ Γ out t, µ Γ in = χ, µ Γ + χ, µ Γ out µ Λ (6) s, λ λ Γ in = s Λ, where χ and χ are the fluxes on Γ associated to u f and uf, respectively, already defined in (55). For brevity, we name (6) extended interface problem (XIP). We can formally write system (6) in terms of a block-matrix operator as with find λ Λ Λ Λ A α DDλ, µ = χ, µ µ Λ Λ Λ (6) αs in I λ µ I I t A α DD λ χ µ = = = = I λ in χ in S out S α λ out χ out +χ out s µ in µ out (63) 5

16 and where the upper-script in (out, resp.) denotes the restriction of the function to Γ in (Γ out, resp.). Remark 5 Problem (6) is well-posed and λ is indeed the trace of the unique solution of (34). The lower-script DD stands for Dirichlet-Dirichlet. This notation is motivated by noting that, if λ and λ are known trace functions on Γ in and Γ, respectively, then the evaluation of the matrix-vector product A α DDλ inside Bi-CGStab iterations requires the approximate solution of two Dirichlet problems like (5) (in Ω and Ω, respectively) and the numerical evaluation of the fluxes across the interface Γ. In view of the numerical discretization of problem (34) via interface problems like (6), a natural question arising from the analysis of the block-matrix system (6) concerns the possibility of replacing the operator A α DD with another one involving Neumann interface conditions instead of Dirichlet ones. To this aim, thanks to Theorems and 3, we observe that both S in is invertible, and, if the smallness assumption (4) on b is satisfied, S is invertible too. Under assumption (4), by defining the (unknown) fluxes (H / (Γin )) φ = S in λ, (H / (Γ)) φ = S α λ, (64) we can split the unknown functions u and u as u = ũ φ + uf, u = ũ φ + uf, where u f and uf still denote the solutions of problems (49), while ũφ and ũ φ are the solutions of the Neumann problems A ũ φ = in Ω ũ φ = on ( Ω \ Γ) in b nũ φ = φ on Γ in, A ũ φ = in Ω ũ φ = on Ω \ Γ ν ũφ b n ũ φ n on Γ out (65) ν ũφ αb n ũ φ n = φ on Γ in Problem (59) can be reformulated in terms of the (unknown) flux variables φ and φ. It reads: with find φ Λ Λ Λ A α NN φ, µ = χ, µ µ Λ Λ Λ (66) 6

17 (S in ) A α NN = αi I I (S α ) Γ in I φ = φ t φ in S out (S in ) I φ out while µ and χ have the same meaning as above. In this case, the evaluation of the matrix vector product A α NNφ requires the solution of two Neumann problems (like (65)) and then the evaluation of the trace of ũ φ (ũ φ, resp.) on Γ in (Γ, resp.), together with the evaluation of the flux b nũ φ on Γ out. For this reason, the formulation (66) is named Neumann-Neumann. Remark 6 When the smallness assumption (4) is not satisfied, the elliptic problem in (65) is not always well-posed and instabilities can develop (see Fig. 5). The same drawback characterizes the next Dirichlet-Neumann form, too. The survey can be completed by considering either Dirichlet-Neumann or Neumann-Dirichlet formulations for problem (59), depending on the choice of either Dirichlet or Neumann unknowns at the interface. More precisely, we name Dirichlet-Neumann (Neumann-Dirichlet, resp.) the formulation in which we decompose the unknown functions u and u as u = u λ +uf, u = ũ φ +uf (u = ũ φ + uf, u = u λ + uf, resp.). More precisely, we name Dirichlet-Neumann (Neumann- Dirichlet, resp.) the formulation in which we decompose the unknown functions u and u as u = u λ + uf, u = ũ φ + uf (u = ũ φ + uf, u = u λ + uf, resp.). The corresponding systems at the interface reads: and find ψ DN Λ Λ Λ A α DNψ DN, µ = χ, µ µ Λ Λ Λ (67) find ψ ND Λ Λ Λ A α ND ψ ND, µ = χ, µ µ Λ Λ Λ (68) with ψ DN = [λ, t, (φ in, φ out )] t, ψ ND = [φ, t, (λ in, λ out )] t and αs in I I (S α ) Γ in (S in ) αi I I A α DN = A α ND = I I I S out I S out (S in ) S α 7

