Analysis of Evolution Macro-Hybrid Mixed Variational Problems

Int. Journal of Math. Analysis, Vol. 2, 2008, no. 14, 663 - 708

Analysis of Evolution Macro-Hybrid

Mixed Variational Problems

Gonzalo Alduncin

Departamento de Recursos Naturales, Instituto de GeofısicaUniversidad Nacional Autonoma de Mexico

Mexico, D.F. C.P. 04510, [email protected]

Abstract

The purpose of this study is to apply composition duality methods inthe qualitative analysis of evolution linear mixed variational problems.Primal and dual evolution mixed formulations are considered, as well ascorresponding macro-hybrid variational models for parallel computing.The well-posedness, stability and convergence analysis of macro-hybridmixed semi-discrete approximations is performed.

Mathematics Subject Classification: 35J50, 65M60, 74S05

Keywords: Evolution macro-hybrid mixed problems, Composition dualitymethods, Primal-dual analysis

1 Introduction

Primal and dual mixed variational formulations in mechanics have played animportant role in the qualitative and numerical analysis of computational fluidand solid problems. They may be alternative variational formulations for a sin-gle problem, offering complementary approaches for analysis and computation,or they may be specific models of different problems. For instance, primal anddual mixed formulation of diffusion and Darcy problems are classical varia-tional alternatives [15], while primal mixed formulations for advection-diffusionprocesses and dual mixed formulations for Stokes and elasticity problems areusual [7, 8].

Composition duality principles have been proposed recently in [1] for thequalitative analysis of primal and dual mixed variational inequalities or inclu-sions, in the context of constrained boundary valued problems. Such principles

664 G. Alduncin

state solvability equivalence between mixed variational problems and corre-sponding primal or dual variational problems, via compositional dualization,whenever primal or dual coupling compatibility conditions are respectively sat-isfied. In fact, such compatibility conditions resemble in an operational sensethe classical (LBB) inf-sup conditions for mixed problems [15, 7]. On the otherhand, related macro-hybrid variational formulations, based on hybrid nonover-lapping domain decomposition methods, have been incorporated into the com-position duality methodology in [2] for purposes of parallel computing. Then,problems of large scale, complex geometry, multiscale and multiphysics behav-ior can be efficiently analyzed and numerically handled. These two fundamen-tal methodologies have been further presented in one more work, [3], where inaddition “mass”-preconditioned augmented or exactly penalized macro-hybridmixed variational problem are treated, in conjuction with proximal-point mul-tidomain algorithms. A relevant aspect of these multidomain algorithms isthat they may be interpreted as proximation implementation of some implicitand semi-implicit time marching schemes for macro-hybrid mixed evolutionproblems.

Our interest in this study is to apply composition duality methods in thequalitative and approximation analysis of evolution linear mixed variationalproblems. Following [4] and [5], we consider primal and dual evolution mixedformulations, as well as corresponding macro-hybrid variational models forparallel computing. As evolution primal and dual compatibility conditions,classical duality conditions from convex optimization [9] are introduced, es-tablishing duality principles for mixed qualitative analysis on the basis of cor-responding primal and dual evolution variational equations. Macro-hybridmixed semi-discrete approximations are further analyzed, demonstrating theirstrong convergence, and performing a regularity in time analysis for determin-ing error estimates suitable for finite element implementation.

A counterpart of this work for elliptic linear mixed problems has beenpresented in [6], where the composition duality methods of [3] are applied inthe analysis of augmented three-field macro-hybrid mixed variational problems.Such an elliptic mixed analysis may be of importance for the stationary macro-hybrid mixed subproblems, encountered at each time step of numerical timemarching fully discrete algorithms (cf. [5]).

2 Evolution Mixed Variational Problems

In this section, we start our study with the composition duality principles forprimal and dual evolution linear mixed variational problems, which are estab-lished under classical compatibility conditions for convex dualization (see [4]).

Evolution macro-hybrid mixed variational problems 665

Then, on the basis of such principles, qualitative analysis results are demon-strated on well-posedness and stability. A regularity in time analysis will begiven in Section 6, related to semi-discrete internal variational approximations.

For the stationary framework of the theory, let V and Y be two Hilbertspaces, with topological duals denoted by V ∗ and Y ∗, and let H and Z becorresponding Hilbert pivot spaces; i.e., V ⊂ H � H∗ ⊂ V ∗ and Y ∗ ⊂ Z∗ �Z ⊂ Y with continuous and dense embeddings. Further, let A = ∇F ∈L(V, V ∗) and C∗ = ∇G∗ ∈ L(Y ∗, Y ) be two linear continuous and symmetricoperators, with Gateaux differentiable quadratic potentials F : V → � andG∗ : Y ∗ → �. Also, let Λ ∈ L(V, Y ) be a linear continuous operator, withtranspose denoted by ΛT ∈ L(Y ∗, V ∗).

2.1 The primal evolution mixed variational problem

Let (0, T ] be the time interval of computational interest, T > 0 fixed and arbi-trary. Let V = L2(0, T ; V ) = {v : [0, T ] → V | ‖v‖V = [

∫ T0 ‖v(t)‖2

V dt]1/2 < ∞}be the primal evolution Hilbert space, with topological dual V∗ = L2(0, T ; V ∗),and let W = {v : v ∈ V , dv/dt ∈ V∗} be the primal solution space, en-dowed with the norm ‖v‖W = ‖v‖V + ‖dv/dt‖V∗ , continuously embedded inC([0, T ]; H). Also, let the Hilbert space Y∗ = L2(0, T ; Y ∗) be the dual solu-tion space, with dual Y = L2(0, T ; Y ) (see [11, 16]). Let R(Λ) ⊂ Y denotethe range of the operator Λ. Then, we consider the following primal evolutionmixed variational problem.

(M)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Given f∗ ∈ V∗, g ∈ L2(0, T ;R(Λ)), u0 ∈ H, find (u, p∗) ∈ W × Y∗ :

−ΛTp∗ =du

dt+ Au − f∗, in V∗,

Λu = C∗p∗ + g, in Y,

u(0) = u0.

For the analysis of problem (M), we introduce the classical primal compati-bility condition that turns out to be equivalent to the non-zero dual operatorcondition,

(CG,Λ) intD(G) ∩R(Λ) �= ∅ ⇐⇒ C∗ = ∇G∗ �= 0.

Here, intD(G) denotes the interior of the effective domain of the conjugateG : Y → � ∪ {+∞} of dual quadratic potential G∗. Note that C = ∇G andC∗ = ∇G∗ are indeed the graph inverse of each other. Hence, the equivalencefollows since C∗ �= 0 implies D(G) = Y , and C∗ = 0 implies G = I{0Y },the indicator of the zero-singleton {0Y } ⊂ Y . Consequently, the adoption ofcondition (CG,Λ) excludes the zero case, C∗ = 0, from our study, which seemsto be not usual in mechanics.

666 G. Alduncin

The next compositional result is valid [9].

Lemma 2.1 Under condition (CG,Λ), the compositional operator equality

∇(G ◦ Λ) = ΛT∇G ◦ Λ (1)

is guaranteed.

Thereby, by dualization, the following primal composition duality principle isconcluded.

Theorem 2.2 Under compatibility condition (CG,Λ), primal evolution mixedproblem (M) is solvable if, and only if, the primal evolution problem

(P)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Given f∗ ∈ V∗, g ∈ L2(0, T ;R(Λ)), u0 ∈ H, find u ∈ W :

0 =du

dt+ Au + ∇(G ◦ Λ)(u − rg) − f∗, in V∗,

u(0) = u0,

is solvable, where rg ∈ V is a fixed Λ-preimage of function g: Λrg = g. Thatis, if (u, p∗) ∈ W ×Y∗ is a solution of problem (M) then primal function u isa solution of problem (P) and, conversely, if u ∈ W is a solution of problem(P) then there is a dual function p∗ = C(Λu − g) ∈ Y∗ such that (u, p∗) is asolution of problem (M). Moreover, the solvability is unique if in addition thepotential dual operator is such that

(CC∗) C∗ = ∇G∗ ∈ L(Y ∗, Y ) is positive definite.

Proof The principle follows from the duality relation

Λu = ∇G∗p∗ + g ⇐⇒ p∗ = ∇G(Λu − g) (2)

of the dual equation of problem (M) and Lemma 2.1, with unique solvabilityclearly implied by condition (CC∗).

Furthermore, assuming the primal coercivity condition

(CA+∇(G◦Λ))

{A + ∇(G ◦ Λ) ∈ L(V, V ∗) is V − coercive; i.e.,

∃ α > 0 : 〈(A + ∇(G ◦ Λ))v, v〉V ≥ α‖v‖2V , ∀v ∈ V,

the well-posedness of primal evolution problem (M) is obtained. We recallthat in the context of linear parabolic problems this coercivity condition isequivalent, modulo a change of variable, to the corresponding semi-coercivitycondition or Garding inequality (see, e.g., [13]). Let mA+∇(G◦Λ) and mΛ denotethe boundedness constants of operators A + ∇(G ◦ Λ) ∈ L(V ,V∗) and Λ ∈L(V ,Y), and βC∗ the positive definiteness constant of operator C∗ ∈ L(Y∗,Y).


Theorem 2.3 Let conditions (CG,Λ), (CC∗) and (CA+∇(G◦Λ)) be satisfied.Then primal evolution mixed problem (M) possesses a unique solution (u, p∗) ∈W × Y∗ such that

‖u(τ )‖H ≤ ‖u0‖H + c1

{ ∫ τ

0‖f∗(t) + ΛTCg(t)‖2

V ∗ dt}1/2

, ∀τ ∈ [0, T ], (3)

‖u‖V ≤ c2(α)‖u0‖H + c3(α)‖f∗ + ΛTCg‖V∗, (4)

∥∥∥∥du

dt

∥∥∥∥V∗≤ mA+∇(G◦Λ)c2(α)‖u0‖H + (mA+∇(G◦Λ)c3(α) + 1)‖f∗ + ΛT Cg‖V∗, (5)

and

‖p∗‖Y∗ ≤ mΛc2(α)

βC∗‖u0‖H +

mΛc3(α)

βC∗‖f∗ + ΛTCg‖V∗ +

1

βC∗‖g‖Y . (6)

Consequently, u ∈ L∞(0, T ; H) and the mapping (u0, f∗ +ΛTCg) ∈ H ×V∗ �→

(u, p∗) ∈ W × Y∗ is continuous.

