Composition duality methods for evolution mixed variational inclusions

Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363 www.elsevier.com/locate/nahs

Composition duality methods for evolution mixed variational inclusions Gonzalo Alduncin Departamento de Recursos Naturales, Instituto de Geof´ısica, Universidad Nacional Aut´onoma de M´exico, D.F. C.P. 04510, Mexico Received 30 June 2006; accepted 13 July 2006

Abstract Composition duality methods are presented for the qualitative and discretization analysis of primal and dual evolution mixed variational inclusions in reflexive Banach spaces. Abstract applications to macro-hybrid variational formulations, semi-discrete internal approximations globally nonconforming and time marching schemes implementable as multidomain proximal-point algorithms are studied. Stationary fully discrete inclusions are considered as well as corresponding preconditioned penalty–duality algorithms. To illustrate the theory, a monotone distributed control diffusion problem is treated. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Composition duality principles; Evolution mixed variational inclusions; Macro-hybridization; Macro-hybrid mixed finite elements; Multidomain proximal-point algorithms; Control mixed variational problems

1. Introduction Composition duality principles for the solvability analysis of Lagrangian and mixed variational problems of saddle and primal–dual formulations in mechanics have been proposed in the context of convex analysis [17] and mixed finite element approximations [15,26]. Such a methodology requires compatibility conditions for compositional dualization in order to establish the relation between mixed variational solutions and corresponding primal or dual variational solutions, conditions that orient the course of the analysis. A unifying compatibility condition for compositional dualization was recently presented in [7], and applied to mixed variational inclusions [6] and macro-hybrid variational formulations of constrained boundary value problems [8], as well as in the analysis of augmented three-field macrohybrid mixed finite element schemes [9]. This “novel” compatibility condition asserts that the coupling continuous linear mixed operators should be surjective. In fact, such a coupling surjectivity condition is a variational operator version of the well-recognized Ladyˇsenskaja–Babuˇska–Brezzi inf-sup or (LBB) condition [21,11,14]. The aim of this paper is to present the composition duality principles for primal and dual evolution mixed variational inclusions given in [10], where the above mentioned compatibility conditions for compositional dualization are combined for primal–dual and dual–primal analysis. Furthermore, our interest is to apply and extend such composition duality principles to the analysis of macro-hybrid variational formulations of evolution mixed problems, corresponding semi-discrete internal approximations globally nonconforming, and time marching schemes

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 1751-570X/$ - see front matter doi:10.1016/j.nahs.2006.07.004

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implementable as multidomain proximal-point algorithms. Also, stationary fully discrete inclusions are considered, as well as preconditioned penalty–duality algorithms derived from time integration schemes. In order to illustrate the theory, we analyze a distributed control diffusion problem as a representative mechanical variational dynamical system. 2. Evolution composition duality principles We begin with the presentation of the composition duality principles for primal and dual evolution mixed variational inclusions in accordance with [10], which provide a basis for the qualitative mixed variational analysis of the problems under suitable compatibility conditions. We shall consider as the stationary framework and operators of the theory the ∗ following: V and Y will denote two reflexive Banach spaces with topological duals V ∗ and Y ∗ , ∂ F : V → 2V and ∂G ∗ : Y ∗ → 2Y two monotone subdifferentials with superpotentials, the proper convex lower semicontinuous functionals F : V → R ∪ {+∞} and G ∗ : Y ∗ → R ∪ {+∞}, and Λ ∈ L(V, Y ) a linear continuous operator with transpose ΛT ∈ L(Y ∗ , V ∗ ). Also, we shall assume that H and Z are Hilbert pivot spaces for V and Y ∗ ; i.e., V ⊂ H ' H ∗ ⊂ V ∗ and Y ∗ ⊂ Z ∗ ' Z ⊂ Y with continuous and dense embeddings. 2.1. The primal evolution mixed variational inclusion For (0, T ], the time interval of computational interest, T > 0 fixed and arbitrary, we define by V = L p (0, T ; V ) = RT p {v : [0, T ] → V | kvkV = [ 0 kv(t)kV dt]1/ p < ∞}, 2 ≤ p < ∞, the primal evolution reflexive ∗ Banach space with topological dual V ∗ = L p (0, T ; V ∗ ), p ∗ = p/( p − 1), and the primal solution space by W = {v : v ∈ V, dv/dt ∈ V ∗ } subspace of the H-valued continuous function space C(0, T ; H ), endowed with the norm kvkW = kvkV + kdv/dtkV ∗ . Further, as the dual solution space, we consider the reflexive Banach space ∗ Y ∗ = L p (0, T ; Y ∗ ) with dual Y = L p (0, T ; Y ) (see [22]). Also, let R(Λ) ⊂ Y denote the range of the operator Λ. Then, as an abstract primal evolution problem for mixed models in mechanics, we consider the following.  Given f ∗ ∈ V ∗ , g ∈ L p (0, T ; R(Λ)) and u 0 ∈ H, find (u, p ∗ ) ∈ W × Y ∗ :     T ∗ du + ∂ F(u) − f ∗ , in V ∗ , −Λ p ∈ (M) dt  Λu ∈ ∂G ∗ ( p ∗ ) + g, in Y,   u(0) = u 0 . For analysis purposes of problem (M), we introduce the primal compatibility condition (CG,Λ )

int D(G) ∩ R(Λ) 6= ∅,

where int D(G) stands for the interior of the effective domain of the conjugate G : Y → R ∪ {+∞} of functional G ∗ ; i.e., y ∈ ∂G ∗ (y ∗ ) ⇔ y ∗ ∈ ∂G(y), ∀y ∈ Y, y ∗ ∈ Y ∗ [12]. Under this classical condition applied in convex analysis, the next compositional result holds [17]. Lemma 2.1. Let condition (CG,Λ ) be satisfied. Then the compositional operator equality ∂(G ◦ Λ) = ΛT ∂G ◦ Λ

(1)

is guaranteed. Hence, by dualization, we can conclude the following primal composition duality principle. Theorem 2.2. Under condition (CG,Λ ), the primal evolution mixed problem (M) is solvable if, and only if, the primal evolution problem  Given f ∗ ∈ V ∗ , g ∈ L p (0, T ; R(Λ)) and u 0 ∈ H, find u ∈ W :   du (P) 0 ∈ + ∂ F(u) + ∂(G ◦ Λ)(u − wg ) − f ∗ , in V ∗ ,  dt  u(0) = u 0 ,

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is solvable, where wg ∈ V is a fixed Λ-preimage of function g : Λwg = g. That is, if (u, p ∗ ) ∈ W ×Y ∗ is a solution of problem (M) then primal function u is a solution of problem (P) and, conversely, if u ∈ W is a solution of problem (P) then there is a dual function p ∗ ∈ ∂G(Λu − g) ⊂ Y ∗ such that (u, p ∗ ) is a solution of problem (M). Proof. The principle readily follows from the duality relation Λu ∈ ∂G ∗ ( p ∗ ) + g ⇐⇒ p ∗ ∈ ∂G(Λu − g)

(2)

for the dual inclusion of problem (M) and Lemma 2.1.



2.2. The dual evolution mixed variational inclusion For dual evolution mixed problems, where the evolution is governed by the dual inclusion, we define the dual solution space X ∗ = {q ∗ : q ∗ ∈ Y ∗ , dq ∗ /dt ∈ Y} subspace of the Z ∗ -valued continuous function space C(0, T ; Z ∗ ), normed by kq ∗ kX ∗ = kq ∗ kY ∗ + kdq ∗ /dtkY . We also denote by R(−ΛT ) ⊂ V ∗ the range of the transpose operator −ΛT . Then, for mixed models in mechanics, we consider the following abstract dual evolution problem.  Given f ∗ ∈ L p (0, T ; R(−ΛT )), g ∈ Y and p0∗ ∈ Z ∗ , find (u, p ∗ ) ∈ V × X ∗ :   −ΛT p ∗ ∈ ∂ F(u) − f ∗ , in V ∗ ,  (M∗ ) d p∗  + ∂G ∗ ( p ∗ ) + g, in Y, Λu ∈   dt ∗  ∗ p (0) = p0 . In this dual case, we introduce the dual compatibility condition (C F ∗ ,−ΛT )

int D(F ∗ ) ∩ R(−ΛT ) 6= ∅,

under which the following dual version of Lemma 2.1 is valid [17]. Lemma 2.1∗ . If condition (C F ∗ ,−ΛT ) is fulfilled, then the compositional operator equality ∂(F ∗ ◦ (−ΛT )) = −Λ∂ F ∗ ◦ (−ΛT )

(1*)

holds true. Consequently, the following dual composition duality principle for problem (M∗ ) is established. Theorem 2.2∗ . Let condition (C F ∗ ,−ΛT ) be satisfied. Then dual evolution mixed problem (M∗ ) is solvable if, and only if, the dual evolution problem  ∗ p T ∗ ∗ ∗ ∗  Given f∗ ∈ L (0, T ; R(−Λ )), g ∈ Y and p0 ∈ Z , find p ∈ X : dp (D) 0 ∈ + ∂G ∗ ( p ∗ ) + ∂(F ∗ ◦ (−ΛT ))( p ∗ + r ∗f ∗ ) + g, in Y,   ∗ dt ∗ p (0) = p0 , is solvable, where r ∗f ∗ ∈ Y ∗ is a fixed −ΛT -preimage of function f ∗ : −ΛTr ∗f ∗ = f ∗ . That is, if (u, p ∗ ) ∈ V × X ∗ is a solution of problem (M∗ ) then dual function p ∗ is a solution of problem (D) and, conversely, if p ∗ ∈ X ∗ is a solution of problem (D) then there is a primal function u ∈ ∂ F ∗ (−ΛT p ∗ + f ∗ ) ⊂ V such that (u, p ∗ ) is a solution of problem (M∗ ). Proof. Now the principle follows from the duality relation −ΛT p ∗ ∈ ∂ F(u) − f ∗ ⇐⇒ u ∈ ∂ F ∗ (−ΛT p ∗ + f ∗ ) for the primal inclusion of problem (M ) and Lemma ∗

2.1∗ .

(2*) 

Before passing to the macro-hybridization of the evolution problems and their discretizations, we observe that methodologically Theorems 2.2 and 2.2∗ state a basis for the qualitative analysis of primal and dual evolution mixed

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problems (M) and (M∗ ) in terms of the qualitative analysis of their corresponding primal and dual evolution problems (P) and (D), respectively. Important issues of analysis to be mentioned are: existence and uniqueness of solutions, continuous dependence on the data, Lyapunov stability, evolutionary behavior at infinity, comparison theorems, as well as regularity in time. Also, it is important to stress that primal and dual evolution mixed inclusions (M) and (M∗ ) are not problems in mathematical duality. They are two different mixed variational formulations that, in computational mechanics, may offer two alternative formulations for a single evolution problem with some complementary numerical advantages, or simply they are just appropriate mixed models for some evolution processes. 3. Macro-hybrid variational formulations A fundamental and natural strategy in mechanics for parallel computing is to spatially decompose the evolution processes under analysis into dynamical subsystems, which then interact across their physical interfaces according to continuity transmission conditions. Here, we shall follow our study [8] (see also [6]), on macro-hybrid variational formulations, for the spatial decomposition of evolution mixed problems (M) and (M∗ ). Hence, let the functional framework of the theory be defined relative to a spatial bounded domain Ω ⊂ Rd , d ∈ {1, 2, 3}, decomposed in terms of disjoint and connected subdomains {Ωe } by Ω=

E [

Ω e,

(3)

e=1

with Lipschitz continuous internal boundaries and interfaces Γe = ∂Ωe ∩ Ω ,

e = 1, 2, . . . , E,

Γe f = Γe ∩ Γ f ,

1 ≤ e < f ≤ E.

(4)

Then, as a central hypothesis, we assume that the primal and dual spaces V and Y ∗ are of a local type in the sense ( ) E Y V = V (Ω ) ' {ve } ∈ V({Ωe }) = V (Ωe ) : {πΓe ve } ∈ Q , e=1

Y = Y (Ω ) ' Y ({Ωe }) = ∗

∗

∗

E Y

(5)

Y (Ωe ), ∗

e=1

with pivot spaces H = H (Ω ) ' H({Ωe }) =

E Y

H (Ωe ),

e=1

Z ∗ = Z ∗ (Ω ) ' Z∗ ({Ωe }) =

E Y

Z ∗ (Ωe ).

(6)

e=1

Here, [πΓe ] is the continuous linear internal boundary primal trace operator of the primal product space V({Ωe }) with QE B(Γe ), satisfying the macro-hybrid compatibility condition values in B({Γe }) = e=1 (C[πΓe ] )

πΓe ∈ L(V (Ωe ), B(Γe ))

is surjective, e = 1, 2, . . . , E,

admissibility subspace, with polar, the dual transmission admissibility and Q ⊂ B({Γe }) is the primal transmission QE subspace of the dual B∗ ({Γe }) = e=1 B ∗ (Γe ),  Q∗ = {µ∗e } ∈ B∗ ({Γe }): h{µ∗e }, {µe }i B ∗ ({Γe }) = 0, ∀{µe } ∈ Q . (7) Further, we define the corresponding evolution product spaces V {Ωe } = L p (0, T ; V({Ωe })) and Y ∗{Ωe } = ∗ ∗ L p (0, T ; Y ∗ ({Ωe })), with duals V ∗{Ωe } = L p (0, T ; V ∗ ({Ωe })) and Y {Ωe } = L p (0, T ; Y({Ωe })), as well as the solution spaces W {Ωe } = {{ve } : {ve } ∈ V {Ωe } , {dve /dt} ∈ V ∗{Ωe } } ⊂ C(0, T ; H({Ωe })) and X ∗{Ωe } = {{qe∗ } : {qe∗ } ∈ Y ∗{Ωe } , {dqe∗ /dt} ∈ Y {Ωe } } ⊂ C(0, T ; Z∗ ({Ωe })). Therefore, imposing the primal transmission condition of (5)1 as a variational constraint via the subdifferential ∂(I Q ◦ [πΓe ]), I Q denoting the indicator functional of subspace Q, the macro-hybridized version of primal evolution