18 As for the NN form, DN form is well-posed if the smallness assumption (4) on b is assumed. It is worth repeating that (6), (66), (67) and (68) represent alternative (albeit equivalent) interface formulations for the extended problem (34). Their finite dimensional approximations will however yield different numerical solutions, and the algebraic counterparts will give rise to systems whose computational solution features different levels of complexity, see Sect The discretized problems In this section we describe the discretization of the variational problems discussed in the previous sections and we propose optimal preconditioners to efficiently solve the associated linear systems. 5. Spectral Element discretization The discretization of the differential equation within each subdomain is performed by quadrilateral conformal Spectral Element Methods (SEM). We refer to [9] for a detailed description of these methods, here we briefly recall their basic features. For k =,, let T k = {T k,m } M k m= be a partition of the computational domain Ω k R d (d =, 3), where each element T k,m is obtained by a bijective and differentiable transformation F k,m from the reference (or parent) element ˆΩ d = (, ) d. We suppose that two adjacent elements of T k share either a common vertex or a complete side. On the reference element we define the finite dimensional space ˆQ N = span{ˆx j ˆxj d d : j,..., j d N} and, for any T k,m T k : T k,m = F k,m (ˆΩ d ), set h k,m = diam(t k,m ) and V Nk (T k,m ) = {v : v = ˆv F k,m for some ˆv ˆQ Nk }. The SEM multidimensional space on Ω k (for k =, ) is X k,δk = {v C (Ω k ) : v Tk,m V Nk (T k,m ), T k,m T k } where δ k is an abridged notation for discrete, that accounts for the local geometric sizes {h k,m } and the local polynomial degrees {N k }, for m =,...,M k and k =,. Note that the polynomial degree N k can either coincide or differ along each spatial direction. In the latter case we denote by Nk x, Ny k and Nz k the polynomial degrees along x, y and z directions, respectively. Nk x, Ny k and Nz k can change from one spectral element to another, however both geometric and approximation conformity is guaranteed inside Ω k. The finite dimensional spaces in which we look for the Spectral Element solution are defined as follows (see (6), (7) and (8) for definitions of V k, Vk, Λ k): V k,δk = V k X k,δk, V k,δ k = V k X k,δ k, k =, Λ,δ = {v,δ Γ in : v,δ V,δ }, Λ,δ = {v,δ Γ : v,δ V,δ }. (69) Because of the difficulty to compute integrals exactly, the bilinear forms a, a, d, the duality products between Λ k and Λ k (for k =, ), and the L inner products are all approximated by Legendre-Gauss-Lobatto (LGL) formulas on the grid induced by the finite 8