Proof The unique solvability follows from Theorem 2.2 and the classical resultof primal unique solvability under coercivity condition (CA+∇(G◦Λ)) [11]. Inparticular, taking into account the time integration by parts formula

⟨du

dt, v⟩V

= (u(T ), v(T ))H − (u(0), v(0))H −⟨

dv

dt, u

⟩V, ∀u, v ∈ W, (7)

the uniqueness result is indeed a consequence of condition (CA+∇(G◦Λ)) since,for u1 ∈ W and u2 ∈ W primal solutions of problem (P), necessarily

∫ τ

0

⟨du1

dt(t)− du2

dt(t), u1(t)−u2(t)

⟩V

dt =1

2‖u1(τ )−u2(τ )‖2

H ≤ 0, ∀τ ∈ [0, T ].

Proceeding similarly, for u ∈ W solution of (P) and τ ∈ [0, T ], it follows that∫ τ

0〈f∗(t), u(t)〉V dt

=∫ τ

0

⟨du

dt(t), u(t)

⟩V

dt +∫ τ

0〈(A + ∇(G ◦ Λ))u(t), u(t)〉V dt

≥ 1

2‖u(τ )‖2

H − 1

2‖u0‖2

H + α∫ τ

0‖u(t)‖2

V dt,

668 G. Alduncin

and applying Young’s inequality, with constant b > 0,∫ τ

0〈f∗(t), u(t)〉V dt ≤

∫ τ

0‖f∗(t)‖V ∗‖u(t)‖V dt

≤ 1

2b2

∫ τ

0‖f∗(t)‖2

V ∗ dt +b2

2

∫ τ

0‖u(t)‖2

V dt,

where f∗ = f∗ + ΛTCg. Hence, choosing b > 0 such that c(α) = α− b2/2 > 0,we obtain the estimate

1

2‖u(τ )‖2

H + c(α)∫ τ

0‖u(t)‖2

V dt ≤ 1

2‖u0‖2

H +1

2b2

∫ τ

0‖f∗(t)‖2

V ∗ dt, ∀τ ∈ [0, T ],

(8)

from which (3) and (4) are concluded. Estimate (5) is then derived from thevariational equation of problem (P), as well as estimate (6) from the dualequation of problem (M) using condition (CC∗).

Remark 2.4 Taking into account that the time interval (0, T ] is arbitrarilyfinite, the solutions in W of primal problem (P), under coercivity condition(CA+∇(G◦Λ)), necessarily are stable in the Lyapunov sense; i.e., they dependglobally (in time) continuously upon the initial data from H. In fact, in thiscoercive case such a stability is exponentially asymptotic: if uu0 and uu0

are theprimal solutions of problem (P) with initial conditions u0 and u0, respectively,then

‖uu0(τ )− uu0(τ )‖H ≤ exp−(α/κ)τ ‖u0 − u0‖H , ∀τ ∈ (0,∞), (9)

where κ > 0 is the constant of the continuous embedding V ⊂ H. Indeed,applying formula (7) and using condition (CA+∇(G◦Λ)), it follows that, ∀τ ∈(0,∞),

0 =∫ τ

0

⟨duu0

dt(t) − duu0

dt(t), uu0(t) − uu0

(t)⟩

Vdt

+∫ τ

0〈(A + ∇(G ◦ Λ))(uu0(t) − uu0

(t)), uu0(t) − uu0(t)〉V dt

≥ 1

2‖uu0(τ ) − uu0

(τ )‖2H − 1

2‖u0 − u0‖2

H +α

κ

∫ τ

0‖uu0(t) − uu0

(t)‖2H dt,

from which Lyapunov stability is evident, independently of α (only by positiv-ity). Further, the exponential asymptotic stability (9) is then a consequenceof this integral inequality, which is reducible to a differential inequality ofCaratheodory type [12].


Remark 2.5 Regarding the evolutionary behavior at infinity of primalproblem (P), we observe that due to the coercivity condition (CA+∇(G◦Λ)),

for each fixed time-independent data f∗ = f∗ + ΛTCg ∈ V ∗, there is a uniqueequilibrium or stationary state uus(t) = us ∈ V,∀ t ≥ 0, which in accordanceto the exponential asymptotic stability (9) of the system, and for the corre-sponding motion uu0 ∈ W originating at any u0 ∈ H, is such that

limτ→+∞ ‖uu0(τ ) − us‖H = 0. (10)

Remark 2.6 With respect to the primal stationary mixed system of theevolution problem (M),

(M)

⎧⎪⎪⎨⎪⎪⎩Given f∗

s ∈ V ∗, gs ∈ R(Λ), find (us, p∗s) ∈ V × Y ∗ :

−ΛTp∗s = Aus − f∗s , in V ∗,

Λus = C∗p∗s + gs, in Y,

a dual composition duality principle can be established for its qualitative anal-ysis, as proposed in [1], in terms of the dual compatibility condition

(CΛT ) ΛT ∈ L(Y ∗, V ∗) is surjective,

equivalent to the lower boundedness of its own transpose Λ ∈ L(V, Y ) [18].Indeed, via dualization (graph inversion) and compositional dualization, theprimal equation of problem (M ) is characterized by (see Lemma 2.1∗ of [1])

−ΛTp∗s = ∇Fus − f∗s ⇐⇒ us ∈ ∂F ∗(−ΛTp∗s + f∗

s )

⇐⇒ −Λus ∈ ∂(F ∗ ◦ (−ΛT ))(p∗s + r∗f∗s),

(11)

where F ∗ : V ∗ → �∪{+∞} is the conjugate of the primal quadratic potentialF , ∂F ∗ : V ∗ → 2V standing for its subdifferential, and r∗f∗

s∈ Y ∗ is a fixed −ΛT -

preimage of function f∗s . Then, under condition (CΛT ), the primal stationary

mixed problem (M ) has a solution if, and only if, its dual stationary problem

(D)

{Given f∗

s ∈ V ∗, gs ∈ R(Λ), find p∗s ∈ Y ∗ :

0 ∈ C∗p∗s + ∂(F ∗ ◦ (−ΛT ))(p∗s + r∗f∗s) + gs, in Y,

has a solution [1]. Moreover, the solvability is unique due to the lower bound-edness of Λ operator. This dual stationary composition duality principle playsa fundamental role, specially in the well-posedness and convergence analysis ofassociated semi-discrete internal variational approximations to problem (M)and their mixed finite element implementation (cf. Section 4, below).

670 G. Alduncin

Example 2.7 As an application of the theory, we shall be considering theclassical mixed diffusion problem, in a spatial bounded domain Ω ⊂ �n, n ∈{2, 3}, with a Lipschitz continuous boundary ∂Ω, and along the time interval(0, T ), T > 0, modeled by

du

dt− div w = f∗,

K−1w = grad u,

⎫⎪⎬⎪⎭ in Ω × (0, T ),

u(0) = u0, in Ω.

(12)

Here, u stands for a mass concentration or temperature field of an evolutionarydiffusion process, with flux vector field w, K−1 is the inverse of a symmetric,positive definite diffusion tensor and u0 is a given initial state of the system inΩ. Further, as boundary conditions, we shall consider Dirichlet and Neumannconditions, d and h, prescribed respectively on disjoint and complementaryparts of the boundary:

u = d, on ∂ΩD × (0, T ),

−w·n = h, on ∂ΩN × (0, T ).(13)

Then, the primal mixed variational formulation of the evolution diffusion prob-lem follows as usual, by integration and application of Green’s formula to itsdivergence equation. For primal coercivity, we shall assume that the Dirichletboundary ∂ΩD is not empty.

Find u = u − ud∈ L2(0, T ; H1

0,D(Ω)) with du/dt ∈ L2(0, T ; (H10,D(Ω))∗),

and w ∈ L2(0, T ; L2(Ω)) :

−∫ T

0

∫Ω

w · grad v dΩ dt

=∫ T

0

⟨du

dt, v

⟩H1

0,D(Ω)

dt +∫ T

0

⟨du

d

dt− f∗, v

⟩H1

0,D(Ω)

dt

+∫ T

0〈h, γv〉H1/2(∂ΩN ) dt, ∀v ∈ L2(0, T ; H1

0,D(Ω)),∫ T

0

∫Ω

grad u · v dΩ dt

=∫ T

0

∫Ω

K−1w · v dΩ dt −∫ T

0

∫Ω

grad ud

· v dΩ dt,

∀v ∈ L2(0, T ; L2(Ω)),

u(0) = u0 − ud(0).

Here, the primal space H10,D(Ω) = {v ∈ H1(Ω) : γv = 0 a.e. on ∂ΩD} is the

usual closed subspace of the Hilbert space H1(Ω) = {v ∈ L2(Ω) : grad v ∈L2(Ω)}, with surjective Dirichlet trace operator γ ∈ L(H1(Ω), H1/2(∂Ω)).