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problem (M), for operators and functions of a local type with respect to decompositions (3)–(6), is given by  Find ({u e }, { pe∗ }) ∈ W {Ωe } × Y ∗{Ωe } :       du e  −{ΛTe pe∗ } ∈ + {∂ Fe (u e )} + ∂(I Q ◦ [πΓe ])({u e }) − { f e∗ }, in V ∗{Ωe } , (M M H ) dt   {Λe u e } ∈ {∂G ∗e ( pe∗ )} + {ge }, in Y {Ωe } ,   {u e (0)} = {u 0e }. Similarly, the macro-hybridized version of dual evolution problem (M∗ ) is expressed by  Find ({u e }, { pe∗ }) ∈ V {Ωe } × X ∗{Ωe } :     −{ΛTe pe∗ } ∈ {∂ Fe (u e )} + ∂(I Q ◦ [πΓe ])({u e }) − { f e∗ }, in V ∗{Ωe } , ∗ (M M H ) d pe∗  + {∂G ∗e ( pe∗ )} + {ge }, in Y {Ωe } , {Λe u e } ∈   dt   ∗ { pe (0)} = { p0∗e }. 3.1. Primal macro-hybrid composition duality principles Let us consider the local version of the primal compatibility condition (CG,Λ ) relative to decompositions (3)–(6), (C[G e ,Λe ] )

int D(G e ) ∩ R(Λe ) 6= ∅,

e = 1, 2, . . . , E,

as well as the corresponding Lemma 2.1. Lemma 3.1. Let condition (C[G e ,Λe ] ) be satisfied. Then the local compositional operator equalities ∂(G e ◦ Λe ) = ΛTe ∂G e ◦ Λe ,

e = 1, 2, . . . , E,

(8)

are guaranteed. Then the composition duality principle of problem (M M H ) is concluded by dualization as in Theorem 2.2. Theorem 3.2. Under condition (C[G e ,Λe ] ), the macro-hybridized primal evolution mixed problem (M M H ) is solvable if, and only if, the macro-hybrid primal evolution problem  Find {ue } ∈ W    {Ω e } :  du e + {∂ Fe (u e )} + {∂(G e ◦ Λe )(u e − wge )} + ∂(I Q ◦ [πΓe ])({u e }) − { f e∗ }, in V ∗{Ωe } , (P M H ) {0e } ∈  dt   {u e (0)} = {u 0e }, is solvable, where {wge } ∈ V {Ωe } is a fixed [Λe ]-preimage of function {ge } : {Λe wge } = {ge }. That is, if ({u e }, { pe∗ }) ∈ W {Ωe } × Y ∗{Ωe } is a solution of problem (M M H ) then primal function {u e } is a solution of problem (P M H ) and, conversely, if {u e } ∈ W {Ωe } is a solution of problem (P M H ) then there is a dual function { pe∗ } ∈ {∂G e (Λe u e − ge )} ⊂ Y ∗{Ωe } such that ({u e }, { pe∗ }) is a solution of problem (M M H ). Moreover, problem (P M H ) is the macro-hybridization of global primal problem (P); i.e., primal problems (P M H ) and (P) are equivalent. Furthermore, the macro-hybridization process for primal evolution problem (M) is completed by introducing ∗ the internal boundary dual trace {λ∗e } ∈ ∂I Q ({πΓe u e }) ⊂ B∗{Γe } = L p (0, T ; B∗ ({Γe })) and dualizing in a compositional sense, with respect to compatibility condition (C[πΓe ] ), the primal transmission constraint imposed by ∂(I Q ◦ [πΓe ])({u e }) in problem (M M H ). The [πΓe ]-compositional dualization is performed as follows (see [8]). Lemma 3.3. Let condition (C[πΓe ] ) be satisfied. Then the macro-hybrid compositional dualization {πΓTe λ∗e } ∈ ∂(I Q ◦ [πΓe ])({u e }) ⇐⇒ {πΓe u e } ∈ ∂I Q ∗ ({λ∗e }) holds true.

(9)

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Proof. Taking into account that the conjugate indicator functional (I Q )∗ = I Q ∗ , by dualization {πΓe u e } ∈ ∂I Q ∗ ({λ∗e }) ⇔ {λ∗e } ∈ ∂I Q ({πΓe u e }). Then equivalence (9) is valid since the variational inequalities of the primal inclusions {πΓTe λ∗e } ∈ ∂(I Q ◦ [πΓe ])({u e }) and {λ∗e } ∈ ∂I Q ({πΓe u e }) are equivalent under macro-hybrid compatibility condition (C[πΓe ] ).  Therefore, the macro-hybrid mixed inclusion of problem (M) turns out to be  ∗ Find (u e , pe∗ ) ∈ WΩe × YΩ , for e = 1, 2, . . . , E :  e   du  e ∗   −ΛT p ∗ − πΓTe λ∗e ∈ + ∂ Fe (u e ) − f e∗ , in VΩ ,  e  e e dt ∗ ∗ (MH) Λe u e ∈ ∂G e ( pe ) + ge , in YΩe ,   u e (0) = u 0e ;     and {λ∗e } ∈ B∗{Γe } satisfying the dual synchronizing condition   {πΓe u e } ∈ ∂I Q ∗ ({λ∗e }), in B{Γe } , and the next composition duality principle establishes its solvability. Theorem 3.4. Let compatibility conditions (C[G e ,Λe ] ) and (C[πΓe ] ) be fulfilled. Then the primal evolution macrohybrid mixed problem (MH) has a solution if, and only if, the macro-hybridized primal evolution problem (P M H ) equivalent to the global primal problem (P) has a solution. That is, if ({u e }, { pe∗ }, {λ∗e }) ∈ W {Ωe } × Y ∗{Ωe } × B∗{Γe } is a solution of problem (MH) then primal function {u e } is a solution of problem (P M H ) and, conversely, if {u e } ∈ W {Ωe } is a solution of problem (P M H ) then there are dual functions { pe∗ } ∈ {∂G e (Λe u e − ge )} ⊂ Y ∗{Ωe } and {λ∗e } ∈ ∂I Q ({πΓe u e }) ⊂ B∗{Γe } such that ({u e }, { pe∗ }, {λ∗e }) is a solution of problem (MH). Proof. The necessity follows from Theorem 3.2 since, on the basis of Lemma 3.3, if ({u e }, { pe∗ }, {λ∗e }) ∈ W {Ωe } × Y ∗{Ωe } × B∗{Γe } is a solution of macro-hybrid problem (MH) then ({u e }, { pe∗ }) is a solution of macro-hybridized problem (M M H ). For the sufficiency, let ({u e }, { pe∗ }) ∈ W {Ωe } × Y ∗{Ωe } be a solution of macro-hybridized problem (M M H ). Then there is a functional {we∗ } ∈ {∂ Fe (u e )} ⊂ V ∗{Ωe } such that −{we∗ } − {ΛTe pe∗ } − {du e /dt} + { f e∗ } ∈ ∂(I Q ◦ [πΓe ])({u e }). From the variational inequality of this last inclusion, taking variations {ve } = ±{ve0 } + {u e }, with {ve0 } in the kernel N ([πΓe ]) ⊂ V {Ωe } , it follows that −{we∗ }−{ΛTe pe∗ }−{du e /dt}+{ f e∗ } belongs to the polar subspace N ([πΓe ])◦ ⊂ V ∗{Ωe } . Hence, by condition (C[πΓe ] ), the Closed Range Theorem states that N ([πΓe ])◦ = R([πΓe ]T ) and then there is a [πΓe ]T -preimage {λ∗e } ∈ B∗{Γe } such that {we∗ } = −{ΛTe pe∗ } − {πΓTe λ∗ } − {du e /dt} + { f e∗ }. That is, applying Lemma 3.3, ({u e }, { pe∗ }, {λ∗e }) conforms to a solution of problem (MH).  3.2. Dual macro-hybrid composition duality principles For the macro-hybridized dual problem (M∗M H ), we consider the macro-hybrid version of the dual compatibility condition (C F ∗ ,−ΛT ), (C[F ∗ ∗

|V (Ωe )

,−ΛTe ] )

∗ T int D(F|V ∗ (Ω ) ) ∩ R(−Λe ) 6= ∅, e

e = 1, 2, . . . , E,

and its compositional result. Lemma 3.1∗ . If condition (C[F ∗ ∗

|V (Ωe )

,−ΛTe ] )

is fulfilled, then the local compositional operator equalities

∗ T ∗ T ∂(F|V ∗ (Ω ) ◦ (−Λe )) = −Λe ∂ F|V ∗ (Ω ) ◦ (−Λe ), e e

e = 1, 2, . . . , E,

(8*)

hold true. Here the restriction to V ∗ ({Ωe }) ⊂ V ∗ (Ω ) of the conjugate superpotential F ∗ : V ∗ (Ω ) → R ∪ {+∞} corresponds ∗ ∗ to the conjugate of the macro-hybrid primal superpotential; i.e., F|V ∗ ({Ω }) = ([Fe ] + I Q ◦ [πΓe ]) . Note that the e macro-hybrid primal subdifferential [∂ Fe ] + ∂(I Q ◦ [πΓe ]) = ∂([Fe ] + I Q ◦ [πΓe ]) since D([Fe ]) ∩ V (Ω ) 6= ∅ [17]. Then the composition duality principle of problem (M∗M H ) is obtained by dualization (see Theorem 2.2*). Theorem 3.2∗ . Let condition (C[F ∗ ∗ (M∗M H )

|V (Ωe )

,−ΛTe ] )

be satisfied. Then macro-hybridized dual evolution mixed problem

is solvable if, and only if, the macro-hybrid dual evolution problem

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 Find { pe∗ } ∈ X ∗{Ωe } :    ∗  d pe ∗ T ∗ ∗ (D M H ) {0e } ∈ + {∂G ∗e ( pe∗ )} + ∂(F|V ∗ ({Ω }) ◦ [−Λe ])({ pe + r f ∗ }) + {ge }, e e  dt   ∗ { pe (0)} = { p0∗e },

in Y {Ωe } ,

is solvable, where {r ∗f ∗ } ∈ Y {Ωe } is a fixed [−ΛTe ]-preimage of function { f e∗ } : {−ΛTe r ∗f ∗ } = { f e∗ }. That is, e e if ({u e }, { pe∗ }) ∈ V {Ωe } × X ∗{Ωe } is a solution of problem (M∗M H ) then dual function { pe∗ } is a solution of problem (D M H ) and, conversely, if { pe∗ } ∈ X ∗{Ωe } is a solution of problem (D M H ) then there is a primal function {u e } ∈ ∂F∗|V ∗ ({Ωe }) ({−ΛTe pe∗ + f e∗ }) ⊂ V {Ωe } such that ({u e }, { pe∗ }) is a solution of problem (M∗M H ). Moreover,

problem (D M H ) is the macro-hybridization of global dual problem (D); i.e., dual problems (D M H ) and (D) are equivalent.

The macro-hybrid mixed inclusion that completes the localization process of dual evolution mixed problem (M∗ ) follows as in the previous primal subsection, by applying the macro-hybrid compositional dualization (9) of Lemma 3.3, under compatibility condition (C[πΓe ] ), in the dualization of the primal transmission constraint. Find (u , p ∗ ) ∈ V × X ∗ , for e = 1, 2, . . . , E : e Ω Ωe   T ∗ eT ∗ e ∗   , ∈ ∂ Fe (u e ) − f e∗ , in VΩ λ −Λ p − π  e e e Γ e e  ∗  d pe  ∗ ∗ (MH∗ ) Λe u e ∈ dt + ∂G e ( pe ) + ge , in YΩe ,   pe∗ (0) = p0∗e ;    ∗ ∗  the dual synchronizing condition  and {λe } ∈ B{Γe } satisfying {πΓe u e } ∈ ∂I Q ∗ ({λ∗e }), in B{Γe } . Moreover, the corresponding composition duality principle is established through the same arguments as in Theorem 3.4. Theorem 3.4∗ . Let compatibility conditions (C[F ∗ ∗

|V (Ωe )

,−ΛTe ] )

and (C[πΓe ] ) be satisfied. Then the dual evolution

macro-hybrid mixed problem (MH ) has a solution if, and only if, the macro-hybridized dual evolution problem (D M H ) equivalent to the global dual problem (D) has a solution. That is, if ({u e }, { pe∗ }, {λ∗e }) ∈ V {Ωe } × X ∗{Ωe } × B∗{Γe } is a solution of problem (MH∗ ) then dual function { pe∗ } is a solution of problem (D M H ) and, conversely, if ∗ T ∗ ∗ { pe∗ } ∈ X ∗{Ωe } is a solution of problem (D M H ) then there is a primal function {u e } ∈ ∂ F|V ∗ ({Ω }) ({−Λe pe + f e }) ⊂ e V {Ωe } and a dual function {λ∗e } ∈ ∂I Q ({πΓe u e }) ⊂ B∗{Γe } such that ({u e }, { pe∗ }, {λ∗e }) is a solution of problem (MH∗ ). ∗

4. Semi-discrete internal variational approximations We now present semi-discrete internal approximations, in general globally nonconforming, of the primal and dual evolution macro-hybrid mixed variational inclusions (MH) and (MH∗ ), and we establish their composition duality principles for qualitative analysis. In a similar fashion, semi-discretizations of evolution mixed problems (M) and (M∗ ) could be treated; however we shall concentrate our efforts here on macro-hybrid models motivated by the technological importance of parallel computing. Let us then introduce internal approximations of the local primal and dual spaces V (Ωe ) and Y ∗ (Ωe ), e = 1, 2, . . . , E, through convergent families of finite dimensional subspaces {Vh e }h e >0 and {Yh ∗e }h ∗e >0 , ∀ve ∈ V (Ωe ), ∀ye∗ ∈ Y ∗ (Ωe ),

∃ vh e ∈ Vh e ⊂ V (Ωe ): lim kve − vh e kV (Ωe ) = 0, h e ↓0

∃ yh ∗e ∈ Yh ∗e ⊂ Y ∗ (Ωe ): lim kye∗ − yh ∗e kY ∗ (Ωe ) = 0, ∗

(10)

h e ↓0

as well as internal approximations of the macro-hybrid dual spaces B ∗ (Γe ), e = 1, 2, . . . E, in terms of a family of finite dimensional subspaces {Bh ◦e }h ◦e >0 , ∀ζe∗ ∈ B ∗ (Γe ),

∃ζh ◦e ∈ Bh ◦e ⊂ B ∗ (Γe ): lim kζe∗ − ζh ◦e k B ∗ (Γe ) = 0. ◦ h e ↓0

(11)

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Stationary compatibility conditions for these families will be given for well-posedness and stability in Section 6, when discussing full discretizations and preconditioned penalty–duality algorithms. Hence, we associate to evolution problems (MH) and (MH∗ ) the semi-discrete inclusions  Find (u h e , ph ∗e ) ∈ Wh e × Yh ∗e , for e = 1, 2, . . . , E :      du h e  ∗ ∗ T T   −Λh e ,h ∗e ph ∗e − πh e ,h ◦e λh ◦e ∈ dt + ∂ Fh e (u h e ) − f h e , in Vh e , ∗ (MHh,h ∗ ,h ◦ ) Λh e ,h ∗e u h e ∈ ∂G h ∗e ( ph ∗e ) + gh ∗e , in Yh ∗e ,   u h e (0) = u 0h e ;     and {λh ◦e } ∈ B{h ◦e } satisfying the dual synchronizing condition   {πh e ,h ◦e u h e } ∈ ∂I Q h ◦ ({λh ◦e }), in B∗{h ◦e } , and  Find (u , p ∗ ) ∈ V × Xh ∗e , for e = 1, 2, . . . , E :   T he he T he  ∗ ∗  ∗ ◦ −Λ   h e ,h ∗e ph e − πh e ,h ◦e λh e ∈ ∂ Fh e (u h e ) − f h e , in Vh e ,   d ph ∗e  + ∂G h ∗e ( ph ∗e ) + gh ∗e , in Yh∗∗e , Λh e ,h ∗e u h e ∈ ∗ (MHh,h ∗ ,h ◦ ) dt   p ∗ (0) = p0h ∗ ;   he e   synchronizing condition  and {λh ◦e } ∈ B{h ◦e } satisfying the dual  {πh e ,h ◦e u h e } ∈ ∂I Q h ◦ ({λh ◦e }), in B∗{h ◦e } . ∗