19 dimensional spaces defined above. It is well known (see [8]) that the quadrature error introduced behaves like the approximation error induced by the spectral approximation. The abridged notation SEM-NI stands for Spectral Element Method with Numerical Integration. Coherently, we can define the finite dimensional counterpart of the Steklov-Poincaré operators S, S, etc.. To this aim, for k =,, we denote by N k,γ the set of nodes of T k Γ whose cardinality is N k,γ. Similar notations are used for the nodes lying on either Γ in or Γ out. The finite dimensional basis in Λ,δ is generated by the characteristic Lagrange polynomials in Ω associated to the LGL nodes of N,Γ in, while that in Λ,δ is generated by the characteristic Lagrange polynomials in Ω associated to the LGL nodes of N,Γ. We denote by µ (i), i =,...,N,Γ in the basis functions of Λ,δ and by µ (i), i =,...,N,Γ the basis functions of Λ,δ. To span the dual spaces Λ k,δ k we use the same Lagrange basis of Λ k,δk, respectively for k =,. It is immediate to prove that, under regularity assumptions (3) and (6), Λ k,δ k Λ k, for k =,. Then we set (S in,δ ) ij = S in µ(j), µ(i) Γ in i, j =,...,N,Γ in (S out,δ ) ij = S out µ (j), µ(i) Γ out i =,...,N,Γ out, j =,...,N,Γ in (S,δ ) ij = S µ (j), µ(i) Γ i, j =,...,N,Γ (S,δ ) ij = S µ (j), µ(i) Γ i, j =,...,N,Γ. In general, the subscript δ denotes the finite dimensional counterpart of the corresponding operator defined at the continuous level and, even if it is not specified, we understand that it is constructed following the SEM-NI approximation. For an extensive presentation of these methods we refer to [8, 9]. Remark 7 If the two partitions T and T share the same edges on Γ and the polynomial degrees coincide in the hyperbolic domain Ω and in the elliptic one Ω, we call that conforming discretization or conforming coupling across Γ. In this case N = N,Γ = N,Γ denotes the common polynomial degree and δ = δ = δ denotes the conforming discretization. Otherwise, we call nonconforming discretization or nonconforming coupling across Γ the case in which either the partitions T and T do not share the same edges on Γ or the polynomial degrees N,Γ and N,Γ differ each other. In this work we bound our analysis to conforming couplings, while we refer to [5] for the nonconforming case, where among all the known methods dealing with nonconformity we choose mortar methods (see, e.g., [, ]), since they achieve the constraint equation associated to the XVF in a very natural way. The use of conforming discretization across the interface Γ implies that the approximation of the unknown trace function is defined through a unique set of degrees of freedom on Γ and the same symbol δ(= δ = δ ) will be used. We denote by M δ, Mδ in and Mδ out the mass matrices associated to the interface integrals on Γ, Γ in and Γ out, respectively. Due to the orthogonality of the characteristic Lagrange basis functions with respect to the Gauss-Legendre-Lobatto quadrature formulas, these mass matrices are diagonal. The conforming finite dimensional counterpart of (53) reads: find λ δ Λ,δ : S δ λ δ = χ δ, (7) 9

20 while that of (58) reads: find λ δ Λ,δ : S δ λ δ = χ δ. (7) (χ out After setting χ δ = [,, Mδ inχin,δ, Mout δ,δ +χout,δ )]t, the discretized form corresponding to XIP problems (6), (66), (67) and (68) read, respectively: Dirichlet-Dirichlet find λ δ = [λ,δ, t δ, (λ in,δ, λ out,δ )] t Λ,δ Λ,δ Λ,δ : A α DD,δλ δ = χ δ, (7) Neumann-Neumann find φ δ = [φ,δ, t δ, (φ in,δ, φ out,δ )] t Λ,δ Λ,δ Λ,δ : A α NN,δφ δ = χ δ, (73) Dirichlet-Neumann find ψ DN,δ = [λ,δ, t δ, (φ in,δ, φ out,δ )] t Λ,δ Λ,δ Λ,δ : A α DD,δψ DN,δ = χ δ, (74) Neumann-Dirichlet find ψ ND,δ = [φ,δ, t δ, (λ in,δ, λout,δ )]t Λ,δ Λ,δ Λ,δ : A α ND,δ ψ ND,δ = χ δ, (75) where αs,δ in Mδ in I I A α DD,δ = A α Mδ in NN,δ = S,δ α S out,δ (S,δ in ) Mδ in R in is the restriction matrix from N,Γ to N,Γ in, and S,δ out(sin,δ ) Mδ in αi in δ M in δ M in δ R in (Sα,δ ) M δ M δ (76) αs in,δ M in δ I R in (S α,δ ) M δ (S,δ in ) Mδ in αm in δ M in δ I A α DN,δ = A α = Mδ in ND,δ M δ M in δ S α,δ S out,δ M in δ (77)