Also, 〈·, ·〉H10,D(Ω) and 〈·, ·〉H1/2(∂ΩN ) stand for the duality pairings of (H1

0,D(Ω))∗×H1

0,D(Ω) and H−1/2(∂ΩN )×H1/2(∂ΩN), and the data is such that f∗ ∈ L2(0, T ;(H1

0,D(Ω))∗), K−1 ∈ L∞(Ω) is a uniformly positive definite, symmetric tensor,

d ∈ L2(0, T ; H1/2(∂ΩD)), h ∈ L2(0, T ; H−1/2(∂ΩN )) and u0 ∈ L2(Ω). The aux-iliary primal field u

dis a given function that satisfies the Dirichlet boundary

condition. Hence, such a primal mixed diffusion problem is identified with theabstract mixed problem (M) of the theory according to the field and functionalframework relations

u ∈ V ∼ u ∈ Vdp = L2(0, T ; H10,D(Ω)),

p∗ ∈ Y∗ ∼ w ∈ Y∗dp = L2(0, T ; L2(Ω)),

(14)

and the operator and function identifications, for a.e. t ∈ (0, T ),

ΛTp∗(t) ∼ gradTw(t), Au(t) − f∗(t) ∼(

dud

dt− f∗ + γT

∂ΩNh)(t),

Λu(t) ∼ grad u(t), C∗p∗(t) + g(t) ∼ K−1w(t) − grad ud(t).

(15)

Thus, in a variational sense, the primal operator A = ∇F is the zero operatorfrom H1

0,D(Ω) → 0∗ ∈ (H10,D(Ω))∗, the coupling operator Λ is the gradient

operator grad ∈ L(H10,D(Ω), L2(Ω)) with transpose ΛT the operator gradT ∈

L(L2(Ω), (H10,D(Ω))∗), and the dual operator C∗ = ∇G∗ is the dual diffusion

operator K−1 ∈ L(L2(Ω), L2(Ω)) with primal C = ∇G the diffusion operatorK ∈ L(L2(Ω), L2(Ω)). Also, the dual data −grad u

d∈ L2(0, T ;R(Λ)).

Therefore, since corresponding conditions (CG,Λ) and (CC∗) are fulfilled, fromthe evolution composition duality principle of Theorem 2.2, the primal mixeddiffusion problem

(Mdp)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find u ∈ Wdp and w ∈ Y∗dp :

−gradTw =du

dt+

dud

dt− f∗ + γT

∂ΩNh, in V∗

dp,

grad u = K−1w − grad ud, in Ydp,

u(0) = u0 − ud(0),

is uniquely solvable if, and only if, its primal evolution problem

(Pdp)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find u ∈ Wdp :

0 =du

dt+ gradTK ◦ grad u + gradTK ◦ grad u

d

+du

d

dt− f∗ + γT

∂ΩNh, in V∗

dp,

u(0) = u0 − ud(0),

672 G. Alduncin

is uniquely solvable. Moreover, since operator gradT K ◦ grad ∈ L(H10,D(Ω),

(H10,D(Ω))∗) is H1

0,D(Ω)-coercive, according to Theorem 2.3 and Remarks 2.4and 2.5, problem (Mdp) possesses a unique solution (u, w) ∈ (L∞(0, T ; L2(Ω))∩Wdp) × Y∗

dp, continuously dependent on the data in the sense of (3)-(6),and exponentially asymptotically stable and convergent to the unique equi-librium of primal dynamical system (Pdp) due to (9) and (10). Note thatthe composition gradient in (1) of the primal diffusion operator is given by∇Jdp = ∇(1/2‖K(·)‖2

L2(Ω) ◦ grad) ∈ L(H10,D(Ω), (H1

0,D(Ω))∗). Further, thedual compatibility condition (CΛT ) holds since the coupling operator grad ∈L(H1

0,D(Ω), L2(Ω)) is bounded below and, consequently, the dual stationarycomposition duality principle of Remark 2.6 is valid with respect to the dualstationary diffusion problem

(Ddp)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find ws ∈ Y ∗

dp :

0 ∈ K−1ws + ∂(I{0V ∗dp

} ◦ (−gradT ))(ws + r∗(f∗−γT

∂ΩNh)s

) − grad uds

,

in Y dp.

Here, I{0V ∗dp

} is the indicator functional of the zero subset of the dual space

V ∗dp = (H1

0,D(Ω))∗, and r∗(f∗−γT

∂ΩNh)s

stands for a fixed −gradT -preimage of

function (f∗ − γT∂ΩN

h)s. In an optimization sense, problem (Ddp) statesthe optimality condition to the minimization problem of functional J∗

dp(·) =1/2‖K−1(·)‖2

L2(Ω) − grad uds

· (·) on the convex set S∗dp = {v ∈ L2(Ω) :

gradTv = (f∗ − γT∂ΩN

h)s}.

2.2 The dual evolution mixed variational problem

We next consider the dual evolution mixed problem. Let X ∗ = {q∗ : q∗ ∈Y∗, dq∗/dt ∈ Y} be now the dual solution space, normed by ‖q∗‖X ∗ = ‖q∗‖Y∗ +‖dq∗/dt‖Y , and continuously embedded in C([0, T ]; Z∗). Let R(−ΛT ) ⊂ V ∗

stand for the range of the transpose operator −ΛT . Then, the dual evolutionlinear model of the theory reads as follows.

(M∗)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Given f∗ ∈ L2(0, T ;R(−ΛT )), g ∈ Y, p∗0 ∈ Z∗, find (u, p∗) ∈ V ×X ∗ :

−ΛTp∗ = Au − f∗, in V∗,

Λu =dp∗

dt+ C∗p∗ + g, in Y,

p∗(0) = p∗0.

Note that the evolution of the problem is now governed by the dual variationalequation. Hence, according to our variational theory [4], the correspondingdual compatibility condition to be adopted is given by


(CF∗ ,−ΛT ) intD(F ∗) ∩ R(−ΛT ) �= ∅ ⇐⇒ A = ∇F �= 0,

where F ∗ : V ∗ → �∪{+∞} is the conjugate of the primal quadratic potentialF , and because of such equivalence the primal zero case, A = 0, is excludedfrom the study. This dual evolution mixed problem with zero primal operatorseems to be another not usual model in mechanics.

The following dual compositional result is valid [9].

Lemma 2.1∗ If condition (CF∗,−ΛT ) is satisfied, then the compositionaloperator equality

∇(F ∗ ◦ (−ΛT )) = −Λ∇F ∗ ◦ (−ΛT ) (1∗)

holds true.

Then, as in the previous subsection, the dual composition duality principle forproblem (M∗) is established.

Theorem 2.2∗ Let condition (CF∗,−ΛT ) be satisfied. Then dual evolutionmixed problem (M∗) is solvable if, and only if, the dual evolution problem

(D)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Given f∗ ∈ L2(0, T ;R(−ΛT )), g ∈ Y, p∗0 ∈ Z∗, find p∗ ∈ X ∗ :

0 =dp∗

dt+ C∗p∗ + ∇(F ∗ ◦ (−ΛT ))(p∗ + r∗f∗) + g, in Y,

p∗(0) = p∗0,

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 dualfunction p∗ is a solution of problem (D) and, conversely, if p∗ ∈ X ∗ is asolution of problem (D) then there is a primal function u = A∗(−ΛTp∗ +f∗) ∈V such that (u, p∗) is a solution of problem (M∗). Moreover, the solvability isunique if in addition the potential primal operator is such that

(CA) A = ∇F ∈ L(V, V ∗) is positive definite.

Proof In this dual case the principle is concluded from the duality relation

−ΛTp∗ = ∇Fu− f∗ ⇐⇒ u = ∇F ∗(−ΛTp∗ + f∗) (2∗)

of the primal equation of problem (M∗) and Lemma 2.1∗, the unique solvabil-ity being implied by condition (CA).

Furthermore, the well-posedness of dual evolution problem (M∗) is ob-tained under the dual coercivity condition

674 G. Alduncin

(CC∗+∇(F∗◦(−ΛT )))

⎧⎪⎪⎨⎪⎪⎩C∗ + ∇(F ∗ ◦ (−ΛT )) ∈ L(Y ∗, Y ) is Y ∗ − coercive; i.e.,

∃ α∗ > 0 : 〈y∗, (C∗ + ∇(F ∗ ◦ (−ΛT )))y∗〉Y ≥ α∗‖y∗‖2Y ∗ ,

∀y∗ ∈ Y ∗,

which, as mentioned in the primal case, is in fact equivalent, modulo a change ofvariable, to the corresponding semi-coercivity condition or Garding inequality.Let mC∗+∇(F∗◦(−ΛT )) and m−ΛT denote the boundedness constants of operators

C∗ + ∇(F ∗ ◦ (−ΛT )) ∈ L(Y∗,Y) and −ΛT ∈ L(Y∗,V∗), and βA the positivedefiniteness constant of operator A ∈ L(V ,V∗).

Theorem 2.3∗ Let conditions (CF∗ ,−ΛT ), (CA) and (CC∗+∇(F∗◦(−ΛT ))) befulfilled. Then dual evolution mixed problem (M∗) has a unique solution(u, p∗) ∈ V × X ∗ such that

‖p∗(τ )‖Z∗ ≤ ‖p∗0‖Z∗ + c∗1

{∫ τ

0‖g(t) − ΛA∗f∗(t)‖2

Y dt}1/2

, ∀τ ∈ [0, T ], (3∗)

‖p∗‖Y∗ ≤ c∗2(α∗)‖p∗0‖Z∗ + c∗3(α

∗)‖g − ΛA∗f∗‖Y , (4∗)∥∥∥∥dp∗

dt

∥∥∥∥Y ≤ mC∗+∇(F∗◦(−ΛT ))c∗2(α

∗)‖p∗0‖Z∗

+ (mC∗+∇(F∗◦(−ΛT ))c∗3(α

∗) + 1)‖g − ΛA∗f∗‖Y ,(5∗)

and

‖u‖V ≤ m−ΛT c∗2(α∗)

βA

‖p∗0‖Z∗ +m−ΛT c∗3(α

∗)

βA

‖g − ΛA∗f∗‖Y +1

βA

‖f∗‖V∗. (6∗)

Consequently, p∗ ∈ L∞(0, T ; Z∗) and the mapping (p∗0, g−ΛA∗f∗) ∈ Z∗×Y �→(u, p∗) ∈ V × X ∗ is continuous.