Here, the semi-discrete evolution product spaces are defined by V {h e } = L p (0, T ; V{h e } ), Y {h ∗e } = L p (0, T ; Y{h ∗e } ) ∗ ∗ ∗ ∗ ), Y ∗ ∗ p and B{h ◦e } = L p (0, T ; B{h ◦e } ), with duals V ∗{h e } = L p (0, T ; V{h {h ∗e } = L (0, T ; Y{h ∗e } ) and B{h ◦e } = e} ∗ ∗ p L (0, T ; B{h ◦ } ), and the solution spaces by W {h e } = {{vh e } : {vh e } ∈ V {h e } , {dvh e /dt} ∈ V {h e } } ⊂ C(0, T ; H({Ωe })) e and X {h ∗e } = {{qh ∗e } : {qh ∗e } ∈ Y {h ∗e } , {dqh ∗e /dt} ∈ Y ∗{h ∗ } } ⊂ C(0, T ; Z∗ ({Ωe })). Moreover, the operators and e functionals are the original ones in their restriction discrete sense: for e = 1, 2, . . . , E, ∗

∂ Fh e : Vh e → 2Vh e ,

f h∗e ∈ Vh∗e , ΛTh e ,h ∗e ∈ L(Yh ∗e , Vh∗e ),

Λh e ,h ∗e ∈ L(Vh e , Yh∗∗e ), ∂G h ∗e : Yh ∗e → 2

Yh∗∗ e

,

gh ∗e ∈ Yh∗∗e ,

πΓh e ,h ◦ ∈ L(Vh e , Bh∗◦e ), e

(12)

πhTe ,h ◦e ∈ L(Bh ◦e , Vh∗e ),

and {u 0h e } ∈ H({Ωe }), { p0h ∗ } ∈ Z∗ ({Ωe }) are given discrete initial data. Further, Qh ◦ ⊂ B{h ◦e } ⊂ B∗ ({Γe }) stands e for a discrete version of the dual transmission admissibility subspace Q∗ ⊂ B∗ ({Γe }) of (7) relative to macro-hybrid approximations (11). Then, the polar discrete subspace of Qh ◦ is given by ( ) E

X  ∗ ∗ ∗ ∗ Qh ◦ = {ζe } ∈ B{h ◦e } : ζe , ζe B ◦ = 0, ∀{ζe } ∈ Qh ◦ , (13) he

e=1

and the discrete version of the global primal space V (Ω ) satisfying (5)1 , relative to primal approximations (10)1 , is ( ) E Y ∗ Vh,h ◦ = {vh e } ∈ V{h e } = Vh e : {πΓh e ,h ◦ vh e } ∈ Qh ◦ . (14) e=1

e

Note that, in general, the method is nonconforming, since Vh,h ◦ 6⊂ V (Ω ). As the discrete macro-hybrid compatibility condition, we have (C[πΓ

h e ,h ◦e

])

πΓh e ,h ◦ ∈ L(Vh e , Bh∗◦e ) e

is surjective, e = 1, 2, . . . , E,

with the following corresponding dualization result (see Lemma 3.3).

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Lemma 4.1. Let condition (C[πΓ {πΓT

h e ,h ◦e

h e ,h ◦e

])

be satisfied. Then the discrete macro-hybrid compositional dualization

λh ◦e } ∈ ∂(I Q ∗h ◦ ◦ [πΓh e ,h ◦ ])({u h e }) ⇐⇒ {πΓh e ,h ◦ u h e } ∈ ∂I Q h ◦ ({λh ◦e }) e

e

(15)

holds true. 4.1. Semi-discrete primal macro-hybrid mixed composition duality principle For the analysis of the semi-discrete primal evolution macro-hybrid mixed inclusion (MHh,h ∗ ,h ◦ ), we consider the classical compatibility condition (C[G ∗∗ ,Λh he

∗] e ,h e

)

int D(G ∗h ∗e ) ∩ R(Λh e ,h ∗e ) 6= ∅,

e = 1, 2, . . . , E,

under which the next compositional result holds as in Lemma 3.1. Lemma 4.2. Let condition (C[G ∗∗ ,Λh he

∗] e ,h e

) be satisfied. Then the discrete compositional operator equalities

∂(G ∗h ∗e ◦ Λh e ,h ∗e ) = ΛTh e ,h ∗e ∂G ∗h ∗e ◦ Λh e ,h ∗e ,

e = 1, 2, . . . , E,

(16)

are guaranteed. Hence, by dualization in the sense of Lemmas 4.1 and 4.2, we have the following semi-discrete primal composition duality principle (see Theorems 2.2, 3.2 and 3.4). Theorem 4.3. Let discrete compatibility conditions (C[G ∗∗ ,Λh he

∗] e ,h e

) and (C[πΓ

] h e ,h ◦ e

) be fulfilled. Then the semi-discrete

primal evolution macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ) has a solution if, and only if, the semi-discrete macrohybridized primal evolution problem  Find {u he } ∈ W   {h e } :    {0 } ∈ du h e + {∂ F (u )} + {∂(G ∗ ◦ Λ ∗ )(u − w )} gh ∗ h e ,h e he he he he h ∗e e (P M Hh,h ∗ ,h ◦ ) dt  + ∂(I ∗ ◦ [π ◦ ])({u }) − { f ∗ }, in V ∗ ,   h ,h h Q e e {h e } he e  h◦  {u h e (0)} = {u 0h e }, has a solution, where {wgh ∗ } ∈ V {h e } is a fixed [Λh e ,h ∗e ]-preimage of function {gh ∗e } : {Λh e ,h ∗e wgh ∗ } = {gh ∗e }. That is, e e if ({u h e }, { ph ∗e }, {λh ◦e }) ∈ W {h e } × Y {h ∗e } × B{h ◦e } is a solution of problem (MHh,h ∗ ,h ◦ ) then primal function {u h e } is a solution of problem (P M Hh,h ∗ ,h ◦ ) and, conversely, if {u h e } ∈ W {h e } is a solution of problem (P M Hh,h ∗ ,h ◦ ) then there are dual functions { ph ∗e } ∈ {∂G ∗h ∗ (Λh e ,h ∗e u h e − gh ∗e )} ⊂ Y {h ∗e } and {λh ◦e } ∈ ∂I Q ∗h ◦ ({πh e ,h ◦e u h e }) ⊂ B{h ◦e } such that e ({u h e }, { ph ∗e }, {λh ◦e }) is a solution of problem (MHh,h ∗ ,h ◦ ). 4.2. Semi-discrete dual macro-hybrid mixed composition duality principle Similarly, introducing the classical dual compatibility condition (C[F ∗

T ] h e |V ∗ ,−Λh e ,h ∗ e he

)

int D(Fh∗e |V ∗ ) ∩ R(−ΛTh e ,h ∗e ) 6= ∅, he

e = 1, 2, . . . , E,

the next compositional result is valid. Lemma 4.2∗ . Let condition (C[F ∗

T ] h e |V ∗ ,−Λh e ,h ∗ e he

) be fulfilled. Then the discrete compositional operator equality

∂([Fh∗e ]|V{h∗ } ◦ [−ΛTh e ,h ∗e ]) = [−Λh e ,h ∗e ]∂[Fh∗e ]|V{h∗ } ◦ [−ΛTh e ,h ∗e ], e

holds true.

e

(16*)

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Here, [Fh∗e ]|V{h∗ } = ([Fh e ] + I Q ∗h ◦ ◦ [πh e ,h ◦e ])∗ corresponds to the restriction of the conjugate of primal superpotential e ∗ ∗ . Then, according to Lemmas 4.1 and 4.2*, the semi-discrete dual [Fh e ] : Vh,h ◦ → R ∪ {+∞} to V{h ⊂ Vh,h ◦ e} composition duality principle is obtained (see Theorems 2.2*, 3.2* and 3.4*). Theorem 4.3∗ . Let discrete compatibility conditions (C[F ∗

T ] h e |V ∗ ,−Λh e ,h ∗ e he

) and (C[πΓ

h e ,h ◦ e

])

be satisfied. Then the semi-

discrete dual evolution macro-hybrid mixed problem (MH∗h,h ∗ ,h ◦ ) has a solution if, and only if, the semi-discrete macro-hybridized dual evolution problem  Find { ph ∗e } ∈ X {h ∗e } :       d ph ∗e (D M Hh,h ∗ ,h ◦ ) {0h ∗ } ∈ + {∂G h ∗e ( ph ∗e )} + ∂([Fh∗e ]|V{h∗ } ◦ [−ΛTh e ,h ∗e ])({ ph ∗e + r ∗f ∗ }) + {gh ∗e }, in Y ∗{h ∗e } , e  e he dt   { p ∗ (0)} = { p }, he 0h ∗ e

has a solution, where

{r ∗f ∗ } he

∈ Y {h ∗e } is a fixed [−ΛTh e ,h ∗ ]-preimage of function { f h∗e } : {−ΛTh e ,h ∗ r ∗f ∗ } = { f h∗e }. e

e

he

That is, if ({u h e }, { ph ∗e }, {λh ◦e }) ∈ V {h e } × X {h ∗e } × B{h ◦e } is a solution of problem (MH∗h,h ∗ ,h ◦ ) then dual function { ph ∗e } is a solution of problem (D M Hh,h ∗ ,h ◦ ) and, conversely, if { ph ∗e } ∈ X {h ∗e } is a solution of problem (D M Hh,h ∗ ,h ◦ ) then there is a primal function {u h e } ∈ ∂[Fh∗e ]|V{h∗ } ({−ΛTh e ,h ∗ ph ∗e + f h∗e }) ⊂ V {h e } and a dual function {λh ◦e } ∈ e e ∂I Q ∗h ◦ ({πh e ,h ◦e u h e }) ⊂ B{h ◦e } such that ({u h e }, { ph ∗e }, {λh ◦e }) is a solution of problem (MH∗h,h ∗ ,h ◦ ). 5. Proximation implicit and semi-implicit time marching schemes In this section, we consider full discretizations of the evolution macro-hybrid mixed inclusions (MH) and (MH∗ ) by applying time marching schemes to the semi-discrete models (MHh,h ∗ ,h ◦ ) and (MH∗h,h ∗ ,h ◦ ). Specifically, implicit and semi-implicit numerical time integration schemes, implementable as proximation iterative algorithms, will be considered. Further, in order to get closer to the numerical computational structure of these discrete models, we shall regard coordinate versions of the spatial discretizations, relative to corresponding finite dimensional basis. Convergence analysis to stationary states is provided. In the next section, corresponding stationary well-posedness will be established via composition duality principles, as well as preconditioned penalty–duality interpretations. Here, we follow our work [5] on parallel proximal-point algorithms. Hence, let us first introduce the coordinate expressions of the primal and dual unknowns of problems (MHh,h ∗ ,h ◦ ) and (MH∗h,h ∗ ,h ◦ ) with respect to the internal variational approximations (10) and (11): for e = 1, 2, . . . , E, u he =

m he X i=1 m h∗

ph ∗e =

Xe

k=1 m h ◦e

λh ◦e =

X s=1

αei φei ∈ Vh e = [φe1 , φe2 , . . . , φem h ] ⊂ V (Ωe ), e

δe∗k ϕe∗k ∈ Yh ∗e = [ϕe∗1 , ϕe∗2 , . . . , ϕe∗m ∗ ] ⊂ Y ∗ (Ωe ), he

θe◦s ψe◦s ∈ Bh ◦e = [ψe◦1 , ψe◦2 , . . . , ψe◦m ◦ ] ⊂ B ∗ (Γe ). he

Then semi-discrete problems (MHh,h ∗ ,h ◦ ) and (MH∗h,h ∗ ,h ◦ ) have the coordinate versions, for a.e. t ∈ (0, T ),  m ∗ Find (α e (t), δ ∗e (t)) ∈ Rm h e × R h e , for e = 1, 2, . . . , E :    dα e    −ΛTe δ ∗e (t) − π Te θ ◦e (t) ∈ Me (t) + ∂Fe (α e (t)) − fe∗ (t),   dt ∗ ∗ (t) (MHh,h ∗ ,h ◦ ) Λe α e (t) ∈ ∂Ge (δ e (t)) + ge (t),   α e (0) = α 0e ;   ◦ m h◦   satisfying the dual synchronizing condition  and {θ e (t)} ∈ R {π e α e (t)} ∈ ∂I Q ◦ ({θ ◦e (t)}),

(17)

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and  m ∗ Find (α e (t), δ ∗e (t)) ∈ Rm h e × R h e , for e = 1, 2, . . . , E :     −ΛTe δ ∗e (t) − π Te θ ◦e (t) ∈ ∂Fe (α e (t)) − fe∗ (t),   ∗   ∗ dδ e ∗(t) (t) + ∂G∗e (δ ∗e (t)) + ge (t), Λ α (t) ∈ M e e e (MHh,h ∗ ,h ◦ ) dt  ∗ ∗  δ e (0) = δ 0e ;   ◦ m h◦  satisfying the dual synchronizing condition  and {θ e (t)} ∈ R {π e α e (t)} ∈ ∂I Q ◦ ({θ ◦e (t)}). Here, the primal and dual local discrete superpotentials are defined, for e = 1, 2, . . . , E, by ! m he X Fe (β e ) = Fe βei φei , β e ∈ Rm h e , i=1 m h∗

G ∗e (µ∗e )

=

G ∗e

Xe

(18)

! µ∗ek

ϕe∗k

,

µ∗e

∈R

m h∗ e

,

k=1

and the local matrices and vectors, for 1 ≤ i, j ≤ m h e , 1 ≤ k, l ≤ m h ∗e , 1 ≤ s ≤ m h ◦e , by Me, ji = (φei , φe j ) H (Ωe ) ,

Λe,ki = hϕe∗k , Λe φei iY (Ωe ) ,

∗ Me,lk = (ϕe∗l , ϕe∗k ) Z (Ωe ) ,

∗ f e,i = h f e∗ , φei iV (Ωe ) ,

π e,si = hψe◦s , πΓe φei i B(Γe ) , ge,k = hϕe∗k , ge iY (Ωe ) .