21 Remark 8 Let us consider the discrete counterpart of (34): seek u δ V δ, t δ Λ,δ : { a(uδ, v δ ) + b(v δ, t δ ) = F(v δ ) v δ V δ b(u δ, s δ ) = s δ Λ,δ, (78) where Vδ = V,δ V δ. When the conforming discretization is considered across the interface, which is the case of the present work, the following inclusions hold: Vδ V, Λ,δ Λ, Z δ = {v δ Vδ : v,δ = v,δ on Γ in } Z, where Z δ is the kernel of the discrete counterpart B δ of B, that is Λ s δ, B δ v δ Λ = b(v δ, s δ ), for any v δ Vδ and s δ Λ,δ. The latter inclusion is crucial for the proof that the discrete operator πa δ is an isomorphism from Z δ onto Z δ (πa δ is defined by V πa δ u δ, v δ V = a(u δ, v δ ), for any u δ, v δ Vδ ) and it can be obtained by using the same arguments of i) in the proof of Theorem 4. Finally, in order that the discrete inf-sup condition corresponding to (35) holds, the surjectivity of B δ can be proved as in step ii) of the proof of Theorem 4. All the finite dimensional interface equations (7) (75) can be solved by a Krylov method for non-symmetric systems (e.g. Bi-CGStab). 5. Comparison among formulations. Numerical results In this section we compare the Steklov-Poincaré approaches SP (7) and SP (7) with the extended interface problems (XIP) (7), (73), (74) and (75) presented in this work in terms of Bi-CGStab iterations (see [4]) needed to solve the corresponding interface equations, versus the discretization parameters, both the polynomial degree N and the number of spectral elements M in each subdomain. More precisely, Steklov-Poincaré approach SP is related to the four XIP with α =, since all these forms provide interface conditions (); similarly Steklov-Poincaré approach SP is related to XIP with α =, since all these forms provide interface conditions (9). Let us consider the following test cases. Test case #. The computational domain Ω = (, ) is split in Ω = (,.8) (, ) and Ω = (.8, ) (, ). The interface is Γ = {.8} (, ). The data of the problem are: b = [y, ] t, b =, f = and the inflow interface is Γ in = {.8} (, ). Dirichlet boundary conditions are imposed on the vertical sides of Ω, precisely u = on ( Ω \Γ) in = { } (, ), u = on {} (, ), while the homogeneous Neumann condition u n = is imposed on ((.8, ) { }) ((.8, ) {}) (see Fig. left). Test case #. The computational domain Ω = (, ) is split in Ω = (,.8) (, ) and Ω = (.8, ) (, ). The interface is Γ = {.8} (, ). The data of the problem are: b = [5y, x] t, b =, f = and the inflow interface is Γ in = {.8} (, ). The imposed Dirichlet boundary conditions are: u = on ((,.8) { }) ({ } (, )), u = on {} (, ), u = on (.8, ) { }, while the homogeneous Neumann condition u n = is imposed on (.8, ) {} (see Fig. center). Test case #3. The computational domain Ω = (, ) is split in Ω = (,.8) (, ) and Ω = (.8, ) (, ). The interface is Γ = {.8} (, ). The data of the problem