Proof These results follow as in the proof of primal Theorem 2.3, but nowtaking into account the dual composition duality principle of Theorem 2.2∗,and the dual time integration by parts formula⟨

y∗,dp∗

dt

⟩Y

= (p∗(T ), y∗(T ))Z∗ − (p∗(0), y∗(0))Z∗ −⟨p∗,

dy∗

dt

⟩Y,

∀p∗, y∗ ∈ X ∗.(7∗)

Then, the concluded general dual estimate is given by, with Young’s constantb > 0 chosen such that c(α∗) = α∗ − b2/2 > 0,

1

2‖p∗(τ )‖2

Z∗ + c(α∗)∫ τ

0‖p∗(t)‖2

Y ∗ dt ≤ 1

2‖p∗0‖2

Z∗ +1

2b2

∫ τ

0‖g(t)‖2

Y dt,

∀τ ∈ [0, T ].(8∗)


Here, g = g − ΛA∗f∗.

As dual versions of Remarks 2.4, 2.5 and 2.6, we have the following.

Remark 2.4∗ Under coercivity condition (CC∗+∇(F∗◦(−ΛT ))), the dual so-lutions of problem (D) are necessarily stable in the Lyapunov sense. In fact,they are exponentially asymptotically stable: if p∗p∗0 and p∗

p∗0are the solutions

of dual problem (D) with initial conditions p∗0 and p∗0, respectively, then

‖p∗p∗0(τ ) − p∗p∗0

(τ )‖Z∗ ≤ exp−(α∗/κ∗)τ ‖p∗0 − p∗0‖Z∗ , ∀τ ∈ (0,∞), (9∗)

κ∗ > 0 standing for the continuous embedding constant of Y ∗ ⊂ Z∗.

Remark 2.5∗ Due to the coercivity condition (CC∗+∇(F∗◦(−ΛT ))), dual prob-lem (D) possesses a unique equilibrium or stationary state p∗p∗s(t) = p∗s ∈Y ∗, ∀ t ≥ 0, for each fixed time-independent data g = g − ΛA∗f∗ ∈ Y . Then,from the exponential asymptotic stability (9∗) of the system, the evolutionarybehavior at infinity for corresponding motions p∗p∗0 ∈ X ∗ originating at anyp∗0 ∈ Z∗, is such that

limτ→+∞ ‖p∗p∗0(τ ) − p∗s‖Z∗ = 0. (10∗)

Remark 2.6∗ Regarding the qualitative analysis of the dual stationarymixed system of evolution problem (M∗),

(M∗)

⎧⎪⎪⎨⎪⎪⎩Given f∗

s ∈ R(−ΛT ), gs ∈ Y, find (us, p∗s) ∈ V × Y ∗ :

−ΛTp∗s = Aus − f∗s , in V ∗,

Λus = C∗p∗s + gs, in Y,

a corresponding primal composition duality principle can be established bymeans of the primal compatibility condition

(CΛ) Λ ∈ L(V, Y ) is surjective,

equivalent to the lower boundedness of its transpose ΛT ∈ L(Y ∗, V ∗). Thatis, by dualization and compositional dualization, the dual equation of problem(M ∗) is such that (see Lemma 2.1 of [1])

Λus = ∇G∗p∗s + gs ⇐⇒ p∗s ∈ ∂G(Λus − gs)

⇐⇒ ΛTp∗s ∈ ∂(G ◦ Λ)(us − rgs),(11∗)

where G : Y → � ∪ {+∞} is the conjugate of the dual quadratic potentialG∗, with subdifferential ∂G : Y → 2Y ∗

, and rgs ∈ V is a fixed Λ-preimage of

676 G. Alduncin

function gs. Then, under condition (CΛ), the dual stationary mixed problem(M ∗) has a solution if, and only if, its primal stationary problem

(P )

{Given f∗

s ∈ R(−ΛT ), gs ∈ Y, find us ∈ V :

0 ∈ Aus + ∂(G ◦ Λ)(us − rgs) − f∗s , in V ∗,

has a solution [1]. Moreover, the solvability is unique due to the lower bounded-ness of ΛT operator. Also, this primal stationary composition duality principleis of fundamental importance, in particular in the well-posedness and conver-gence analysis of associated semi-discrete internal variational approximations,but now to dual problem (M∗) and their mixed finite element implementation(see Section 4, below).

Example 2.7∗ Continuing with the analysis of the classical mixed diffusionproblem, modeled by equations (12) and (13), let us next consider its dualvariational formulation. This dual formulation follows as before, but regardingthe gradient equation as the primal of the mixed variational system, to whichthe boundary conditions are incorporated. For dual coercivity, we shall assumethat the Neumann boundary ∂ΩN �= ∅.

Find w = w − wh∈ L2(0, T ; H0,N(div; Ω)), and u ∈ L2(0, T ; L2(Ω))

with du/dt ∈ L2(0, T ; L2(Ω)) :

−∫ T

0

∫Ω

u div v dΩ dt =∫ T

0

∫Ω

K−1w · v dΩ dt +∫ T

0

∫Ω

K−1wh

· v dΩ dt

−∫ T

0

∫∂ΩD

d div v dΩ dt, ∀ v ∈ L2(0, T ; H0,N(div; Ω)),∫ T

0

∫Ω

div w v dΩ dt =∫ T

0

∫Ω

du

dtv dΩ dt −

∫ T

0

∫Ω

div wh

v dΩ dt

−∫ T

0

∫Ω

f∗ v dΩ dt, ∀ v ∈ L2(0, T ; L2(Ω)),

u(0) = u0.

Now the primal space is the closed subspace H0,N(div; Ω) = {v ∈ H(div; Ω) :δv = 0 in H−1/2(∂ΩN)} of the usual Hilbert space H(div; Ω) = {v ∈ L2(Ω) :div v ∈ L2(Ω)}, with surjective Neumann trace operator δ ≡ (·)·n ∈ L(H(div; Ω), H−1/2(∂Ω)). Also, the data is such that K−1 ∈ L∞(Ω) is a uniformlypositive definite, symmetric tensor, f∗ ∈ L2(0, T ; L2(Ω)), d ∈ L2(0, T ; H1/2

(∂ΩD)), h ∈ L2(0, T ; H−1/2(∂ΩN)) and u0 ∈ L2(Ω). The auxiliary primal fieldw

his a given function that satisfies the Neumann boundary condition. Thus,

the dual evolution mixed diffusion problem is an (M∗)-type problem, underthe field and functional framework relations


u ∈ V ∼ w ∈ Vdp∗ = L2(0, T ; H0,N(div; Ω)),

p∗ ∈ Y∗ ∼ u ∈ Y∗dp∗ = L2(0, T ; L2(Ω)),

(14∗)

and the operator and function identifications, for a.e. t ∈ (0, T ),

ΛTp∗(t) ∼ divT u(t), Au(t) − f∗(t) ∼ K−1w(t) + (K−1wh− divT d)(t),

Λu(t) ∼ div w(t), C∗p∗(t) + g(t) ∼ (−div wh− f∗)(t).

(15∗)

Hence, in a variational sense, the primal operator A = ∇F is the dual diffu-sion operator K−1 ∈ L(H0,N(div; Ω), (H0,N(div; Ω))∗) with dual A∗ = ∇F ∗

the diffusion operator K ∈ L((H0,N(div; Ω))∗, H0,N(div; Ω)), the coupling op-erator Λ is the divergence operator div ∈ L(H0,N (div; Ω), L2(Ω)) with trans-pose ΛT the operator divT ∈ L(L2(Ω), (H0,N(div; Ω))∗), and the dual operatorC∗ = ∇G∗ is the zero operator from L2(Ω) → 0 ∈ L2(Ω). Therefore, the dualevolution composition duality principle of Theorem 2.2∗ is valid in this casesince the corresponding conditions (CF∗,−ΛT ) and (CA) are satisfied; i.e., thedual mixed diffusion problem

(M∗dp)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find w ∈ Vdp∗ and u ∈ X ∗dp∗ :

−divTu = K−1w + K−1wh− divT d, in V∗

dp∗ ,

div w =du

dt− div w

h− f∗, in Ydp∗ ,

u(0) = u0,

is uniquely solvable if, and only if, its dual evolution problem

(Ddp)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Find u ∈ X ∗

dp∗ :

0 =du

dt+ divK divTu − f∗ − divKdivT d, in Ydp∗ ,

u(0) = u0,

is uniquely solvable. Hence, since operator divK divT ∈ L(L2(Ω), L2(Ω)) isL2(Ω)-coercive, from Theorem 2.3∗ problem (M∗

dp) possesses a unique solu-tion (w, u) ∈ (L∞(0, T ; L2(Ω)) ∩ Vdp) × X ∗

dp, continuously dependent on thedata in the sense of (3∗) − (6∗). Moreover, from Remarks 2.4∗ and 2.5∗, sucha solution is exponentially asymptotically stable and convergent to the uniqueequilibrium of dual dynamical system (Ddp∗) in accordance with (9∗) and (10∗).Note that the composition gradient version (1∗)1 of this dual diffusion oper-ator is given by ∇Jdf∗ = ∇(1/2‖K∗(·)‖2

H(div;Ω) ◦ divT ) ∈ L(L2(Ω), L2(Ω)).On the other hand, the primal stationary composition duality principle givenin Remark 2.6∗ applies in this case since the primal positive definiteness con-dition (CA) is satisfied, as well as the primal surjectivity-compatibility condi-tion (CΛ) for the divergence coupling operator div ∈ L(H0,N(div; Ω), L2(Ω)).