Also, the discrete dual synchronizing indicator superpotential is )! ( m h◦ Xe ◦ ◦ ◦ , {ν ◦e } ∈ Rm h ◦ . {νes ψes } I Q ◦ ({ν e }) = I Q h ◦

(19)

(20)

s=1

5.1. A primal time marching scheme (t)

For semi-discrete primal evolution problem (MHh,h ∗ ,h ◦ ), we consider the following numerical time integration scheme. Let r > 0 denote the time marching step and let m ≥ 0 be the step number. The primal semi-implicit Euler scheme QE m ∗ Given {α 0e } ∈ e=1 D(Fe ) ⊂ Rm h with {π e α 0e } ∈ Q∗◦ , find (α m+1 , δ ∗m+1 ) ∈ Rm h e × R h e , for e = 1, 2, . . . , E: e e α m+1 − αm e e ∗m + ∂Fe (α m e ) − fe , r ∈ ∂G∗e (δ ∗m+1 ) + gm+1 ; e e

−ΛTe δ ∗m+1 − π Te θ ◦m+1 ∈ Me e e Λe α m+1 e

and {θ ◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition e {π e α m+1 } ∈ ∂I Q ◦ ({θ ◦m+1 }), e e where Q∗◦ ⊂ Rm h ◦ denotes the polar of the primal synchronizing subspace Q◦ defined in (20). With the primal compatibility conditions of Theorem 4.3 in force, the discrete primal problem of the above scheme at the m + 1 ≥ 1 time step is given by  E Y   0  D(Fe ) ⊂ Rm h with {π e α 0e } ∈ Q∗◦ , find α m+1 ∈ Rm h e , for e = 1, 2, . . . , E :  e Given {α e } ∈ e=1   (Pm+1 M Hh,h ∗ ,h ◦ )  m m ∗m m+1 {M α } ∈ r {∂F  e e e (α e )} − r {fe } + [Me ] + r [∂(G e ◦ Λe )(· − wge )] + r ∂(I Q ∗◦ ◦ [π e ])    × ({α m+1 }). e Then, introducing the [Me ]-resolvent of the primal subdifferential ∂Sm+1 = [∂(G e ◦Λe )(·−wm+1 ge )]+∂(I Q ∗◦ ◦[π e ]) = r m+1 m+1 −1 ) , which is a single valued firm ∂({(G e ◦ Λe )(· − wg−e )} + I Q ∗◦ ◦ [π e ]) defined by J[M ],∂ S m+1 ≡ ([Me ] + r ∂S e

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

contraction [19], and has the proximation characterization [3], for β ∈ Rm h , r J[M

e ],∂ S

m+1

◦ [Me ](β) = Prox[Me ],r S m+1 (β)    1 m+1 ≡ arg inf [Me ](ξ − β)·(ξ − β) + r S (ξ ) , ξ ∈R m h 2

(21)

the proximation form of the primal semi-implicit Euler scheme is obtained as a multidomain proximal-point algorithm (t) for problem (MHh,h ∗ ,h ◦ ). : Algorithm ALG0MQ E m+1 }, for −{ξ m } ∈ Given {α 0e } ∈ e=1 D(Fe ) ⊂ Rm h with {π e α 0e } ∈ Q∗◦ , known {α m e e }, m ≥ 0, calculate {α m ∗m {∂Fe (α e )} − {fe }, such that {α m+1 } = Prox[Me ],r {G e ◦Λe (·−wm+1 )}+I e ge

◦[πe ] Q∗ ◦

−1 m ({α m e } + r {Me ξ e }).

Thereby, convergence of the scheme to stationary states of the primal dynamical system is given as follows. Proposition 5.1. Let the discrete compatibility conditions (C[G ∗∗ ,Λh he

∗] e ,h e

) and (C[πΓ

] h e ,h ◦ e

) be satisfied. Let the primal

operator [∂Fe ] = [Ae ] be a single valued, Lipschitz continuous and strongly monotone operator, with uniform constants a and c, respectively. Then, for time-independent data {fe∗m } = {fe∗ }, {wm+1 ge } = {wge }, m ≥ 0, algorithm m+1 ALG0M evolves, as m → ∞, to a (P M Hh,h ∗ ,h ◦ )-stationary state of the semi-discrete primal evolution problem (t)

(MHh,h ∗ ,h ◦ ), whenever 0 < r < 2c/a 2 .

(22)

Proof. In this case algorithm ALG0M takes the resolvent form r {α m+1 } = J[M e e ],∂({(G e ◦Λe )(·−wg

◦[πe ]) e )}+I Q ∗ ◦

∗ ({(Me − r Ae )α m e + r fe }),

(23)

and the convergence is concluded, taking into account the firm contraction property of the resolvent operator (21),  since the operator [Me ] − r [Ae ] turns out to be an strict contraction (see Theorem 6.1 of [19]). 5.2. Dual time marching schemes We next consider Euler and operator splitting time integration schemes for the semi-discrete dual evolution problem ∗(t) (MHh,h ∗ ,h ◦ ), which are implementable as multidomain proximal-point algorithms. 5.2.1. The dual semi-implicit Euler scheme QE m h∗ m ∗m+1 ) ∈ Rm h e × Rm h ∗e , for e = 1, 2, . . . , E: ∗ and {θ ◦0 Given {δ ∗0 e } ∈ Q◦ , find (α e , δ e e }∈ e=1 D(G e ) ⊂ R T ◦m m ∗m −ΛTe δ ∗m e − π e θ e ∈ ∂Fe (α e ) − fe , ∗ Λe α m e ∈ Me

δ ∗m+1 − δ ∗m e e + ∂G∗e (δ ∗m+1 ) + gem+1 ; e r

} ∈ Rm h ◦ satisfying the dual synchronizing condition and {θ ◦m+1 e m ◦m+1 {θ ◦m } + ∂I Q ◦ ({θ ◦m+1 }), e } + r {π e α e } ∈ {θ e e

where the synchronization is implemented as an Uzawa approximation. r r Introducing the proximation characterizations of the resolvent operators J[M , relative to the primal ∗ ∗ and J∂ I Q◦ e ],[∂G e ] functionals {G e } and I Q ∗◦ [3] (see (21)), r ∗ J[M = I − [Me−∗ ]Prox[Me−∗ ],r {G e ◦(1/r )Ie } ◦ [Me∗ ], ∗ ∗ ◦ [Me ] = Prox[Me∗ ],r {G ∗ e} e ],[∂G e ]

J∂r I Q ◦ = Proj Q ◦ = I − Proj Q ∗◦ ,

(24)

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

where [Me−∗ ] stands for the inverse of the “mass” diagonal matrix [Me∗ ] and r {G e ◦ (1/r )Ie } is the conjugate of functional {r G ∗e }, the next proximal realization of the dual semi-implicit Euler scheme is obtained, as a parallel algorithm. ∗: Algorithm ALG4MQ E ∗m ◦m m h∗ ∗ m Given {δ ∗0 } ∈ and {θ ◦0 e e } ∈ Q◦ , known {δ e } and {θ e }, m ≥ 0, calculate in parallel α e , e=1 D(G e ) ⊂ R and δ e∗m+1 , e = 1, 2, . . . , E, such that T ◦m m ∗m −ΛTe δ ∗m e − π e θ e ∈ ∂Fe (α e ) − fe , −∗ m m+1 δ ∗m+1 = (I − Me−∗ Prox Me−∗ ,r G e ◦(1/r )Ie ◦ Me∗ )(δ ∗m )); e e + r Me (Λe α e − ge

and {θ ◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition e m {θ ◦m+1 } = (I − Proj Q ∗◦ )({θ ◦m e + r π e α e }). e

For the stationary convergence analysis of this dual algorithm, we define the dual operator of the primal operator Af ∗ = [∂ Fe ](·) − {fe∗ }, relative to the coupling operator Ξ = ([Λe ], [πe ]) in the sense of Mosco [24]: for (µ∗ , ν ◦ ) ∈ Rm h ∗ × Rm h ◦ , A∗f ∗ ,Ξ (µ∗ , ν ◦ ) ≡ (A∗f ∗ ,Λ (µ∗ ), A∗f ∗ ,π (ν ◦ )) = {(ξ ∗ , ζ ◦ ) ∈ Rm h ∗ × Rm h ◦ : ∃β ∈ Rm h , ξ ∗ = −[Λe ]β, ζ ◦ = −[π e ]β, −[ΛTe ]µ∗ − [π Te ]ν ◦ ∈ A f ∗ (β)}.

(25)

Then the discrete macro-hybrid dual problem of the dual semi-implicit Euler scheme at the m + 1 ≥ 1 time step turns out to be  E Y  ∗m+1 Given {δ ∗0 } ∈  D(G ∗e ) ⊂ Rm h ∗ and {θ ◦0 }, {θ ◦m+1 }) ∈ Rm h ∗ × Rm h ◦ :  e e } ∈ Q◦ , find ({δ e e e=1 (Dm+1 h,h ∗ ,h ◦ )  ∗ ∗m ∗ ∗m+1 ∗ ∗m ∗ {M δ } ∈ r A }) + r {gm+1 },  e e e f ∗m ,Λ ({δ e }) + ([Me ] + r [∂Ge ])({δ e   ◦m ◦m+1 ∗ ◦m ∗ {θ e } ∈ r A f ∗m ,π ({θ e }) + ([Ie ] + ∂I Q ◦ )({θ e }), and a convergence to stationary dual states is concluded as follows. Proposition 5.2. Let the dual operator A∗f ∗ ,Ξ defined by (25) be single valued, Lipschitz continuous and strongly monotone with uniform constants a ∗ and c∗ , respectively. Then, for time-independent data f ∗m = f ∗ , wm+1 = wg , g m+1 algorithm ALG4M∗ evolves, as m → ∞, to a (Dh,h ∗ ,h ◦ )-stationary state of the semi-discrete dual evolution problem ∗(t)

(MHh,h ∗ ,h ◦ ), whenever 0 < r < 2c∗ /a ∗2 . Proof. Expressing the equations of dual problem takes the form

(26) (Dm+1 h,h ∗ ,h ◦ )

in terms of resolvent operators (24), algorithm ALG4M∗

r ∗ ∗ ∗m {δ ∗m+1 } = J[M ∗ ∗ (([Me ] − r AΛ ){δ e }), e e ],[∂G e ]

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{θ ∗m+1 } = J∂r I Q ∗ (([Ie∗ ] − r A∗π ){θ ∗m e e }),

and the convergence follows, as for algorithm ALG0M , from the firm contraction property of resolvent operators (24) and condition (26) for parameter r .  ∗(t)

As a second numerical integration scheme for the dual evolution problem (MHh,h ∗ ,h ◦ ), we consider the following. 5.2.2. The dual implicit Euler scheme QE m h∗ ∗ m+1 , δ ∗m+1 ) ∈ Rm h e × Rm h ∗e , for e = 1, 2, . . . , E: Given {δ ∗0 and {θ ◦0 e }∈ e } ∈ Q◦ , find (α e e e=1 D(G e ) ⊂ R −ΛTe δ ∗m+1 − π Te θ ◦m+1 ∈ ∂Fe (α m+1 ) − fe∗m+1 , e e e

G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

Λe α m+1 ∈ Me∗ e

349

δ ∗m+1 − δ ∗m e e + ∂G∗e (δ ∗m+1 ) + gem+1 ; e r

and {θ e◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition m+1 {θ ◦m } ∈ {θ ◦m+1 } + ∂I Q ◦ ({θ ◦m+1 }). e } + r {π e α e e e

Proceeding as for the dual semi-implicit Euler scheme, we have the next multidomain proximal-point algorithm. Algorithm ALG5M∗ : QE ∗m ∗m m h∗ ∗ m+1 }, and, in Given {δ ∗0 and {θ ◦0 e } ∈ Q◦ , known {δ e } and {θ e }, m ≥ 0, calculate {α e e }∈ e=1 D(G e ) ⊂ R parallel, δ ∗m+1 , e = 1, 2, . . . , E, such that e −∗ m+1 −ΛTe (Ie − Me−∗ Prox Me−∗ ,r G e ◦(1/r )Ie ◦ Me∗ )(δ ∗m − gm+1 )) e + r Me (Λe α e e m+1 − [π Te ]([Ie ] − Proj Q ∗◦ )({θ ◦m })e ∈ ∂Fe (α m+1 ) − fe∗m+1 , e f + rπ f α f −∗ m+1 δ ∗m+1 = (Ie − Me−∗ Prox Me−∗ ,r G e ◦(1/r )Ie ◦ Me∗ )(δ ∗m − gm+1 )); e e + r Me (Λe α e e

and {θ ◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition e m+1 } = ([Ie ] − Proj Q ∗◦ )({θ ◦m {θ ◦m+1 }). e + r π e αe e

An alternative proximal realization of the implicit Euler scheme is obtained by introducing the intermediate family of vectors ({ζ ∗e }, {κ ◦e }) ∈ Rm h ∗ × Rm h ◦ defined by the following, and also by dualization, such that ζ ∗e ∈ ∂G∗e (δ ∗e ) ⇐⇒ δ ∗e ∈ ∂Ge (ζ ∗e ), e = 1, 2, . . . , E, {κ ◦e } ∈ ∂I Q ◦ ({θ ◦e }) ⇐⇒ {θ ◦e } ∈ ∂I Q ∗◦ ({κ ◦e }).

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−∗ m+1 − gm+1 − ζ ∗m+1 )} ∈ Indeed, expressing the corresponding subdifferential equations {δ ∗m e } + r {Me (Λe α e e e ◦m ∗m+1 m+1 ◦m+1 ◦m+1 ∗ {∂Ge (ζ e )} and {θ e } + r {π e α e − κe } ∈ ∂I Q ◦ ({κ e )} in terms of the resolvent-proximation-projection 1/r 1/r relations J[M −∗ ],{∂G } ◦ [Me−∗ ] = Prox[Me−∗ ],(1/r ){G e } and J∂ I ∗ = Prox I Q ∗ = Proj Q ∗◦ , the following three-field ◦ Q◦ e e proximal-point algorithm is concluded.