22 layer layer Ω Ω Ω Ω Ω Ω layer Figure : Data for test cases # (left), # (center) and # 3 (right). The cyan (red, resp.) line denotes the interface Γ out (Γ in, resp.) are: b = [y, x] t, b =, f = and the inflow interface is Γ in = {.8} (,.). The imposed Dirichlet boundary condition are: u (x, y) = ( x)/ on (, ) { }, u = on { } (, ), u (x, y) = /π arctan(( x)) on (,.8) {}, u (x, y) = /π arctan(( x)) on (.8, ) {}, u = on {} (, ). Finally the homogeneous Neumann condition u n = is imposed on (.8, ) { } (see Fig. right). For all the test cases, the viscosity will be specified below. We discretize the problem using the conforming SEM-NI approach, presented in the previous Section. In each subdomain we define a mesh of quadrilaterals that can be either uniform or not. We remind that in this section and in the following one we consider conforming discretizations across the interface between the hyperbolic and the elliptic subdomains. Because of the presence of a boundary layer near the right vertical side, the mesh is finer there and a high polynomial degree along x direction is used to prevent the numerical solution to be affected by spurious oscillations. In Fig. 3 we compare the number of Bi-CGStab iterations for the Test case # needed to solve the extended interface problems (7), (73), (74), (75) and the heterogeneous Steklov- Poincarè equations (7), (7) versus the spectral polynomial degree and the number of spectral elements, respectively. Denoting by r (k) the residual of the linear system at the k th Bi-CGStab iteration, the iterations are stopped when r (k+) / r () ǫ, with ǫ =. For each formulation, we have computed a least-square fit of a law like #it Cx q, where x can indicate either the viscosity ν, the spectral polynomial degree N or the number of spectral elements M along each direction and in each subdomain. In each legend we report the estimated value of q. The convergence rate of the XIP approaches depends on the parameter α, i.e. on the set of interface conditions chosen, (9) for α = and () for α =. As a matter of fact when α =, DD (as SP) is the best approach, while ND is the worst one for what concerns the dependence on polynomial degree and number of spectral elements (see Fig. 3, top). On the contrary, when α =, with the exception of DN form, all the other XIP forms outperform in a similar way, but the number of iterations are always lower for the Bi-CGStab solution applied to the classical (non-extended) interface problems (7) (cf. curve). (see Fig. 3, bottom). In Fig. 4 we report the number of Bi-CGStab iterations versus the viscosity. It noticeably grows when ν and α = (Fig. 4, right) for both NN and DN approaches. This is

23 Bicgstab it 4 3 SP, q=.36 DD, q=.38 NN, q=.35 DN, q=. ND, q=.64 Bicgstab it 5 5 SP, q=.3 DD, q=.34 NN, q=.34 DN, q=.3 ND, q=.46 5 Bicgstab it SP, q=.3 DD, q=.53 NN, q=. DN, q=.6 ND, q=.85 N Bicgstab it SP, q=.3 DD, q=.4 NN, q=.59 DN, q=.74 ND, q=.5 Mx = My N Mx = My Figure 3: Test case #. Bi-CGStab iterations needed to solve the extended interface problems DD (7), NN (73), DN (74), ND (75) (at top with α =, at bottom with α = ) and the Steklov Poincaré equations SP (7) and SP (7). The viscosity is ν =.. At left, the spectral element mesh is formed by 3 6 (4 6, resp.) elements in Ω (Ω, resp.), while the polynomial degree is N in each element of each domain. At right, the polynomial degree is N = 6 in all elements, while the number of spectral elements in each Ω k (k =, ) is M = M x M y. 3

24 Bicgstab it 3 SP, q=.8 DD, q=.33 NN, q=.79 DN, q=. ND, q=.3 Bicgstab it 3 SP, q=.8 DD, q=. NN, q=.88 DN, q=.3 ND, q= ν 4 3 ν Figure 4: Test case #. Bi-CGStab iterations needed to solve the XIP DD (7), NN (73), DN (74), ND (75) (at left with α =, at right with α = ) and the Steklov Poincaré equations SP (7) and SP (7). The spectral element mesh is generated by 3 4 elements with N = 6 in Ω, and by 6 4 elements with in Ω. N = 6 in all elements of Ω with the exception of those close to the boundary layer (where N x = 7 N y = 6). due to the fact that condition (4) is not satisfied when ν 3 and instabilities affect the numerical solution, see also Remark 6. In Fig. 5 the numerical solution of Test # is shown for ν = 4, unstable (stable, resp.) for NN (DD, resp.) form and α =. In some situations, the use of a good preconditioner will stabilize the solution, as we will see in the next section. For what concerns the dependence on the viscosity, the convergence rate of all approaches is independent of ν, with the exception of XIP approaches NN and DN when α =. Also in this case, the number of iterations are always lower for the Bi-CGStab solution applied to the classical (non-extended) interface problems (7) and (7) (cf. SP and SP curves) x y x y Figure 5: Test case #. Numerical solution provided by NN formulation (66) (at left) and by DD form (6) (at right) for ν = 4 when α = 4