678 G. Alduncin

Then, the corresponding dual stationary mixed system (M∗dp) of evolution dif-

fusion problem (M∗dp) is uniquely solvable if, and only if, the primal stationary

diffusion problem

(P dp∗)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find ws ∈ V dp∗ :

0 ∈ K−1ws + ∂(I{0Ydp∗} ◦ div)(ws − r(−div wh−f∗)s

+K−1whs

− divT ds, in V ∗dp∗ ,

is uniquely solvable, where I{0Ydp∗} denotes the indicator functional of the zero

subset of the primal space Ydp∗ = L2(Ω), and r(−div wh−f∗)s is a fixed div-

preimage of function (−div wh− f∗)s. By potentiality, problem (P dp∗) corre-

sponds to the optimality condition of functional Jdp∗(·) = 1/2‖K−1(·)‖2(H(div;Ω))∗

on the convex set Sdp∗ = {v ∈ H(div; Ω) : divv = (div wh

+ f∗)s}.

3 Evolution Macro-Hybrid Mixed Variational

Problems

We now pass to macro-hybridize the primal and dual evolution mixed varia-tional problems (M) and (M∗) in the sense of [2] (see [5] for general mono-tone variational inclusions), and to perform their qualitative analysis. Suchformulations, based on hybrid domain decompositions with primal continu-ity transmission constraints dualized, are very suitable for parallel computing,allowing the treatment of dynamical multisystems.

Hence, let the primal and dual functional frameworks be defined relative toa spatial bounded domain Ω ⊂ �d, d ∈ {2, 3}, which is decomposed in termsof disjoint and connected subdomains {Ωe},

Ω =E⋃

e=1

Ωe, (16)

with internal boundaries and interfaces

Γe = ∂Ωe ∩ Ω, e = 1, 2, ..., E,

Γef = Γe ∩ Γf , 1 ≤ e < f ≤ E,(17)

assumed to be Lipschitz continuous. Then, we consider primal and dual spacesV and Y ∗ of a local type in the sense

V = V (Ω) �{{ve} ∈ V ({Ωe}) =

E∏e=1

V (Ωe) : {πΓeve} ∈ Q},

Y ∗ = Y ∗(Ω) � Y ∗({Ωe}) =E∏

e=1

Y ∗(Ωe),

(18)


with pivot spaces

H = H(Ω) � H({Ωe}) =E∏

e=1

H(Ωe),

Z∗ = Z∗(Ω) � Z∗({Ωe}) =E∏

e=1

Z∗(Ωe),

(19)

where Q ⊂ B({Γe}) =∏E

e=1 B(Γe) stands for the primal transmission ad-missibility subspace, relative to the linear continuous, internal boundary pri-mal trace operator [πΓe] ∈ L(V ({Ωe}), B({Γe})). Thus, as correspondingprimal and dual evolution product spaces, we consider the vector Hilbertspaces V{Ωe} = L2(0, T ; V ({Ωe})) and Y∗

{Ωe} = L2(0, T ; Y ∗({Ωe})), with du-als V∗

{Ωe} = L2(0, T ; V ∗({Ωe})) and Y{Ωe} = L2(0, T ; Y ({Ωe})), as well asthe solution spaces W{Ωe} = {{ve} : {ve} ∈ V{Ωe}, {dve/dt} ∈ V∗

{Ωe}} ⊂C([0, T ]; H({Ωe})) and X ∗

{Ωe} = {{q∗e} : {q∗e} ∈ Y∗{Ωe}, {dq∗e/dt} ∈ Y{Ωe}} ⊂

C([0, T ]; Z∗({Ωe})).Consequently, the macro-hybridization of the evolution mixed problems

consists in imposing the primal transmission condition of (18)1 as a variationalconstraint in terms of the subdifferential ∂(IQ ◦ [πΓe]), where IQ denotes theindicator functional of subspace Q. That is, for operators and functionals of alocal type with respect to decompositions (16)-(19); i.e., for {ve} ∈ V ({Ωe})and {y∗

e} ∈ Y ∗({Ωe}),

A{ve} � {Aeve},f∗ � {f∗

e },Λ{ve} � {Λeve},

C∗{y∗e} � {C∗

ey∗e},

g � {ge},

(20)

the macro-hybridized version of primal evolution problem (M) is given by

(MMH)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find ({ue}, {p∗e}) ∈ W{Ωe} × Y∗{Ωe} :

−{ΛTe p∗e} ∈

{due

dt

}+ {Aeue} + ∂(IQ ◦ [πΓe])({ue}) − {f∗

e },in V∗

{Ωe},

{Λeue} = {C∗ep

∗e} + {ge}, in Y{Ωe},

{ue(0)} = {u0e},

and the macro-hybridized version of dual evolution problem (M∗) by

680 G. Alduncin

(M∗MH)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find ({ue}, {p∗e}) ∈ V{Ωe} × X ∗{Ωe} :

−{ΛTe p∗e} ∈ {Aeue} + ∂(IQ ◦ [πΓe])({ue}) − {f∗

e }, in V∗{Ωe},

{Λeue} ={

dp∗edt

}+ {C∗

ep∗e} + {ge}, in Y{Ωe},

{p∗e(0)} = {p∗0e}.

Furthermore, the compositional dualization of the variational macro-hybridconstraint operator ∂(IQ◦[πΓe])({ue}), is accomplished under the macro-hybridcompatibility condition

(C [πΓe ]) πΓe ∈ L(V (Ωe), B({Γe})) is surjective, e = 1, 2, ..., E,

and the introduction of the dual transmission admissibility subspace, of thedual internal boundary space B∗({Γe}) =

∏Ee=1 B∗(Γe),

Q∗ ={{μ∗

e} ∈ B∗({Γe}) : [{μ∗e}, {μe}]B∗({Γe}) = 0,∀{μe} ∈ Q

}. (21)

Hence, Q∗ is the polar subspace of the primal transmission admissibility sub-space Q, and defining the internal boundary dual trace vector {λ∗

e} ∈ ∂IQ

({πΓeue}) ⊂ B∗{Γe} = L2(0, T ; B∗({Γe})), the next dualization result can be

concluded [2].

Lemma 3.1 Let compatibility condition (C [πΓe ]) be fulfilled. Then the macro-hybrid compositional dualization

{πTΓe

λ∗e} ∈ ∂(IQ ◦ [πΓe])({ue}) ⇐⇒ {πΓeue} ∈ ∂IQ∗({λ∗

e}) (22)

holds true.

Proof Since the conjugate indicator functional (IQ)∗ = IQ∗ , by convex du-alization {πΓeue} ∈ ∂IQ∗({λ∗

e}) ⇔ {λ∗e} ∈ ∂IQ({πΓeue}). Then, under con-

dition (C [πΓe ]), equivalence (22) follows directly from the equivalence of thevariational inequalities of primal inclusions {πT

Γeλ∗

e} ∈ ∂(IQ ◦ [πΓe])({ue}) and{λ∗

e} ∈ ∂IQ({πΓeue}).

3.1 The primal evolution macro-hybrid mixed

variational problem

Proceeding as in Section 2, but now at a local level, we consider the primalcompatibility condition

(C [Ge,Λe]) intD(Ge) ∩R(Λe) �= ∅ ⇐⇒ C∗e = ∇G∗

e �= 0, e = 1, 2, ..., E,

under which Lemma 2.1 is valid locally.


Lemma 3.2 Let compatibility condition (C [Ge,Λe]) be satisfied. Then thelocal compositional operator equalities

∇(Ge ◦ Λe) = ΛTe ∇Ge ◦ Λe, e = 1, 2, ..., E, (23)

are guaranteed.

Moreover, the composition duality principle of hybridized problem (MMH)reads as follows (cf. Theorem 2.2).

Theorem 3.3 Under condition (C [Ge,Λe]), the macro-hybridized primal evo-lution mixed problem (MMH) is solvable if, and only if, the macro-hybrid pri-mal evolution problem

(PMH)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find {ue} ∈ W{Ωe} :

{0e} ∈{

due

dt

}+ {Aeue} + {∇(Ge ◦ Λe)(ue − rge)}

+ ∂(IQ ◦ [πΓe])({ue}) − {f∗e }, in V∗

{Ωe},

{ue(0)} = {u0e},

is solvable, where {rge} ∈ V{Ωe} is a fixed [Λe]-preimage of function {ge}:{Λerge} = {ge}. That is, if ({ue}, {p∗e}) ∈ W{Ωe} × Y∗

{Ωe} is a solution ofproblem (MMH) then primal function {ue} is a solution of problem (PMH)and, conversely, if {ue} ∈ W{Ωe} is a solution of problem (PMH) then thereis a dual function {p∗e} = {Ce(Λeue − ge)} ∈ Y∗

{Ωe} such that ({ue}, {p∗e})is a solution of problem (MMH). Moreover, problem (PMH) is the macro-hybridization of global primal problem (P); i.e., primal problems (PMH) and(P) are equivalent, and the solvability is unique if in addition the local potentialdual operators are such that

(C [C∗e ]) C∗

e = ∇G∗e ∈ L(Y ∗

e , Ye) is positive definite, e = 1, 2, ..., E.

Remark 3.4 We observe that the local conditions (C [Ge,Λe]) and (C [C∗e ]) are

indeed equivalent to the corresponding global conditions (CG,Λ) and (CC∗) dueto the assumed decomposition properties of spaces and operators (18)-(20).

We are now in a position to conclude the macro-hybrid formulation ofprimal evolution mixed problem (M) and to determine its well-posedness. Bydualization in the sense of Lemma 3.1, we have the next macro-hybrid versionof problem (MMH).

682 G. Alduncin

(MH)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (ue, p∗e) ∈ WΩe × Y∗

Ωe, for e = 1, 2, ..., E :

−ΛTe p∗e − πT

Γeλ∗

e =due

dt+ Aeue − f∗

e , in V∗Ωe

,

Λeue = C∗ep

∗e + ge, in YΩe ,

ue(0) = u0e;

and {λ∗e} ∈ B∗

{Γe} satisfying the dual synchronizing condition

{πΓeue} ∈ ∂IQ∗({λ∗e}), in B{Γe}.