Algorithm ALG1M∗ : QE ∗m ◦m m h∗ ∗ ◦m+1 } satisfying Given {δ ∗0 and {θ ◦0 e } ∈ Q◦ , known {δ e } and {θ e }, m ≥ 0, calculate {κ e e }∈ e=1 D(G e ) ⊂ R the primal synchronizing condition   1 ◦m ◦m+1 m+1 {κ e } = Proj Q ∗◦ θ + π e αe , r e and ζ ∗m+1 , α m+1 , e = 1, 2, . . . , E, such that e e   1 ∗ ∗m m+1 m+1 ∗m+1 M δ + Λe α e − ge , ζe = Prox Me−∗ ,(1/r )G e r e e −∗ ∗m+1 ◦m+1 −ΛTe (δ ∗m ) − π Te (θ ◦m ) e − r Me ζ e e − rκe

∈ (∂Fe + r ΛTe Me−∗ Λe + r π Te π e )(α m+1 ) − fe∗m+1 − r ΛTe Me−∗ gm+1 ; e e and δ ∗m+1 , θ e◦m+1 , e = 1, 2, . . . , E, e −∗ m+1 δ ∗m+1 = δ ∗m − gm+1 − ζ ∗m+1 ), e e + r Me (Λe α e e e m+1 θ e◦m+1 = θ ◦m − κ ◦m+1 ). e + r (π e α e e

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

In order to establish the convergence of these algorithms to stationary states, we note that the discrete macro-hybrid dual problem of the implicit Euler scheme at the m + 1 ≥ 1 time step, upon dualization in terms of the dual operator (25), is given by  E Y  ∗0 ∗m+1   Given {δ } ∈ D(G ∗e ) ⊂ Rm h ∗ and {θ ◦0 }, {θ ◦m+1 }) ∈ Rm h ∗ × Rm h ◦ : e e } ∈ Q◦ , find ({δ e e   e=1 m+1 (D^ h,h ∗ ,h ◦ )  ∗ ∗m ∗ ∗ ∗ ∗m+1 {M δ } ∈ ([M }) + r {gm+1 },  e e e ] + r A f ∗m+1 ,Λ + r [∂Ge ])({δ e e    ◦m ∗ ∗ ◦m+1 }). {θ e } ∈ ([Ie ] + r A f ∗m+1 ,π + ∂I Q ◦ )({θ e Then the following result is valid. Proposition 5.3. Let the dual operators A∗Λ + [∂G∗e ] and A∗π + ∂I Q ◦ be maximal monotone. Then, for timem+1 independent data f ∗m = f ∗ , wm+1 = w , algorithms ALG5 ∗ and ALG1 ∗ evolve, as m → ∞, to a (D^ )g

g

M

M

∗(t)

h,h ∗ ,h ◦

stationary state of the semi-discrete dual evolution problem (MHh,h ∗ ,h ◦ ). m+1 Proof. In this case, the equations of dual problem (D^ h,h ∗ ,h ◦ ) have the resolvent expressions r {δ ∗m+1 } = J[M ∗ e e ],A ∗

f ,Λ

r } = JA {θ ◦m+1 ∗ e ∗ f

∗m +[∂G ∗e ] ({δ e }),

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◦m +∂ I Q ◦ ({θ e }), ,π

and the stationary convergence holds true from the firm contraction property of the resolvent operators (see Theorem 2.1 in [19]).  5.2.3. The operator splitting schemes ∗(t)

For the semi-discrete dual evolution problem (MHh,h ∗ ,h ◦ ), we finally consider the next two dual semi-implicit time marching schemes, which result in fully parallel versions of algorithm ALG1M∗ with an initial primal synchronization. The Douglas–Rachford scheme: QE m h∗ ∗ ∗ m ∗m+1 ) ∈ Rm h e × Rm h ∗e , e = 1, 2, . . . , E, such Given {δ ∗0 and {θ ◦0 e } ∈ Q , find (α e , δ e e } ∈ e=1 D(G e ) ⊂ R that T ◦m m ∗m −ΛTe δ ∗m e − π e θ e ∈ ∂Fe (α e ) − fe , ∗ Λe α m e ∈ Me

δ ∗m+1 − δ ∗m e e + ∂G∗e (δ ∗m+1 ); e r

and {θ ◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition e m ◦m+1 {θ ◦m } + ∂I Q ◦ ({θ ◦m+1 }); e } + r {π e α e } ∈ {θ e e

find (α m+1 , δ ∗m+1 ) ∈ Rm h e × R e e

m h∗ e

, e = 1, 2, . . . , E, such that

−ΛTe δ ∗m+1 − π Te θ ◦m+1 ∈ ∂Fe (α m+1 ) − fe∗m+1 , e e e Λe α m+1 ∈ Me∗ e

δ e∗m+1 − δ ∗m e + ∂G∗e (δ e∗m+1 ); r

and {θ ◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition e m+1 }). {θ ◦m } ∈ {θ ◦m+1 } + ∂I Q ◦ ({θ ◦m+1 e } + r {π e α e e e

G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

351

The Peaceman–Rachford scheme: QE m h∗ ∗ ∗ m ∗m+1/2 ) ∈ Rm h e × Rm h ∗e , e = 1, 2, . . . , E, such Given {δ ∗0 and {θ ◦0 e }∈ e } ∈ Q , find (α e , δ e e=1 D(G e ) ⊂ R that T ◦m m ∗m −ΛTe δ ∗m e − π e θ e ∈ ∂Fe (α e ) − fe , ∗m+1/2

∗ Λe α m e ∈ Me ◦m+1/2

and {θ e

δe

− δ ∗m ∗m+1/2 e + ∂G∗e (δ e ); r/2

} ∈ Rm h ◦ satisfying the dual synchronizing condition ◦m+1/2

m {θ ◦m e } + r/2{π e α e } ∈ {θ e

find (α m+1 , δ ∗m+1 ) ∈ Rm h e × R e e

m h∗ e

◦m+1/2

} + ∂I Q ◦ ({θ e

});

, e = 1, 2, . . . , E, such that

−ΛTe δ ∗m+1 − π Te θ ◦m+1 ∈ ∂Fe (α m+1 ) − fe∗m+1 , e e e ∗m+1/2

Λe α m+1 ∈ Me∗ e

δ ∗m+1 − δe e r/2

∗m+1/2

+ ∂G∗e (δ e

);

and {θ e◦m+1 } ∈ Rm h ◦ satisfying the dual synchronizing condition ◦m+1/2

{θ e

◦m+1/2

} ∈ {θ ◦m+1 } + ∂I Q ◦ ({θ e } + r/2{π e α m+1 e e

}).

As for algorithm ALG1M∗ , we introduce an intermediate family of vectors ({ζ ∗e }, {κ ◦e }) ∈ Rm h ∗ × Rm h ◦ such that ζ ∗e ∈ ∂G∗e (δ ∗e ) ⇐⇒ δ ∗e ∈ ∂Ge (ζ ∗e ), {κ ◦e }

∈

∂I Q ◦ ({θ ◦e })

⇐⇒

{θ ◦e }

e = 1, 2, . . . , E,

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∈ ∂I Q ∗◦ ({κ ◦e }).

Then the Douglas–Rachford scheme has the next proximal realization. Algorithm ALG2M∗ : QE QE ∗m m h∗ ∗ m Given {α 0e } ∈ e=1 D(Fe ) ⊂ Rm h , {δ ∗0 and {θ ◦0 e } ∈ e } ∈ Q◦ , known {α e }, {δ e } and e=1 D(G e ) ⊂ R ◦m ◦m+1 {θ e }, m ≥ 0, calculate {κ e } satisfying the primal synchronizing condition   1 ◦m m θ , + π α } = Proj {κ ◦m+1 ∗ e e Q◦ e r e and, in parallel, ζ ∗m+1 , α m+1 , δ e∗m+1 , θ ◦m+1 , e = 1, 2, . . . , E, such that e e e   1 ∗ ∗m m m −∗ M δ + Λ α − g , ζ ∗m+1 = Prox e e e e Me ,(1/r )G e r e e −∗ ∗m+1 ◦m+1 −ΛTe (δ ∗m ) − π Te (θ ◦m ) e − r Me ζ e e − rκe

∈ (∂Fe + r ΛTe Me−∗ Λe + r π Te π e )(α m+1 ) − fe∗m+1 − r ΛTe Me−∗ gm+1 , e e −∗ m+1 δ ∗m+1 = δ ∗m − gm+1 − ζ ∗m+1 ), e e + r Me (Λe α e e e m+1 θ e◦m+1 = θ ◦m − κ ◦m+1 ). e + r (π e α e e

In the same manner, for the Peaceman–Rachford scheme, in terms of the intermediate vector relations ∗m+1/2

∗ ζ ∗m+1 = Λe α m e e ∈ ∂Ge (δ e

{κ e◦m+1 }

=

{πe α m e }

∈

∗m+1/2

) ⇐⇒ δ e

◦m+1/2 ∂I Q ◦ ({θ e })

⇐⇒

∈ ∂Ge (ζ ∗m+1 ), e

◦m+1/2 {θ e }

the following parallel proximal-point algorithm is concluded.

e = 1, 2, . . . , E,

∈ ∂I Q ∗◦ ({κ ◦m+1 }), e

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

∗: Algorithm ALG3MQ QE E ∗m m h∗ 0 ∗ m Given {α e } ∈ e=1 D(Fe ) ⊂ Rm h , {δ ∗0 and {θ ◦0 e } ∈ e } ∈ Q◦ , known {α e }, {δ e } and e=1 D(G e ) ⊂ R ◦m ◦m+1 {θ e }, m ≥ 0, calculate {κ e } satisfying the primal synchronizing condition   2 ◦m m ◦m+1 θ + π e αe ; {κ e } = Proj Q ∗◦ r e

∗m+1/2

and, in parallel, ζ ∗m+1 , δe e

◦m+1/2

, θe , α m+1 , δ ∗m+1 , θ ◦m+1 , e = 1, 2, . . . , E, such that e e e   2 ∗ ∗m m ζ ∗m+1 = Prox Me−∗ ,(1/r )G e M δ + Λe α m , e e − ge r e e r −∗ ∗m+1/2 m m ∗m+1 δe = δ ∗m ), e + Me (Λe α e − ge − ζ e 2 r ◦m+1/2 m ◦m+1 ), θe = θ ◦m e + (π e α e − κ e 2     r r ◦m+1/2 ∗m+1/2 − π Te θ e − κ ◦m+1 −ΛTe δ e − Me−∗ ζ ∗m+1 e e 2 2  r T −∗ r T  m+1 r ∗m+1 ∈ ∂Fe + Λe Me Λe + π e π e (α e ) − fe − ΛTe Me−∗ Λe gm+1 , e 2 2 2 r ∗m+1/2 δ ∗m+1 = δe + Me−∗ (Λe α m+1 − gm+1 − ζ ∗m+1 ), e e e e 2 r ◦m+1/2 θ e◦m+1 = θ e + (π e α m+1 − κ ◦m+1 ). e e 2 The convergence of these two algorithms to dual stationary states can be established as before via resolvent characterizations. To this end, we introduce the auxiliary dual supervector s∗ ∈ Rm h ∗ × Rm h ◦ defined by

M∗ s∗ ∈ (M∗ + r A∗f ∗ ,Ξ )({δ ∗e }, {θ ◦e }) ⇐⇒ ({δ ∗e }, {θ ◦e }) = JM∗ ,A∗ ∗ (M∗ s∗ ), (32) f ,Ξ   ∗ [0e ] ∗ e] where M∗ = [M [0e ] [Ie ] and A f ∗ ,Ξ is the dual operator (25). Then the discrete macro-hybrid dual problem of the Douglas–Rachford scheme at the m + 1 ≥ 1 time step is expressed in a compact form by  E Y  ∗−1 Given {δ ∗0 } ∈  D(G ∗e ) ⊂ Rm h ∗ , {θ ◦0 ∈ Rm h ∗ × Rm h ◦ ,  e } ∈ Q◦ and s e   ^ e=1 m+1 ∗m+1 (D^ }, {θ ◦m+1 }), s∗m ∈ Rm h ∗ × Rm h ◦ : h,h ∗ ,h ◦ ) find ({δ e e  ∗ ∗m ∗  M s ∈ (M + r A∗f ∗ ,Ξ )({δ e∗m+1 }, {θ ◦m+1 }),  e   ∗ ∗m ◦m ∗ ∗m−1 ∗ ◦m ∗m 2M ({δ e }, {θ e }) − M s ∈ (M + r ∂G ∗ )(({δ ∗m − s∗m−1 ). e }, {θ e }) + s Similarly, the discrete macro-hybrid dual problem of the Peaceman–Rachford scheme is expressed by  E Y  ∗−1 Given {δ ∗0 } ∈ D(G ∗e ) ⊂ Rm h ∗ , {θ ◦0 ∈ Rm h ∗ × Rm h ◦ ,  e e } ∈ Q◦ and s    e=1  ^  ◦m+1 ∗m m h∗ ^ find ({δ ∗m+1 }, × Rm h ◦ : ^ m+1 e  {θ e r }), s ∈ R (Dh,h ∗ ,h ◦ )  }, {θ ◦m+1 M∗ s∗m ∈ M∗ + A∗f ∗ ,Ξ ({δ ∗m+1 }),  e e   2       2M∗ ({δ ∗m }, {θ ◦m }) − M∗ s∗m−1 ∈ M∗ + r ∂G ∗ ({δ ∗m }, {θ ◦m }) + 1 s∗m − 1 s∗m−1 . e e e e 2 2 2 Here the dual subdifferential ∂G ∗ = ({∂G∗e }, ∂I Q ◦ ). Proposition 5.4. Let the dual operators A∗Λ + [∂G∗e ] and A∗π + ∂I Q ◦ be maximal monotone. Then, for time^ m+1 independent data f ∗m = f ∗ , wm+1 = wg , algorithms ALG2M∗ and ALG3M∗ evolve, as m → ∞, to a (D^ g h,h ∗ ,h ◦ )- and ^ ^ ∗(t) m+1 a (D^ h,h ∗ ,h ◦ )-stationary state of the semi-discrete dual evolution problem (MH h,h ∗ ,h ◦ ), respectively.

G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

353

^ ^ ^ ^ m+1 m+1 r Proof. Expressing dual problems (Dh,h ∗ ,h ◦ ) and (D^ h,h ∗ ,h ◦ ) in terms of the resolvent operators JM∗ ,A∗f ∗ ,Ξ and ∗m+1 r }, {θ ◦m+1 }) their resolvent versions are obtained: JM ∗ ,∂ G ∗ , by elimination of ({δ e e r r ∗ r s∗m+1 = JM ◦ M∗ − I)(s∗m ) + (I − JM )(s∗m ), ∗ ,∂ G ∗ ◦ M ◦ (2JM∗ ,A∗ ∗ ,A ∗ f ∗ ,Ξ f ∗ ,Ξ

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r ∗ r s∗m+1 = (2JM ∗ ,∂ G ∗ ◦ M − I) ◦ (2JM∗ ,A∗ ∗

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f ,Ξ

◦ M∗ − I)(s∗m ).