25 5.3 Optimal preconditioners for the interface problems Starting from well known results about preconditioning of the discrete Steklov-Poincaré operator Sδ (57), in this Section we propose optimal preconditioners for the finite dimensional counterpart of the extended interface problems DD, NN, DN and ND. It is well known (see, e.g. [8, pag.6],[]) that S,δ is an optimal preconditioner of S δ, in the sense that there exists a positive constant C independent of δ, but depending on problem data (e.g. Ω, ν, b, b ) such that K((S,δ ) S δ ) C, (79) where K(A) = (max i λ i (A) )/(min i λ i (A) ) is the iterative condition number of a generic square real matrix A whose eigenvalues are λ i (A) C. Following [8] it is possible to prove that S,δ is an optimal preconditioner of S δ with respect to the discretization, i.e., there exists C > independent of δ, but depending on problem data (e.g. Ω, ν, b, b ) such that K((S,δ) S δ ) C. (8) Numerical results that confirm (79) and (8) are shown in Tables, 3, 5 (columns SP and SP, respectively), where few Preconditioned Bi-CGStab (PBi-CGStab) iterations are sufficient to converge to the required tolerance for all used discretizations. Experimental results show that in general C C. Let us now consider the extended interface Dirichlet-Dirichlet problem (7) (76). We take a -block decomposition of A α DD,δ, as drawn by the continuous lines in (76). The construction of an optimal preconditioner for problem (7) is based on the construction of an optimal preconditioner P ΣDD,δ for the Schur complement Σ α DD,δ of the matrix Aα DD,δ, with respect to the (,)-block S,δ α. Such idea has been proposed in [, Sect..3.] in the context of domain decomposition methods for elliptic problems. Since Mδ in is non-singular, it is easy to see that [ ] αs Σ α DD,δ = S,δ α in +,δ S,δ out = αs δ + ( α)sδ, (8) where S δ and Sδ are the matrices introduced in (7) and (7), respectively. Thus, we set P ΣDD,δ = S,δ and, owing to both (79) and (8), P Σ DD,δ is an optimal preconditioner for Σ α DD,δ, i.e., K(P Σ DD,δ Σ α DD,δ) C α, for α =,. (8) Now, we proceed as follows. Since A α DD,δ = Lα δ Uα DD,δ, with I L α δ = I I αs,δ in I S,δ out I, Uα DD,δ = αs,δ in Mδ in I I Σ α DD,δ, (83) 5

26 we define the left preconditioner PDD,δ α for Aalpha P α DD,δ = L α δ Ũα DD,δ, with Ũ α DD,δ = DD,δ as αs,δ in Mδ in I I P ΣDD,δ. (84) In fact, Ũα DD differs from Uα DD only in the (,)-block, where the Schur complement matrix Σ α DD,δ has been replaced by its preconditioner P Σ DD,δ. The following theorem holds: Theorem 5 Let A R n n be non-singular, L, U R n n be ( )-block triangular matrices lower and upper, respectively, such that A = LU and [ ] U U U =, U where U, U are squared blocks. If there exists a square non-singular matrix P of the same dimension of U and a positive constant C such that K(P U ) C, (85) then the matrix P = LŨ, with Ũ = [ ] U U P satisfies the bound K(P U) C. (86) Proof. Since A is non-singular, both U and U kk (for k =, ) are non-singular and [ Ũ U = U U P ] [ P, P A = Ũ I U U = U (I P U ) P U that is, the spectrum of P A coincides with that of P U plus the eigenvalue λ =. Indeed λ = is an eigenvalue of P U, too. As a matter of fact, the eigenvector v of P A associated to λ = satisfies the matrix equation Av = Pv Uv = Ũv ], if and only if [ U U U ] [ v v ] [ U U = P ][ v v ] U v = P v, that is λ = is an eigenvalue of P U, provided that v is not null. It follows that λ i (P A) = λ i (P U ) and K(P A) = K(P U ). (87) 6