Moreover, the following macro-hybrid composition duality principle holds [5].

Theorem 3.5 Let compatibility conditions (C [Ge ,Λe]) and (C [πΓe ]) be ful-filled, as well as condition (C [C∗

e ]). Then the primal evolution macro-hybridmixed problem (MH) has a unique solution if, and only if, the hybridizedprimal evolution problem (PMH) equivalent to the global primal problem (P)has a unique solution. That is, if ({ue}, {p∗e}, {λ∗

e}) ∈ W{Ωe} × Y∗{Ωe} × B∗

{Γe}is the solution of problem (MH) then primal function {ue} is the solution ofproblem (PMH) and, conversely, if {ue} ∈ W{Ωe} is the solution of problem(PMH) then there are unique dual functions {p∗e} = {∇Ge(Λeue−ge)} ∈ Y∗

{Ωe}and {λ∗

e} ∈ ∂IQ({πΓeue}) ⊂ B∗{Γe} such that ({ue}, {p∗e}, {λ∗

e}) is the solutionof problem (MH).

Proof The principle follows from Theorem 3.3 and Lemma 3.1. The necessityis readily obtained. On the other hand, let ({ue}, {p∗e}) ∈ W{Ωe} × Y∗

{Ωe}be a solution of problem (MMH). Then {w∗

e} = −{ΛTe p∗e} − {due/dt} −

{Aeue} + {f∗e } ∈ ∂(IQ ◦ [πΓe])({ue}). But this means in terms of its vari-

ational inequality, taking variations {ve} = ±{ve0} + {ue}, with {ve0} inthe kernel N ([πΓe]) ⊂ V{Ωe}, that {w∗

e} is an element of the polar sub-space N ([πΓe])

◦ ⊂ V∗{Ωe}. Then, under condition (C [πΓe ]), from the Closed

Range Theorem N ([πΓe ])◦ = R([πΓe]

T ) and, consequently, there is a [πΓe]T -

preimage, {λ∗e} ∈ B∗

{Γe}, such that {πTΓe

λ∗e} = {w∗

e}. Hence, applying Lemma3.1, ({ue}, {p∗e}, {λ∗

e}) is a solution of problem (MH), and by Theorem 3.3the sufficiency of the principle holds. The uniqueness of the macro-hybridvariable {λ∗

e} ∈ B∗{Γe} is a consequence of condition (C [πΓe ]) that implies the

injectivity of the transpose primal trace operators πTΓe

∈ L(B∗({Γe}), V ∗(Ωe)),e = 1, 2, ..., E.

Next, the well-posedness of problem (MH) is concluded as follows (cf.Theorem 2.3). Let mA and mΛT denote the boundedness constants of operatorsA ∈ L(V ,V∗) and ΛT ∈ L(Y∗,V∗). Also, under condition (C [πΓe ]) that isequivalent to the lower boundedness of the transpose primal trace operator


[πTΓe

] ∈ L(B∗{Γe},V∗

{Ωe}) [18], let βπTΓ

be the corresponding lower boundednessconstant.

Theorem 3.6 Let conditions (CG,Λ), (CC∗), (CA+∇(G◦Λ)) and (C [πΓe ]) besatisfied. Then primal evolution macro-hybrid mixed problem (MH) has aunique solution ({ue}, {p∗e}, {λ∗

e}) ∈ W{Ωe}×Y∗{Ωe}×B∗

{Γe} such that estimates(3)−(6) hold in the corresponding product sense, with the macro-hybrid stabilityestimate

‖{λ∗e}‖B∗

{Ωe}≤ 1

βπTΓ

(mΛT mΛ

βC∗+ mA+∇(G◦Λ) + mA

)c2(α)‖u0‖H

+1

βπTΓ

((mΛT mΛ

βC∗+ mA+∇(G◦Λ) + mA

)c3(α) + 1

)‖f∗ + ΛTCg‖V∗

+1

βπTΓ

‖f∗‖V∗ +mΛT

βπTΓβC∗

‖g‖Y .

(24)

Consequently, {ue} ∈ L∞(0, T ; H) and the mapping (u0, f∗ + ΛTCg) ∈ H ×

V∗ �→ ({ue}, {p∗e}, {λ∗e}) ∈ W{Ωe} ×Y∗

{Ωe} × B∗{Γe} is continuous.

Proof Considering Remark 3.4, this result is a corollary of Theorems 3.5,3.3, 2.2 and 2.3. Estimate (24) is obtained from the lower boundedness of thetranspose operator [πT

Γe], equivalent to the surjectivity of primal trace operator

[πΓe], in conjuction with the local primal equations of problem (MH).

Example 3.7 The macro-hybrid mixed variational formulation of the pri-mal evolution mixed diffusion problem (Mdp) of Example 2.7 is written as

(MHdp)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (ue, w∗e) ∈ Wdp;Ωe × Y∗

dp;Ωe, for e = 1, 2, ..., E :

−gradTe we − πT

ΓewnΓe

=due

dt+

dude

dt− f∗

e + γT∂ΩNe

he, in V∗dp;Ωe

,

grade ue = K−1e we − grade u

de, in Ydp;Ωe ,

ue(0) = u0e − ude

(0),

and {wnΓe} ∈ B∗

dp;{Γe} satisfying the dual synchronizing condition

{πΓeue} ∈ ∂IQ∗dp

({wnΓe}), in Bdp;{Γe},

where {wnΓe} corresponds to the internal boundary normal flux synchroniz-

ing field, element of the internal boundary Neumann trace product spaceB∗

dp;{Γe} = L2(0, T ; H−1/2({Γe})), dual of the internal boundary Dirichlet trace

684 G. Alduncin

product space Bdp;{Γe} = L2(0, T ; H1/2({Γe})). Hence, the primal and dualtransmission admissibility subspaces are defined by

Qdp ={{μe} ∈ H1/2({Γe}) : μe = μf a.e. on Γef , 1 ≤ e < f ≤ E

},

Q∗dp =

{{μ∗

e} ∈ H−1/2({Γe}) : 〈{μ∗e}, {μe}〉Bdp({Γe}) = 0,∀{μe} ∈ Qdp

},

(25)one being the polar of the other, establishing as the primal transmission condi-tion the Dirichlet trace continuity across the interfaces, and as the dual trans-mission condition the Neumann trace continuity in the H−1/2-weak sense. Fur-ther, the macro-hybrid compatibility condition (C [πΓe ]) is satisfied due to thesurjectivity of the local Dirichlet trace operators γe ∈ L(H1(Ωe), H

1/2(∂Ωe)),e = 1, 2, ..., E. Therefore, since the local versions of the primal evolutioncompatibility condition (CG,Λ) and the positive definiteness condition (CC∗)are satisfied, from Theorem 3.5 macro-hybrid mixed problem (MHdp) has aunique solution if, and only if, its primal problem (P cd) has a unique solution,which is indeed uniquely solvable.

3.2 The dual evolution macro-hybrid mixed

variational problem

For the dualization of the primal equation of macro-hybridized dual problem(M∗

MH), we observe that its macro-hybrid primal subdifferential [∇Fe]+∂(IQ◦[πΓe]) = ∂([Fe]+IQ◦[πΓe]), since D([Fe])∩V (Ω) �= ∅. Then the required conju-gate for dualization is the functional ([Fe]+IQ◦ [πΓe])

∗ = F ∗|V ∗({Ωe}), the restric-

tion to V ∗({Ωe}) ⊂ V ∗(Ω) of the conjugate primal potential F ∗ : V ∗(Ω) → �.Hence, the macro-hybrid version of the dual compatibility condition (CF∗ ,−ΛT )is given by

(C [F ∗|V ∗(Ωe)

,−ΛTe ])

⎧⎨⎩ intD(F ∗|V ∗(Ωe)

) ∩ R(−ΛTe ) �= ∅, e = 1, 2, ..., E,

⇐⇒ Ae = ∇Fe �= 0, e = 1, 2, ..., E,

and the corresponding compositional result is the following.

Lemma 3.2∗ If condition (C [F ∗|V ∗(Ωe)

,−ΛTe ]) is satisfied, then the local compo-

sitional operator equalities

∇(F ∗|V ∗(Ωe) ◦ (−ΛT

e )) = −Λe∇F ∗|V ∗(Ωe) ◦ (−ΛT

e ), e = 1, 2, ..., E, (23∗)

hold true.

Then, by dualization (see Theorem 2.2∗), the dual composition duality princi-ple of problem (M∗

MH) is obtained.


Theorem 3.3∗ Let condition (C [F ∗|V ∗(Ωe)

,−ΛTe ]) be satisfied. Then macro-

hybridized dual evolution mixed problem (M∗MH) is solvable if, and only if, the

macro-hybrid dual evolution problem

(DMH)

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find {p∗e} ∈ X ∗{Ωe} :

{0e} ={

dp∗edt

}+ {C∗

e p∗e} + ∇(F ∗|V ∗({Ωe}) ◦ [−ΛT

e ])({p∗e + r∗f∗e})

+ {ge}, in Y{Ωe},

{p∗e(0)} = {p∗0e},

is solvable, where r∗f∗e∈ Y{Ωe} is a fixed [−ΛT

e ]-preimage of function {f∗e }:

{−ΛTe r∗f∗

e} = {f∗

e }. That is, if ({ue}, {p∗e}) ∈ V{Ωe} × X ∗{Ωe} is a solution

of problem (M∗MH) then dual function {p∗e} is a solution of problem (DMH)

and, conversely, if {p∗e} ∈ X ∗{Ωe} is a solution of problem (DMH) then there

is a primal function {ue} = ∇F �|V ∗({Ωe})({−ΛTe p∗e + f∗

e }) ∈ V{Ωe} such that({ue}, {p∗e}) is a solution of problem (M∗

MH). Moreover, problem (DMH) isthe macro-hybridization of global dual problem (D); i.e., dual problems (DMH)and (D) are equivalent, and the solvability is unique if in addition the potentialprimal operator satisfies condition (CA).