Then algorithms ALG2M∗ and ALG3M∗ are convergent due to the contraction property of these resolvent compositions (see [23] and [18]).  6. Stationary composition duality principles and exact penalization For the computational implementation of the above proximation time marching schemes, well-posedness of the stationary macro-hybrid mixed discrete inclusions encountered at each time step has to be guaranteed. Moreover, as demonstrated, an intrinsic important application of such proximal-point algorithms is provided by their convergence, for time-independent data, as iterative “mass” preconditioned augmented procedures in calculating stationary numerical states. Hence, regarding the common stationary discrete problem of the semi-discrete primal and dual (t) ∗(t) evolution problems (MHh,h ∗ ,h ◦ ) and (MHh,h ∗ ,h ◦ ),  m ∗ Find (α e , δ ∗e ) ∈ Rm h e × R h e , for e = 1, 2, . . . , E :    T ∗ T ◦  −Λe δ e − π e θ e ∈ ∂Fe (α e ) − fe∗ , (MHh,h ∗ ,h ◦ ) Λe α e ∈ ∂G∗e (δ ∗e ) + ge ;   and {θ ◦e } ∈ Rm h ◦ satisfying the dual synchronizing condition    {π e α e } ∈ ∂I Q ◦ ({θ ◦e }), we next present corresponding dual and primal stationary composition duality principles, following [7], as well as preconditioned penalty–duality algorithmic interpretations according to [2]. 6.1. Dual composition duality principle and exact penalization For compositional dualization, complementary to the discrete macro-hybrid compatibility condition (C[πΓ ◦ ] ) of h e ,h e Lemma 4.1, we consider the dual stationary compatibility condition (C[ΛT

] h e ,h ∗ e

)ΛTh e ,h ∗e ∈ L(Yh ∗e , Vh∗e )

is surjective, e = 1, 2, . . . , E,

in terms of which the macro-hybridized primal equation of discrete problem (MHh,h ∗ ,h ◦ ) is dualized and its dual stationary composition duality principle follows (see the proofs of Lemma 3.3 and Theorem 3.4). Lemma 6.1. Let compatibility condition (C[ΛT ∗ ] ) be satisfied. Then the discrete dual stationary compositional h e ,h e dualization −{Λe α e } ∈ ∂([Fe∗ ]|Vh∗ ◦ [−ΛTe ])(δ ∗e + r∗fe∗ ) ⇐⇒ −{ΛTe δ ∗e } ∈ ∂([Fe ] + I Q ∗◦ ◦ [π e ])({α e }) − {fe∗ }

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holds true, where [Fe∗ ]|Vh∗ = ([Fe ] + I Q ∗◦ ◦ [π e ])∗ . Theorem 6.2. Let discrete compatibility conditions (C[ΛT

] h e ,h ∗ e

) and (C[πΓ

] h e ,h ◦ e

) be fulfilled. Then the discrete primal

stationary macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ) has a solution if, and only if, the discrete macro-hybridized dual problem  Find {δ ∗e } ∈ Rm h ∗ : (D M Hh,h ∗ ,h ◦ ) {0e } ∈ {∂G∗e (δ ∗e )} + ∂([Fe∗ ]|Vh∗ ◦ [−ΛTe ])({δ ∗e + r ∗fe∗ }) + {ge }

354

G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

has a solution, where {r∗f ∗ } ∈ Rm h ∗ is the coordinate vector of a fixed [−ΛTh e ,h ∗ ]-preimage of function { f h∗e } : e

e

{−ΛTh e ,h ∗ r ∗f ∗ } = { f h∗e }. That is, if ({α e }, {δ ∗e }, {θ ◦e }) ∈ Rm h × Rm h ∗ × Rm h ◦ is a solution of problem (MHh,h ∗ ,h ◦ ) e

he

then dual coordinate vector {δ ∗e } is a solution of problem (D M Hh,h ∗ ,h ◦ ) and, conversely, if {δ ∗e } ∈ Rm h ∗ is a solution of problem (D M Hh,h ∗ ,h ◦ ) then there is a primal coordinate vector {α e } ∈ ∂[Fe∗ ]|Vh∗ ({−ΛTe δ ∗e + fe∗ }) ⊂ Rm h and a dual coordinate vector {θ ◦e } ∈ ∂I Q ∗◦ ({π e α e }) ⊂ Rm h ◦ such that ({α e }, {δ ∗e }, {θ ◦e }) is a solution of problem (MHh,h ∗ ,h ◦ ). Proof. First, applying the macro-hybrid compositional dualization process as in the proof of Theorem 3.4, the macrohybridized mixed equations of problem (MHh,h ∗ ,h ◦ ) turn out to be −{ΛTe δ ∗e } ∈ ∂([Fe ] + I Q ∗◦ ◦ [π e ])({α e }) − {fe∗ }, {Λe α e } ∈ {∂G∗e (δ ∗e )} + {ge }.

(36)

Then the necessity of the theorem is implied by Lemma 6.1. Conversely, for {δ ∗e } ∈ Rm h ∗ a solution of problem (D M Hh,h ∗ ,h ◦ ), there is a vector {ζ e } ∈ {∂G∗e (δ ∗e )} such that −{ζ e } − {ge } ∈ ∂([Fe∗ ]|Vh∗ ◦ [−ΛTe ])({δ ∗e + r ∗f ∗ }). e From the variational inequality of this dual inclusion with variations {µ∗e } = ±{µ∗e0 } + {δ ∗e + r ∗f ∗ } for any e

{µ∗e0 } ∈ N ([−ΛTe ]), it follows that −{ζ e } − {ge } belongs to the kernel-polar subspace N ([−ΛTe ])◦ . Consequently, under condition (C[ΛT ∗ ] ), from the Closed Range Theorem N ([−ΛTe ])◦ = R([−Λe ]) and there is a primal h e ,h e

coordinate vector {α e } ∈ Rm h such that {−Λe α e } = −{ζ e } − {ge }. Therefore, {Λα e } − {ge } ∈ {∂G∗e (δ ∗e )} and {−Λα e } ∈ ∂([Fe∗ ]|Vh∗ ◦ [−ΛTe ])({δ ∗e + r ∗f ∗ }) ⇔ {α e } ∈ ∂[Fe∗ ]|Vh∗ ({−ΛTe δ ∗e + fe∗ }), where the equivalence is due to e Lemma 6.1. That is, ({α e }, {δ ∗e }) ∈ Rm h × Rm h ∗ solves problem (36). Further, as in the proof of Theorem 3.4, there is a macro-hybrid coordinate vector {θ ◦e } ∈ Rm h ◦ such that {π Te θ ◦e } ∈ ∂(I Q ∗◦ ◦ [π e ])({α e }) ⇔ {θ ◦e } ∈ ∂I Q ∗◦ ({π e α e }), with equivalence due to Lemma 4.1, and ({α e }, {δ ∗e }, {θ ◦e }) is a solution of problem (MHh,h ∗ ,h ◦ ).  The preconditioned penalty–duality algorithmic interpretation of primal algorithm ALG0M is given as follows. Proposition 6.3. Let discrete compatibility conditions (C[G ∗∗ ,Λh he

∗] e ,h e

) and (C[πΓ

h e ,h ◦ e

])

be fulfilled. Then algorithm

ALG0M , for time-independent data, corresponds to a penalty–duality iterative approximation process of discrete primal stationary macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ), defined for its primally M-preconditioned augmented version, exactly penalized with parameter r > 0,  ∗ ◦ mh × Rm h ∗ × Rm h ◦ :  e }, {δ e }, {θ e }) ∈ R Find T({α  −{Λe δ ∗e } − {π Te θ ◦e } ∈ {∂Fe (α e )} − {fe∗ }, (MHrM h,h ∗ ,h ◦ )  {M α } + r {ΛTe δ ∗e } + r {π Te θ ◦e }   e e ∈ {Me α e } + r {∂(G e ◦ Λe )(α e − wge )} + ∂(I Q ∗◦ ◦ [πe ])({α e }), where {wge } ∈ Rm h is the coordinate vector of a fixed [Λh e ,h ∗e ]-preimage of function {gh ∗e }: {Λh e ,h ∗e wgh ∗ } = {gh ∗e }. e

Proof. Expressing the vector {ξ e } of algorithm ALG0M by {ξ e } = {ΛTe δ ∗e } + {π Te θ ◦e } and taking into account the resolvent-proximation relation (21), the interpretation T ◦m m ∗ −{ΛTe δ ∗m e } − {π e θ e } ∈ {∂Fe (α e )} − {fe }, T ∗m T ◦m {Me α m e } + r {Λe δ e } + r {π e θ e }

∈

{Me α m+1 } + r {∂(G e e

◦ Λe )(α m+1 e

(37) − wge )} + ∂(I Q ∗◦ ◦ [πe ])({α m+1 }), e

is readily concluded, which, in fact, defines a penalty–duality iterative procedure for problem (MHh,h ∗ ,h ◦ ), with primally M-preconditioned augmented dual equations on the basis of Lemmas 4.2 and 4.1, and exact penalization parameter r > 0.  Remark 6.4. In general, the surjectivity of a linear continuous operator defined on a reflexive Banach space is equivalent to the lower boundedness of its transpose operator and this, in turn, is also equivalent to its transpose injectivity in conjunction with the transpose closed range property [27]. Hence, primal and dual compatibility

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conditions (C[πΓ

] h e ,h ◦e

) and (C[ΛT

h e ,h ∗ e

])

are indeed respectively equivalent to the transpose compatibility conditions

 T ∗ [πΓ ◦ ] ∈ L(B{h ◦e } , V{h ) is bounded below; i.e., for e = 1, 2, . . . , E,   e} h e ,h e   ∃ c T > 0: ∀µh ◦ ∈ Bh ◦ , πe e e T (C[π ) Γh ,h ◦ ]  hπΓT ◦ µh ◦e , vh e iVh e e e  h e ,h e T   sup ≥ cπeT kµh ◦e k Bh ◦ , kπΓh e ,h ◦e µh ◦e kVh∗e = e kv k vh e ∈Vh e \{0h e }

h e Vh e

and  [Λh e ,h ∗e ] ∈ L(V{h e } , Y∗{h ∗e } ) is bounded below; i.e., for e = 1, 2, . . . , E,    ∃ cΛ > 0: ∀vh ∈ Vh , e e e (C[TΛT ] ) hΛh e ,h ∗e vh e , qh ∗e iYh ∗ ∗  e h e ,h e  ∗ v h kY ∗ = ≥ cΛe kvh e kVh e , kΛ sup  h ,h e e e  h∗ kqh ∗ kY ∗ e q ∗ ∈Y ∗ \{0 ∗ } he

he

e

he

he

which correspond to the primal macro-hybrid mixed Ladyˇsenskaja–Babuˇska–Brezzi inf-sup condition [21,11,14]. 6.2. Primal composition duality principle and exact penalizations Considering now discrete problem (MHh,h ∗ ,h ◦ ) as a dual stationary macro-hybrid mixed problem, we introduce the primal complementary compatibility condition (C[Λh e ,h ∗ ] ) e

Λh e ,h ∗e ∈ L(Vh e , Yh∗∗e ) is surjective, e = 1, 2, . . . , E,

for the compositional dualization of its dual equation and primal composition duality principle. Lemma 6.1∗ . Under compatibility condition (C[Λh e ,h ∗ ] ), the discrete primal stationary compositional dualization e

{ΛTe δ ∗e } ∈ {∂(G e ◦ Λe )(α e − wge )} ⇐⇒ {Λe α e } ∈ {∂G∗e (δ ∗e )} + {ge }

(35*)

is valid. Theorem 6.2∗ . Let discrete compatibility conditions (C[Λh e ,h ∗ ] ) and (C[πΓ ◦ ] ) be satisfied. Then the discrete dual h e ,h e e stationary macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ) has a solution if, and only if, the discrete macro-hybridized primal problem  Find {α e } ∈ Rm h : (P M Hh,h ∗ ,h ◦ ) {0e } ∈ {∂Fe (α e )} + {∂(G e ◦ Λe )(α e − wge )} + ∂(I Q ∗◦ ◦ [π e ])({α e }) − {fe∗ } has a solution, where {wge } ∈ Rm h is the coordinate vector of a fixed [Λh e ,h ∗e ]-preimage of function {gh ∗e } : {Λh e ,h ∗e wgh ∗ } = {gh ∗e }. That is, if ({α e }, {δ ∗e }, {θ ◦e }) ∈ Rm h × Rm h ∗ × Rm h ◦ is a solution of problem (MHh,h ∗ ,h ◦ ) e then primal coordinate vector {α e } is a solution of problem (P M Hh,h ∗ ,h ◦ ) and, conversely, if {α e } ∈ Rm h is a solution of problem (P M Hh,h ∗ ,h ◦ ) then there are dual coordinate vectors {δ ∗e } ∈ {∂Ge (Λe α e − ge )} and {θ ◦e } ∈ ∂I Q ∗◦ ({π e α e }) ⊂ Rm h ◦ such that ({α e }, {δ ∗e }, {θ ◦e }) is a solution of problem (MHh,h ∗ ,h ◦ ). Proof. The necessity follows, from the macro-hybridized mixed Eq. (36) of problem (MHh,h ∗ ,h ◦ ), by applying Lemma 6.1* to the dual equation. The sufficiency is demonstrated through the same reasoning as for Theorem 6.2.  In this dual stationary case, algorithms ALG4M∗ and ALG5M∗ have the next preconditioned penalty–duality interpretation. Proposition 6.3∗ . Let discrete compatibility conditions (C[F ∗

T ] h e |V ∗ ,−Λh e ,h ∗ e he

) and (C[πΓ

] h e ,h ◦ e

) be satisfied. Then

algorithms ALG4M∗ and ALG5M∗ , for time-independent data, correspond to penalty–duality iterative approximation

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

processes of discrete dual stationary macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ), defined for its dually M∗ preconditioned augmented version, exactly penalized with parameter r > 0,  Find ({α e }, {δ ∗e }, {θ ◦e }) ∈ Rm h × Rm h ∗ × Rm h ◦ :    −{ΛTe δ ∗e } − {π Te θ ◦e } ∈ {∂Fe (α e )} − {fe∗ }, (MHrM ∗ h,h ∗ ,h ◦ ) {M∗ δ ∗ } + r {Λe α e } ∈ {Me∗ δ ∗e } + r {∂G∗e (δ ∗e )} + r {ge },    ◦e e {θ e } + r {π e α e } ∈ {θ ◦e } + ∂I Q ◦ ({θ ◦e }). Proof. Utilizing the resolvent-proximation relations (24), algorithms ALG4M∗ and ALG5M∗ take respectively the forms T ◦m m ∗ −ΛTe δ ∗m e − π e θ e ∈ ∂Fe (α e ) − fe , m ∗ ∗m+1 Me∗ δ ∗m + r ∂G∗e (δ ∗m+1 ) + r ge , e + r Λe α e ∈ Me δ e e m {θ ◦m e } + r {π e α e }

∈

(38)

}), {θ ◦m+1 } + ∂I Q ◦ ({θ ◦m+1 e e

and − π Te θ ◦m+1 −ΛTe δ ∗m+1 ∈ ∂Fe (α m+1 ) − fe∗ , e e e m+1 Me∗ δ ∗m ∈ Me∗ δ ∗m+1 + r ∂G∗e (δ ∗m+1 ) + r ge , e + r Λe α e e e

(39)

m+1 }), {θ ◦m } ∈ {θ ◦m+1 } + ∂I Q ◦ ({θ ◦m+1 e e } + r {π e α e e

recovering, in fact, the original expressions of the dual semi-implicit and implicit Euler schemes, with a direct interpretation of penalty–duality iterative approximations of problem (MHh,h ∗ ,h ◦ ), dually M∗ -preconditioned and with exact penalization parameter r > 0.  Furthermore, algorithms ALG1M∗ , ALG2M∗ and ALG3M∗ can also be regarded as preconditioned penalty–duality approximations of problem (MHh,h ∗ ,h ◦ ) in a dual sense. Proposition 6.3∗∗ . Algorithms ALG1M∗ , ALG2M∗ and ALG3M∗ , for time-independent data, correspond to penalty–duality iterative approximation processes of discrete dual stationary macro-hybrid mixed problem (MHh,h ∗ ,h ◦ ), defined respectively for its macro-hybrid dually M∗ -preconditioned augmented versions, exactly penalized with parameter r > 0,  ∗ ◦ m ∗ m ◦ Find ({δ e }, {θ e }) ∈ R h × R h : ∗ ∗ ∗ ∗ {M δ } ∈ (Me + r A f ∗ ,Λ + r [∂G∗e ])({δ ∗e }) + r {ge }, (DrM^ ∗ h,h ∗ ,h ◦ )  ◦e e {θ e } ∈ (Me∗ + r A∗f ∗ ,π + ∂I Q ◦ )({θ ◦e }),  ∗ ◦ m ∗ m ◦ Find ({δ e }, {θ e }) ∈ R h × R h : ^ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ r ^ (D M ∗ h,h ∗ ,h ◦ ) 2{Me δ e } ∈ (Me + r A f ∗ ,Λ )({δ e }) + (Ie + r ∂Ge )({δ e }),  ◦ ∗ ∗ ◦ ∗ ◦ 2{θ e } ∈ (Me + r A f ∗ ,π )({θ e }) + (Ie + ∂I Q ◦ )({θ e }), and  ∗ ◦ m h∗ ×Rm h ◦ :    e }) ∈ R  Find ({δ e }, {θ  r r 2{M∗ δ ∗e } ∈ Me∗ + A∗f ∗ ,Λ ({δ ∗e }) + Ie∗ + ∂G∗e ({δ ∗e }), ) ◦ ,h 2   r 2   ◦  2{θ e } ∈ Me∗ + A∗f ∗ ,π ({θ ◦e }) + (Ie∗ + ∂I Q ◦ )({θ ◦e }). 2

^ ^ (Dr^ ∗ ∗ M h,h

Proof. The penalty–duality interpretations are direct consequences respectively of discrete macro-hybrid dual ^ ^ ^ ^ ^ m+1 m+1 m+1 problems (D^ ), ( D ) and ( D  ∗ ◦ ∗ ◦ h,h ,h h,h ,h h,h ∗ ,h ◦ ).