As a dual version of Remark 3.4, we have the following.

Remark 3.4∗ We note that the local dual condition (C [F ∗|V ∗(Ωe)

,−ΛTe ]) is in

fact equivalent to the global condition (CF∗ ,−ΛT ).

Therefore, the macro-hybrid formulation of the dual evolution mixed prob-lem (M∗) is concluded through macro-hybrid compositional dualization, ac-cording to Lemma 3.1, of the primal transmission constraint.

(MH∗)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (ue, p∗e) ∈ VΩe × X ∗

Ωe, for e = 1, 2, ..., E :

−ΛTe p∗e − πT

Γeλ∗

e = Aeue − f∗e , in V∗

Ωe,

Λeue =dp∗edt

+ C∗e p∗e + ge, in YΩe ,

p∗e(0) = p∗0e;

and {λ∗e} ∈ B∗

{Γe} satisfying the dual synchronizing condition

{πΓeue} ∈ ∂IQ∗({λ∗e}), in B{Γe}.

Moreover, the dual macro-hybrid composition duality principle can then beestablished, reasoning as for Theorem 3.5.

Theorem 3.5∗ Let compatibility conditions (CF∗,−ΛT ) and (C [πΓe ]) be sat-isfied, as well as condition (CA). Then the dual evolution macro-hybrid mixed

686 G. Alduncin

problem (MH∗) has a unique solution if, and only if, the macro-hybridizeddual evolution problem (DMH) equivalent to the global dual problem (D) has aunique solution. That is, if ({ue}, {p∗e}, {λ∗

e}) ∈ V{Ωe} × X ∗{Ωe} × B∗

{Γe} is thesolution of problem (MH∗) then dual function {p∗e} is the solution of problem(DMH) and, conversely, if {p∗e} ∈ X ∗

{Ωe} is the solution of problem (DMH)then there are unique functions {ue} = ∂F ∗

|V ∗({Ωe})({−ΛTe p∗e + f∗

e }) ∈ V{Ωe}and {λ∗

e} ∈ ∂IQ({πΓeue}) ⊂ B∗{Γe} such that ({ue}, {p∗e}, {λ∗

e}) is the solutionof problem (MH∗).

Furthermore, the well-posedness of problem (MH∗) follows from Theo-rems 3.5∗, 3.3∗, 2.2∗ and 2.3∗ (see Theorem 3.6).

Theorem 3.6∗ Let conditions (CF∗,−ΛT ), (CA), (CC∗+∇(F∗◦(−ΛT ))) and(C [πΓe ]) be fulfilled. Then dual evolution macro-hybrid mixed problem (MH∗)has a unique solution ({ue}, {p∗e}, {λ∗

e}) ∈ V{Ωe} × X ∗{Ωe} × B∗

{Γe} such thatestimates (3∗)− (6∗) hold in the corresponding product sense, with the macro-hybrid stability property

‖{λ∗e}‖B∗

{Ωe}≤ 1

βπTΓ

(mΛT +

mAm−ΛT

βA

)c∗2(α

∗)‖p∗0‖Z∗

+1

βπTΓ

(mΛT +

mAm−ΛT

βA

)c∗3(α

∗)‖g − ΛA∗f∗‖V∗ +mA

βπTΓβA

‖f∗‖V∗ .

(24∗)

Consequently, {p∗e} ∈ L∞(0, T ; Z∗) and the mapping (p∗0, g − ΛA∗f∗) ∈ Z∗ ×Y �→ ({ue}, {p∗e}, {λ∗

e}) ∈ V{Ωe} × X ∗{Ωe} ×B∗

{Γe} is continuous.

Example 3.7∗ The macro-hybrid mixed variational formulation of the dualevolution mixed diffusion problem (M∗

dp) of Example 2.7∗ is expressed by

(MH∗dp)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (we, ue) ∈ Vdp∗;Ωe × X ∗dp∗;Ωe

, for e = 1, 2, ..., E :

−divTe ue − πT

ΓewnΓe

= K−1e we + K−1

e whe

− divTe de, in V∗

dp∗;Ωe,

dive we =due

dt− dive w

he− f∗

e , in Ydp∗;Ωe ,

ue(0) = u0e,

and {uΓe} ∈ B∗dp∗;{Γe} satisfying the dual synchronizing condition

{πΓewe} ∈ ∂IQ∗dp

({uΓe}), in Bdp∗;{Γe}.

Here, {uΓe} corresponds to the internal boundary dual synchronizing field thatbelongs to the internal boundary Dirichlet trace product space B∗

dp∗;{Γe} =

L2(0, T ; H1/2({Γe})), dual of the internal boundary Neumann trace product


space Bdp∗;{Γe} = L2(0, T ; H−1/2({Γe})). Hence, the primal and dual trans-mission admissibility subspaces, one the polar of the other, turn out to be

Qdp∗ ={{μ∗

e} ∈ H−1/2({Γe}) : 〈{μ∗e}, {μe}〉Bdp∗({Γe}) = 0,∀{μe} ∈ Q∗

dp∗}

,

Q∗dp∗ =

{{μe} ∈ H1/2({Γe}) : μe = μf a.e. on Γef , 1 ≤ e < f ≤ E

}.

(25∗)

In this dual case, the primal transmission condition is the Neumann trace conti-nuity across the interfaces in the H−1/2-weak sense, and the dual transmissioncondition is the Dirichlet trace continuity (cf. (25)). Also, the correspond-ing macro-hybrid compatibility condition (C [πΓe ]) is satisfied since the localNeumann trace operators of normal fluxes δe ∈ L(H(div; Ωe), H

−1/2(∂Ωe)),e = 1, 2, ..., E, are surjective. Therefore, being the local versions of the dualevolution compatibility condition (C [F ∗

|V ∗(Ωe),−ΛT

e ]) and the primal positive def-

initeness condition (CA) satisfied, according to Theorem 3.5∗ dual macro-hybrid mixed problem (MH∗

dp∗) is uniquely solvable due to the unique solv-ability of its dual problem (Ddp).

4 Semi-Discrete Internal Variational

Approximations

In this section, we proceed to introduce semi-discrete internal variational ap-proximations of the primal and dual evolution macro-hybrid mixed problems(MH) and (MH∗) and, through composition duality principles, we estab-lish their well-posedness. In general, this type of discretizations are of a globalnonconforming character, allowing implementations of local finite elements de-fined on independent subdomain as well as interface meshes, a fundamentalnumerical strategy in computational mechanics for dynamical multisystems.

The semi-discretization process consists in defining a discrete stationaryframework in terms of internal approximations {Vhe}he>0 and {Y ∗

h∗e}h∗

e>0 ofthe product local primal and dual spaces V ({Ωe}) and Y ∗({Ωe}); i.e., fore = 1, 2, ..., E,

∀ve ∈ V (Ωe), ∃ vhe ∈ Vhe ⊂ V (Ωe) : limhe↓0

‖ve − vhe‖V (Ωe) = 0,

∀y∗e ∈ Y ∗(Ωe), ∃ y∗

h∗e∈ Y ∗

h∗e⊂ Y ∗(Ωe) : lim

h∗e↓0

‖y∗e − y∗

h∗e‖Y ∗(Ωe) = 0.

(26)

Similarly, internal approximations {B∗h◦

e}h◦

e>0 of the product macro-hybrid dualspace B∗({Γe}) are defined such that, for e = 1, 2, ...E ,

688 G. Alduncin

∀ζ∗e ∈ B∗(Γe), ∃ ζ∗

h◦e∈ B∗

h◦e⊂ B∗(Γe) : lim

h◦e↓0

‖ζ∗e − ζ∗

h◦e‖B∗(Γe) = 0. (27)

In this manner, the semi-discrete evolution product spaces are definedby V{he} = L2(0, T ; V {he} =

∏E1 Vhe), Y∗

{h∗e} = L2(0, T ; Y ∗

{h∗e} =

∏E1 Y ∗

h∗e)

and B∗{h◦

e} = L2(0, T ; B∗{h◦

e} =∏E

1 B∗h◦

e), with duals V∗

{he} = L2(0, T ; V ∗{he}),

Y{h∗e} = L2(0, T ; Y {h∗

e}) and B{h◦e} = L2(0, T ; B{h◦

e}), and the solution spacesby W{he} = {{vhe} : {vhe} ∈ V{he}, {dvhe/dt} ∈ V∗

{he}} ⊂ C([0, T ]; H({Ωe}))and X ∗

{h∗e} = {{q∗h∗

e} : {q∗h∗

e} ∈ Y∗

{h∗e}, {dq∗h∗

e/dt} ∈ Y{h∗

e}} ⊂ C([0, T ]; Z∗({Ωe})).Then, we associate to evolution problems (MH) and (MH∗) the semi-discrete problems

(MHh,h∗,h◦)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (uhe , p∗h∗

e) ∈ Whe × Y∗

h∗e, for e = 1, 2, ..., E :

−ΛThe,h∗

ep∗h∗

e− πT

Γhe,h◦eλ∗

h◦e

=duhe

dt+ Aheuhe − f∗

he, in V∗

he,

Λhe,h∗euhe = C∗

h∗ep∗h∗

e+ gh∗

e, in Yh∗

e,

uhe(0) = u0he;

and {λ∗h◦

e} ∈ B∗

{h◦e} satisfying the dual synchronizing condition

{πΓhe,h◦euhe} ∈ ∂IQ∗

h◦ ({λ∗h◦

e}), in B{h◦

e},

and

(MH∗h,h∗,h◦)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find (uhe , p∗h∗

e) ∈ Vhe × X ∗

h∗e, for e = 1, 2, ..., E :

−ΛThe,h∗

ep∗h∗

e− πT

Γhe,h◦eλ∗

h◦e

= Aheuhe − f∗he

, in V∗he

,

Λhe,h∗euhe =

dp∗h∗e

dt+ C∗

h∗ep∗h∗

e+ gh∗

e, in Yh∗

e,

p∗h∗e(0) = p∗0h∗

e;

and {λ∗h◦

e} ∈ B∗

{h◦e} satisfying the dual synchronizing condition

{πΓhe,h◦euhe} ∈ ∂IQ∗

h◦ ({λ∗h◦

e}), in B{h◦

e}.