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

Remark 6.4∗ . As in Remark 6.4, we observe that the primal compatibility condition (C[Λh e ,h ∗ ] ) is equivalent to the e transpose condition  T ∗ [Λh e ,h ∗e ] ∈ L(Y{h ∗e } , V{h ) is bounded below; i.e., for e = 1, 2, . . . , E,  e}   ∃ cΛT > 0: ∀qh ∗ ∈ Yh ∗ , e e e (C[TΛ ∗ ] ) hΛTh e ,h ∗ qh ∗e , vh e iVh e h e ,h e  e  kΛTh e ,h ∗ qh ∗e kV ∗ = sup ≥ cΛTe kqh ∗e kYh ∗ ,  he e e kvh e kVh e vh e ∈Vh e \{0h ∗ } e

which corresponds to the dual macro-hybrid mixed Ladyˇsenskaja–Babuˇska–Brezzi inf-sup condition [21,11,14]. 6.3. Optimization interpretations Another relevant aspect of these algorithms is their optimization interpretation (see [4,6]). Indeed, due to the potentiality of the stationary problem (MHh,h ∗ ,h ◦ ), algorithms ALG4M∗ and ALG5M∗ can be identified as M∗ preconditioned Uzawa approximations of the minimax problem associated to the two-field augmented Lagrangian, for {β e } ∈ Rm h , ({µ∗e }, {η◦e }) ∈ Rm h ∗ × Rm h ◦ , L rM ∗ ({β e }, ({µ∗e }, {η◦e })) =

E X

Fe (β e ) +

e=1

+ (I Q ∗◦ )1/r



E X

 G Me−∗ ,1/r

e=1

1 ∗ ∗ M µ + Λe β e − ge r e e



E E E X X 1 ◦ 1 1 ◦ 2 X {ηe } + {π e β e } − kµ∗e k2Me∗ − kηe k − fe∗ ·β e , r 2r 2r e=1 e=1 e=1



(40)

where G Me−∗ ,1/r (µ∗e ) =

inf

nr

νe∗ ∈D (G e )

( (I

Q ∗◦

)1/r ({η◦e })

=

inf

{νe◦ }∈Q ∗◦

2

o kν ∗e − µ∗e k2M −∗ + G e (ν ∗e ) , e

E X r e=1

2

e = 1, 2, . . . , E, (41)

) kν ◦e

− η◦e k2

,

are the Fr´echet differentiable convex potentials of the Moreau–Yosida approximations of ∂Ge and ∂I Q ∗◦ , ∂G Me−∗ ,1/r ≡ 1/r

1/r

r (M−∗ e − J∂G e ) and ∂I Q ∗◦,1/r ≡ r (I − J∂ I ∗ ), respectively [13]. Here, k · k Me∗ and k · k Me−∗ denote the norms induced Q◦

by the local Me∗ - and Me−∗ -mass inner products, respectively. In fact, this functional is the potential of the M∗ preconditioned augmented problem (MHrM ∗ h,h ∗ ,h ◦ ) and corresponds to the augmented functional of the potential of problem (MHh,h ∗ ,h ◦ ), L({β e }, ({µ∗e }, {η◦e })) =

E X

Fe (β e ) −

e=1

+

E X

E X

G ∗e (µ∗e ) − I Q ◦ ({η◦e })

e=1

µ∗e ·(Λe β e − ge ) −

e=1

E X e=1

η◦e ·π e β e −

E X

fe∗ ·β e .

(42)

e=1

Also algorithm ALG1M∗ , and its variants ALG2M∗ and ALG3M∗ , correspond to M∗ -preconditioned Uzawa approximations of the minimax problem associated to the augmented Lagrangian, for {β e } ∈ Rm h , ({ν ∗e }, {ξ ◦e }) ∈ Rm h ∗ × Rm h ◦ , ({µ∗e }, {η◦e }) ∈ Rm h ∗ × Rm h ◦ , LrM ∗ ({β e }, ({ν ∗e }, {ξ ◦e }), ({µ∗e }, {η◦e })) = L({β e }, ({ν ∗e }, {ξ ◦e }), ({µ∗e }, {η◦e })) +

E X r e=1

2

kΛe β e − ν ∗e k2Me +

E X r e=1

2

kπ e β e − ξ ◦e k2 ,

(43)

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G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

of the three-field Lagrangian, L({β e }, ({ν ∗e }, {ξ ◦e }), ({µ∗e }, {η◦e })) =

E X

Fe (β e ) +

e=1

+

E X

E X

G e (ν ∗e − ge ) + I Q ∗◦ ({ξ ◦e })

e=1

µ∗e ·(Λe β e − ν ∗e ) +

e=1

E X e=1

η◦e ·(π e β e − ξ ◦e ) −

E X

fe∗ ·β e .

(44)

e=1

This Lagrangian (44) can be identified as the potential of the three-field equivalent formulation of discrete macrohybrid mixed problem (MHh,h ∗ ,h ◦ ) given by (see (28)) Find (α , ζ ∗ ) ∈ Rm h e × Rm h ∗e , for e = 1, 2, . . . , E : e e   T ∗  −Λ δ − π Te θ ◦e ∈ ∂Fe (α e ) − fe∗ ,  e e   ∗  δ ∈ ∂Ge (ζ ∗e − ge );    e◦ {κ e } ∈ Rm h ◦ satisfying the primal synchronizing condition (MH3h,h ∗ ,h ◦ ) {θ ◦ } ∈ ∂I ∗ ({κ ◦ });    e ∗ Q◦ ◦ e m h ∗e ◦  and (δ e , θ e ) ∈ R × Rm h e , for e = 1, 2, . . . , E :    ∗ ∗   Λe α e − ζ ◦e ∈ ∂0(δ◦e ), π e α e − κ e ∈ ∂0(θ e ). Moreover, augmented Lagrangian (43) is then the potential of the M∗ -preconditioned augmented version of three-field problem (MH3h,h ∗ ,h ◦ ), Find (α , ζ ∗ ) ∈ Rm h e × Rm h ∗e , for e = 1, 2, . . . , E : e e    −ΛTe (δ ∗e − r Me−∗ ζ ∗e ) − π Te (θ ◦e − r κ ◦e ) ∈ (∂Fe + r ΛTe Me−∗ Λe + r π Te π e )(α e ) − fe∗ ,     δ ∗ + r Me−∗ Λe α e ∈ ∂Ge (ζ ∗e − ge ) + r Me−∗ ζ ∗e ;    e◦ {κ e } ∈ Rm h ◦ satisfying the primal synchronizing condition (MH3r ) ∗ ∗ ◦ M h,h ,h {θ ◦e } + r π e α e ∈ ∂I Q ∗◦ ({κ ◦e }) + r κ ◦e ;   m ∗ ◦  and (δ ∗e , θ ◦e ) ∈ R h e × Rm h e , for e = 1, 2, . . . , E :    δ ∗e = δ ∗e + r Me−∗ (Λe α e − ζ ∗e ),  θ ◦e = θ ◦e + r (π e α e − κ ◦e ). 7. An application: Distributed control mixed diffusion problem Let Ω ⊂ Rn , n ∈ {2, 3}, be a spatial bounded domain with a Lipschitz continuous boundary ∂Ω , and let (0, T ), T > 0, be the time interval. As a representative evolution mixed model from mechanics, we present here the distributed control diffusion problem considered in [10],  du − div w = − p ∗ ,  dt in Ω × (0, T ), −1 K w = grad u,  (45)  ∗ ∗ u ∈ ∂φ ( p ), u(0) = u 0 ,

in Ω ,

where u denotes the mass concentration or temperature field of a diffusion process, with flux vector w, and controlled by a distributed source p ∗ . Also, K is a symmetric positive definite diffusion tensor, ∂φ ∗ is the monotone subdifferential of a dual control mechanism [16], and u 0 is a given initial condition in Ω . Further, as boundary conditions, we consider Dirichlet and Neumann conditions, db and b h, prescribed respectively on disjoint and complementary parts of the boundary: b u ∈ ∂ψ ∗ (−w·n) = {d}, on ∂Ω D × (0, T ), −w·n ∈ ∂ψ(u) = {b h}, on ∂Ω N × (0, T ).

(46)

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Here, ∂ψ stands for the monotone boundary subdifferential, with dual ∂ψ ∗ , that relates variationally Dirichlet and Neumann traces according to −w·n ∈ ∂ψ(u) ⇔ u ∈ ∂ψ ∗ (−w·n) [25,1]. 7.1. The primal evolution mixed variational inclusion problem For convenience, we shall assume that the Dirichlet boundary data are homogeneous, db = 0

on ∂Ω D × (0, T ),

(47)

in order to satisfy the dual data range condition g ∈ L p (0, T ; R(Λ)). Then, the primal mixed variational formulation of the evolution control problem follows as usual, by integration and application of Green’s formula to its divergence equation, and by considering the variational inequality of the distributed control dual subdifferential equation (see Remark 4.1 of [7]).  1 1 Find u ∈ L 2 (0, T ; H0,D (Ω )) with du/dt ∈ L 2 (0, T ; (H0,D (Ω ))∗ ),    2 2 ∗ 2 1 ∗  w ∈ L (0, T ; L (Ω )) and p ∈ L (0, T ; (H0,D (Ω )) ) :   Z T Z TZ      w · grad v dΩ dt − h p ∗ , vi H 1 (Ω ) dt −   0,D  0 Ω 0   Z Z  T T  1 = hdu/dt, vi H 1 (Ω ) dt − hb h, γ vi H 1/2 (∂ Ω N ) dt, ∀v ∈ L 2 (0, T ; H0,D (Ω )), (Mcd ) 0,D  0 0  Z Z Z Z  T T    grad u · v dΩ dt = K −1 w · v dΩ dt, ∀v ∈ L 2 (0, T ; L2 (Ω )),    0 Ω 0 Ω  Z T Z T Z T    ∗ ∗ ∗ ∗ 1  Φ (q ) dt ≥ Φ ( p ) dt + hq ∗ − p ∗ , ui H 1 (Ω ) dt, ∀q ∗ ∈ L 2 (0, T ; (H0,D (Ω ))∗ ),   0,D  0 0  0 u(0) = u 0 . 1 (Ω ) = {v ∈ H 1 (Ω ) : γ v = 0 a.e. on ∂Ω } is a closed subspace of the Sobolev Hilbert Here, the primal space H0,D D 1 2 space H (Ω ) = {v ∈ L (Ω ): grad v ∈ L2 (Ω )}, with surjective Dirichlet trace operator γ ∈ L(H 1 (Ω ), H 1/2 (∂Ω )), 1 (Ω ))∗ → R ∪ {+∞} is the control dual superpotential, conjugate of the primal convex superpotential Φ ∗ : (H0,D R 1 (Ω ) → R ∪ {+∞} assumed to be proper and lower semicontinuous, and h·, ·i Φ(·) = Ω φ(·) dΩ : H0,D H 1 (Ω ) and 0,D

1 (Ω ))∗ × H 1 (Ω ) and H −1/2 (∂Ω ) × H 1/2 (∂Ω ), respectively. h·, ·i H 1/2 (∂ Ω N ) denote the duality pairings of (H0,D N N 0,D −1 ∞ Moreover, the data are such that K ∈ L (Ω ) is a uniformly positive definite symmetric tensor, db = 0 ∈ L 2 (0, T ; H 1/2 (∂Ω D )), b h ∈ L 2 ((0, T ); H −1/2 (∂Ω N )) and u 0 ∈ L 2 (Ω ). Hence, by inspection, distributed control diffusion variational problem (Mcd ) can be identified with the abstract mixed problem (M) of the theory under the field and functional framework relations 1 u ∈ V ∼ u ∈ L 2 (0, T ; H0,D (Ω )),

(48)

1 p ∗ ∈ Y ∗ ∼ (w, p ∗ ) ∈ L 2 (0, T ; L2 (Ω )) × L 2 (0, T ; (H0,D (Ω ))∗ ),

as well as the operator and function identifications, for a.e. t ∈ (0, T ), ΛT p ∗ (t) ∼ gradT w(t) + p ∗ (t), Λu(t) ∼ (grad u(t), u(t)) ,

∂ F(u(t)) − f ∗ (t) ∼ ∂0(u(t)) + γ∂TΩ N b h(t),   ∂G ∗ ( p ∗ (t)) + g(t) ∼ K −1 w(t), ∂Φ ∗ ( p ∗ (t)) .

(49)

1 (Ω ) → {0∗ } ⊂ (H 1 (Ω ))∗ Thus, in this primal example, the primal operator ∂ F is the zero subdifferential ∂0 : H0,D 0,D 1 (Ω ))∗ → 2 H0,D (Ω ) , the coupling operator Λ with dual ∂ F ∗ the zero-indicator subdifferential ∂ I{0∗ } : (H0,D 1 (Ω ), L2 (Ω ) × H 1 (Ω )) with transpose ΛT the operator is the gradient-identity operator (grad, I ) ∈ L(H0,D 0,D 1 (Ω ))∗ , (H 1 (Ω ))∗ ), and the dual operator ∂G ∗ is the dual diffusion-control (gradT , I ∗ ) ∈ L(L2 (Ω ) × (H0,D 0,D 1

1 (Ω ))∗ → L2 (Ω ) × 2 H0,D (Ω ) with primal ∂G the diffusion-control operator operator (K −1 , ∂Φ ∗ ) : L2 (Ω ) × (H0,D 1

1 (Ω ) → L2 (Ω ) × 2(H0,D (Ω )) . (K, ∂Φ) : L2 (Ω ) × H0,D 1

∗

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Therefore, under the corresponding primal evolution compatibility condition (CG,Λ ) that turns out to be (CG,Λcd )

int D(Φ) 6= ∅,

the primal evolution composition duality principle of Theorem 2.2 states that the distributed control primal mixed diffusion problem  Find u ∈ W and (w, p ∗ ) ∈ Y ∗ :     −gradT w − p ∗ ∈ du + ∂0(u) + γ T b h, in V ∗ ,  dt  ∂ ΩN (Mcd )   (grad u, u) ∈ K −1 w , ∂Φ ∗ ( p ∗ ) , in Y,    u(0) = u 0 , is solvable if, and only if, its primal evolution problem  Find u ∈ W :   du (P cd ) 0 ∈ + ∂0(u) + gradT K ◦ grad(u) + ∂Φ(u) + γ∂TΩ N b h,  dt  u(0) = u 0 ,

in V ∗ ,

is solvable. The macro-hybrid variational formulation of this primal evolution mixed control diffusion problem is given on the basis of corresponding decompositions (3)–(7), with primal dual transmission admissibility subspaces QE Q E and −1/2 Qcd ⊂ Bcd ({Γe }) = e=1 H 1/2 (Γe ) and Q∗cd ⊂ B∗cd ({Γe }) = e=1 H (Γe ), one being the polar of the other, defined by  Qcd = {µe } ∈ Bcd ({Γe }): µe = µ f a.e. on Γe f , 1 ≤ e < f ≤ E , (50)  Q∗cd = {µ∗e } ∈ B∗cd ({Γe }): h{µ∗e }, {µe }i Bcd ({Γe }) = 0, ∀{µe } ∈ Qcd . (50*) That is, the primal transmission condition is the Dirichlet trace continuity across the interfaces, while the dual transmission condition is the Neumann trace continuity in the H −1/2 -weak sense. Moreover, the macro-hybrid compatibility condition (C[πΓe ] ) is satisfied due to the surjectivity of the local Dirichlet trace operators γe ∈ L(H 1 (Ωe ), H 1/2 (∂Ωe )), e = 1, 2, . . . , E. Also, the local version of primal evolution compatibility condition (CG,Λ ) takes the form (C[G e ,Λe ]cd )

int D(Φe ) 6= ∅,

e = 1, 2, . . . , E.

Hence, under this condition, Theorem 3.4 establishes that the corresponding macro-hybrid mixed problem (MHcd ) has a solution if, and only if, its primal problem (P cd ) has a solution. With respect to the semi-discrete internal variational discretizations of Section 4, we only stress that finite element implementations of mixed and macro-hybrid internal approximations have been deeply analyzed [15,26] and, nowadays, they still constitute an active and important topic in computational mechanics. Further, relative to proximation time marching schemes, the primal algorithm ALG0M offers a direct numerical time integration, in contrast to other schemes that require at each time step the solution of elliptic discrete variational inequalities, to which, in fact, the dual algorithms ALG1–5M∗ of Section 5.2 may be appropriate. On the other hand, regarding the corresponding stationary discrete macro-hybrid mixed problem (MHh,h ∗ ,h ◦ cd ), assume that the dual [ΛTh e ,h ∗ ]-surjectivity compatibility condition (C[ΛT ∗ ]cd ), equivalent to the transpose [Λh e ,h ∗e ]e

h e ,h e

lower boundedness condition (C[TΛT ] ), is fulfilled. Then, the dual composition duality principle of Theorem 6.2 cd h e ,h ∗ e applies, and problem (MHh,h ∗ ,h ◦ cd ) has a solution if, and only if, its dual stationary problem (D M Hh,h ∗ ,h ◦ cd ) has a solution. Also, from Proposition 6.3, under compatibility conditions (C[G ∗∗ ,Λh ,h ∗ ]cd ) and (C[πΓ ◦ ]cd ), it follows that he

e e

h e ,h e

algorithm ALG0 M defines a penalty–duality iterative procedure for problem (Mh,h ∗ ,h ◦ cd ) as an approximation of its M-preconditioned augmented formulation (MrM h,h ∗ ,h ◦ cd ), with exact penalization parameter r > 0.

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361

7.2. The dual evolution mixed variational inclusion problem Finally, we consider the dual mixed variational formulation of the distributed control diffusion problem (45) and (46). For convenience, we shall assume that both Dirichlet and Neumann boundary data are homogeneous, db = 0

on ∂Ω D × (0, T )

and b h=0

on ∂Ω N × (0, T ),

(51)

in order to guarantee the primal data range condition f ∗ ∈ L p (0, T ; R(−ΛT )). The variational mixed model follows as before, but now by incorporating the boundary conditions to the gradient equation via Green’s formula, and by considering the primal control subdifferential equation: p ∗ ∈ ∂φ(u), in Ω × (0, T ).  Find w ∈ L 2 (0, T ; H0,N (div; Ω )), and u ∈ L 2 (0, T ; L 2 (Ω ))     with du/dt ∈ L 2 (0, T ; L 2 (Ω )) :   Z TZ Z TZ      u div v dΩ dt = K −1 w · v dΩ dt, ∀v ∈ L 2 (0, T ; H0,N (div; Ω )), −    0 Ω 0 Ω Z T Z TZ (M∗cd ) Z T   Φ(v) dt ≥ Φ(u) dt − du/dt (v − u) dΩ dt    0 Z Z 0 0 Ω   T    + div w (v − u) dΩ dt, ∀v ∈ L 2 (0, T ; L 2 (Ω )),    0 Ω  u(0) = u 0 . In this dual formulation, the primal space H0,N (div; Ω ) = {v ∈ H(div; Ω ): δv = 0 in H −1/2 (∂Ω N )} is the closed subspace of the usual Hilbert space H(div; Ω ) = {v ∈ L2 (Ω ) : div v ∈ L 2 (Ω )}, with surjective Neumann trace operator δ ≡ (·)·n ∈ L(H(div; Ω ), H −1/2 (∂Ω )), and Φ is the control primal convex superpotential, assumed to be proper and lower semicontinuous, but now defined in L 2 (Ω ). Thus, dual evolution mixed problem (M∗cd ) is a particular case of the dual evolution mixed inclusion of the theory (M∗ ), with field and functional framework relations u ∈ V ∼ w ∈ L 2 (0, T ; H0,N (div; Ω )),

p ∗ ∈ Y ∗ ∼ u ∈ L 2 (0, T ; L 2 (Ω )),

(52)

and operator and function identifications, for a.e. t ∈ (0, T ), ΛT p ∗ (t) ∼ divT u(t), Λu(t) ∼ div w(t),

∂ F(u(t)) − f ∗ (t) ∼ K −1 w(t), ∂G ∗ ( p ∗ (t)) + g(t) ∼ ∂Φ(u(t)).

(53)

Hence, for our dual mixed example, the primal operator ∂ F is the dual diffusion operator K −1 ∈ L(H0,N (div; Ω ), (H0,N (div; Ω ))∗ ) with dual ∂ F ∗ the diffusion operator K ∈ L((H0,N (div; Ω ))∗ , H0,N (div; Ω )), the coupling operator Λ is the divergence operator div ∈ L(H0,N (div; Ω ), L 2 (Ω )) with transpose ΛT the operator divT ∈ L(L 2 (Ω ), (H0,N (div; Ω ))∗ ), and the dual operator ∂G ∗ is the primal control subdifferential ∂Φ : L 2 (Ω ) → 2 2 2 L (Ω ) with primal ∂G the dual control subdifferential ∂Φ ∗ : L 2 (Ω ) → 2 L (Ω ) . Therefore, the dual evolution composition duality principle of Theorem 2.2* is valid, since the corresponding compatibility condition (C F ∗ ,−ΛT ) is trivially satisfied; i.e., the distributed control dual mixed diffusion problem cd

 Find w ∈ V and u ∈ X ∗ :    −divT u = K −1 w, in V ∗ , ∗ (Mcd ) du  + ∂Φ(u), in Y, div w ∈   dt  u(0) = u 0 , is solvable if, and only if, its dual evolution problem  Find u ∈ X ∗ :   du (D cd ) 0 ∈ + ∂Φ(u) + div K divT u, in Y,  dt  u(0) = u 0 , is solvable.

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The macro-hybrid variational formulation of this dual evolution mixed problem is given in terms of corresponding (3)–(7), but now with primalQand dual transmission admissibility subspaces Qcd ∗ ⊂ Bcd ∗ ({Γe }) = decompositions QE E −1/2 H 1/2 (Γe ), defined by (Γe ) and Q∗cd ∗ ⊂ B∗cd ∗ ({Γe }) = e=1 e=1 H o n Qcd ∗ = {µ∗e } ∈ Bcd ∗ ({Γe }): h{µ∗e }, {µe }i B ∗ ∗ ({Γe }) = 0, ∀{µe } ∈ Q∗cd ∗ , (54) cd  Q∗cd ∗ = {µe } ∈ B∗cd ∗ ({Γe }): µe = µ f a.e. on Γe f , 1 ≤ e < f ≤ E . (54*) Hence, in contrast with the primal case, the primal transmission condition is now the Neumann trace continuity across the interfaces in the H −1/2 -weak sense, while the dual transmission condition is the Dirichlet trace continuity; the transmission conditions interchange their primal and dual roles. Further, the macro-hybrid compatibility condition (C[πΓe ] ) is now satisfied due to the surjectivity of the local Neumann trace operators δe ≡ (·)·ne ∈ L(H(div; Ωe ), H −1/2 (∂Ωe )), e = 1, 2, . . . , E, and the macro-hybrid version (C[F ∗ ∗ ,−ΛTe ] ) of dual evolution |V (Ωe ) compatibility condition (C F ∗ ,−ΛT ), is trivially satisfied. Consequently, according to Theorem 3.4*, the corresponding macro-hybrid mixed problem (MH∗cd ) has a solution if, and only if, its dual problem (D cd ) has a solution. Relative to semi-discrete internal variational discretizations, dual mixed variational models are more common in computational mechanics, as well as mixed and macro-hybrid finite element implementations [15,26]. Moreover, dual algorithms ALG1–5M∗ of Euler and operator splitting time marching schemes are classical and have been proved to be very effective in fluid and solid mechanics [18,20]. Lastly, for the stationary discrete macro-hybrid mixed control diffusion problem (MHh,h ∗ ,h ◦ cd ), assume that the primal [Λh e ,h ∗e ]-surjectivity compatibility condition (C[Λh e ,h ∗ ]cd ), equivalent in turn to the transpose e

[ΛTh e ,h ∗ ]-lower boundedness condition (C[TΛ ∗ ]cd ), is satisfied. Then, the primal composition duality principle of e h e ,h e Theorem 6.2* states that problem (MHh,h ∗ ,h ◦ cd ) has a solution if, and only if, its primal stationary problem (P M H h,h ∗ ,h ◦ cd ) has a solution. Furthermore, from Proposition 6.3*, under the discrete compatibility conditions (C[F ∗ ∗ ,−ΛT ∗ ]cd ) and (C[πΓ ◦ ]cd ), we can mention that algorithms ALG4M∗ and ALG5M∗ correspond to h e |V he

h e ,h e

h e ,h e

penalty–duality iterative procedures of problem (Mh,h ∗ ,h ◦ cd ) as approximations of its M∗ -preconditioned augmented formulation (MrM ∗ h,h ∗ ,h ◦ cd ), with exact penalization parameter r > 0. Also, from Proposition 6.3**, algorithms ALG1M∗ , ALG2M∗ and ALG3M∗ define penalty–duality iterative processes as approximations of the corresponding ^ r ^ r ^ ^ ^ M∗ -preconditioned augmented dual problems (DrM^ ∗ h,h ∗ ,h ◦ cd ), (D M ∗ h,h ∗ ,h ◦ cd ) and (D M ∗ h,h ∗ ,h ◦ cd ), respectively. References [1] G. Alduncin, Subdifferential and variational formulations of boundary value problems, Comput. Methods. Appl. Mech. Engrg. 72 (1989) 173–186. [2] G. Alduncin, Numerical resolvent methods for constrained problems in mechanics, Approx. Theory Appl. 12 (4) (1996) 1–25. [3] G. Alduncin, On Gabay’s algorithms for mixed variational inequalities, Appl. Math. Optim. 35 (1997) 21–44. [4] G. Alduncin, Numerical resolvent methods for macro-hybrid mixed variational inequalities, Numer. Funct. Anal. Optim. 19 (1998) 667–696. [5] G. Alduncin, Parallel proximal-point algorithms for constrained problems in mechanics, in: L.T. Yang, M. Paprzycki (Eds.), Practical Applications of Parallel Computing, Nova Science, New York, 2003, pp. 69–88. [6] G. Alduncin, Composition duality methods for mixed variational inclusions, Appl. Math. Optim. 52 (2005) 311–348. [7] G. Alduncin, Composition duality principles for mixed variational inequalities, Math. Comput. Modelling 41 (2005) 639–654. [8] G. Alduncin, Macro-hybrid variational formulations of constrained boundary value problems, Numer. Funct. Anal. Optim. 28 (2007) (in press). [9] G. Alduncin, Analysis of augmented three-field macro-hybrid mixed finite element schemes (in press). [10] G. Alduncin, Composition duality principles for evolution mixed variational inclusions, Appl. Math. Lett. 20 (2007) (in press). [11] I. Babuˇska, The finite element methods with Lagrange multipliers, Numer. Math. 20 (1973) 179–192. [12] V. Barbu, Th. Precupanu, Convexity and Optimization in Banach Spaces, D. Reidel, Dordrecht, 1986. [13] H. Brezis, Op´erateurs Maximaux Monotones, North-Holland, Amsterdam, 1973. [14] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO BR-2 (1974) 129–151. [15] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. [16] G. Duvaut, J.-L. Lions, Les In´equations en M´echanique et en Physique, Dunod, Paris, 1972. [17] I. Ekeland, R. Temam, Analyse Convexe et Probl`emes Variationnels, Dunod, Gauthier Villars, Paris, 1974.

G. Alduncin / Nonlinear Analysis: Hybrid Systems 1 (2007) 336–363

363

[18] M. Fortin, R. Glowinski (Eds.), M´ethodes de Lagrangien Augment´e: Applications a` la R´esolution Num´erique de Probl`emes aux Limites, Dunod-Bordas, Paris, 1982. [19] D. Gabay, Application de la m´ethode des multiplicateurs aux in´equations variationnelles, in: M. Fortin, R. Glowinski (Eds.), M´ethodes de Lagrangien Augment´e, Dunod-Bordas, Paris, 1982, pp. 279–307. [20] R. Glowinski, P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989. [21] O. Ladyˇsenskaja, The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon & Breach, New York, 1969. [22] J.L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin`eaires, Dunod, Paris, 1969. [23] P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979) 964–979. [24] U. Mosco, Dual variational inequalities, J. Math. Anal. Appl. 40 (1972) 202–206. [25] P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Convex and Nonconvex Energy Functions, Birkh¨auser, Boston, 1985. [26] J.E. Roberts, J.-M. Thomas, Mixed and hybrid methods, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. II, Elsevier, Amsterdam, 1991, pp. 523–639. [27] K. Yosida, Functional Analysis, Springer-Verlag, New York, 1974.