Here, the operators and functionals are the original ones in their restrictiondiscrete sense: for e = 1, 2, ..., E, Ahe ∈ L(Vhe , V

∗he

), C∗h∗

e∈ L(Y ∗

h∗e, Yh∗

e), Λhe,h∗

e∈

L(Vhe , Yh∗e), ΛT

he,h∗e∈ L(Y ∗

h∗e, V ∗

he), πΓhe,h◦e

∈ L(Vhe , Bh◦e), πT

Γhe,h◦e∈ L(B∗

h◦e, V ∗

he),

f∗he

∈ V ∗he

and gh∗e∈ Yh∗

e. Further, {u0he

} ∈ H({Ωe}) and {p∗0h∗e} ∈ Z∗({Ωe})

are convergent discrete initial data; i.e.,

lim{he}↓0 ‖{u0e} − {u0he}‖H = 0,

lim{h∗e}↓0 ‖{p∗0e

} − {p∗0h∗e}‖Z∗ = 0,

(28)


and Q∗h◦ ⊂ B∗

{h◦e} ⊂ B∗({Γe}) is a discrete version of the dual transmission

admissibility subspace Q∗ ⊂ B∗({Γe}), (21), with discrete polar subspacedefined by

Qh◦ ={{ζe} ∈ B{h◦

e} :E∑

e=1

〈ζ∗e , ζe〉B(Γe)= 0, ∀{ζ∗

e} ∈ Q∗h◦

}. (29)

Hence, in such a semi-discrete context, the discrete version of the primalspace V (Ω), characterized in a product sense by (18)1, is

V h,h◦ ={{vhe} ∈ V {he} =

E∏e=1

Vhe : {πΓhe,h◦evhe} ∈ Qh◦

}, (30)

which, in general, is not necessarily a conforming space; i.e., V h,h◦ �⊂ V (Ω).Moreover, the discrete version of the macro-hybrid compatibility condition(C [πΓe ]) is given by

(C [πΓhe,h◦e])

⎧⎨⎩ [πΓhe,h◦e] ∈ L(V {he}, B{h◦

e}) is surjective,

⇐⇒ [πTΓhe,h◦e

] ∈ L(B∗{h◦

e}, V∗{he}) is injective,

where the equivalence is due to the closedness of the finite dimensional range(R([πT

Γhe,h◦e]), with the next dualization result (cf. Lemma 3.1).

Lemma 4.1 Let condition (C [πΓhe,h◦e]) be fulfilled. Then the discrete macro-

hybrid compositional dualization

{πTΓhe,h◦e

λ∗h◦

e} ∈ ∂(IQh◦ ◦ [πΓhe,h◦e

])({uhe}) ⇐⇒ {πΓhe,h◦euhe} ∈ ∂IQ∗

h◦ ({λ∗h◦

e})(31)

holds true.

Remark 4.2 We recall that the local πΓhe,h◦e-surjectivity conditions, e =

1, 2, ..., E, are in general further equivalent to their transpose compatibilityconditions; i.e., for some βπT

Γhe,h◦e> 0, e = 1, 2, ..., E,

(CTπΓhe,h◦e

)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

πTΓhe,h◦e

∈ L(B∗h◦

e, V ∗

he) is bounded below :

‖πThe,h◦

eη∗

h◦e‖V ∗

he= sup

vhe∈Vhe\{0}

〈πTΓhe,h◦e

η∗h◦

e, vhe〉Vhe

‖vhe‖Vhe

≥ βπTΓhe,h◦e

‖η∗h◦

e‖B∗

h◦e

,

∀ η∗h◦

e∈ B∗

h◦e.

Importantly, these conditions can be recognized as the hybrid Ladysenskaja-Babuska-Brezzi inf-sup conditions (cf. [14])

(LBBπΓhe,h◦e) inf

η∗h◦

e∈B∗

h◦e\{0}

supvhe∈Vhe\{0}

〈η∗h◦

e, πΓhe,h◦e

vhe〉Bh◦e

‖η∗h◦

e‖B∗

h◦e

‖vhe‖Vhe

≥ βπTΓhe,h◦e

.

690 G. Alduncin

4.1 Semi-discrete primal composition duality principle

and well-posedness

For the classical dualization of the semi-discrete primal evolution problem(MHh,h∗,h◦), we consider the compatibility condition

(C [Gh∗e,Λhe,h∗e ]) intD(Gh∗

e)∩R(Λhe,h∗

e) �= ∅ ⇐⇒ C∗

h∗e

= ∇G∗h∗

e�= 0, e = 1, 2, ..., E,

which leads to the following compositional result (cf. Lemma 3.2).

Lemma 4.3 Let condition (C [Gh∗e,Λhe,h∗e ]) be satisfied. Then the discrete

compositional operator equalities

∇(Gh∗e◦ Λhe,h∗

e) = ΛT

he,h∗e∇Gh∗

e◦ Λhe,h∗

e, e = 1, 2, ..., E, (32)

are guaranteed.

Then, introducing the condition for the local potential discrete dual operators

(C [C∗h∗

e]) C∗

h∗e

= ∇G∗h∗

e∈ L(Y ∗

h∗e, Yh∗

e) is positive definite, e = 1, 2, ..., E,

and applying Lemmas 4.1 and 4.3, the semi-discrete primal composition dualityprinciple is obtained like in Theorem 3.5.

Theorem 4.4 Let discrete compatibility conditions (C [Gh∗e,Λhe,h∗e ]) and (C

[πΓhe,h◦e]) be fulfilled, as well as condition (C [C∗

h∗e]). Then the semi-discrete pri-

mal evolution macro-hybrid mixed problem (MHh,h∗ ,h◦) has a unique solutionif, and only if, the semi-discrete macro-hybridized primal evolution problem

(PMHh,h∗,h◦ )

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Find {uhe} ∈ W{he} :

{0he} ∈{

duhe

dt

}+ {Aheuhe} + {∇(Gh∗

e◦ Λhe,h∗

e)(uhe − rgh∗

e)}

+ ∂(IQh◦ ◦ [πΓhe,h◦e])({uhe}) − {f∗

he}, in V∗

{he},

{uhe(0)} = {u0he},

has a unique solution, where {rgh∗e} ∈ V{he} is a fixed [Λhe,h∗

e]-preimage of func-

tion {gh∗e}: {Λhe,h∗

ergh∗

e} = {gh∗

e}. That is, if ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈ W{he} ×

Y∗{h∗

e} × B∗{h◦

e} is the solution of problem (MHh,h∗,h◦) then primal function{uhe} is the solution of problem (PMHh,h∗,h◦ ) and, conversely, if {uhe} ∈ W{he}is the solution of problem (PMHh,h∗,h◦ ) then there are unique dual functions{p∗h∗

e} = {∂Gh∗

e(Λhe,h∗

euhe − gh∗

e)} ∈ Y∗

{he} and {λ∗h◦

e} ∈ ∂IQh◦ ({πΓhe,h◦e

uhe}) ⊂B∗

{h◦e} such that ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) is the solution of problem (MHh,h∗ ,h◦).


Remark 4.5 According to the decomposition properties of spaces and op-erators of Section 3, the discrete conditions (C [Gh∗

e,Λhe,h∗e ]) and (C [C∗

h∗e]) are

indeed implied respectively by the continuous local conditions (C [Ge,Λe]) and(C [C∗

e ]), which in turn are equivalent to the global conditions (CG,Λ) and (CC∗)(cf. Remark 3.4).

Furthermore, assuming the discrete primal coercivity condition

(C [Ahe+∇(Gh∗e◦Λhe,h∗e )])

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[Ahe + ∇(Gh∗e◦ Λhe,h∗

e)] ∈ L(V h,h◦ , V ∗

h,h◦) is

V h,h◦ − coercive; i.e., ∃ {αhe,h◦e} > 0 :

E∑e=1

〈(Ahe + ∇(Gh∗e◦ Λhe,h∗

e))vhe , vhe〉Vhe,h◦e

≥E∑

e=1

αhe,h◦e‖vhe‖2

Vhe,h◦e, ∀{vhe} ∈ V h,h◦

⇐⇒E∑

e=1

〈(Ahe + ∇(Gh∗e◦ Λhe,h∗

e))vhe , vhe〉1/2

Vhe,h◦e

is a norm in V h,h◦ ,

the well-posedness of problem (MHh,h∗,h◦) can be concluded (cf. Theo-rems 2.3 and 3.6). Let βC∗

h∗ be the positive definiteness constant of discrete

dual operator [C∗h∗

e] ∈ L(Y∗

{h∗e}, Y{h∗

e}), and βπTΓh,h◦

= min {βπTΓhe,h◦e

: e =

1, 2, ..., E} the lower boundedness constant of the transpose primal trace op-erator [πT

Γhe,h◦e] ∈ L(B∗

{h◦e}, V

∗{he}).

Theorem 4.6 Let continuous conditions (CG,Λ), (CC∗), and discrete con-ditions (C [πΓhe,h◦e

]) and (C [Ahe+∇(Gh∗e◦Λhe,h∗e )]) be satisfied. Then semi-discrete

primal evolution macro-hybrid mixed problem (MHh,h∗,h◦) has a unique solu-tion ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈ W{he} ×Y∗

{h∗e} × B∗

{h◦e} such that

‖{uhe(τ )}‖H ≤ ‖{u0he}‖H + c1

{ ∫ τ

0‖f∗(t) + ΛTCg(t)‖2

V ∗ dt}1/2

, ∀τ ∈ [0, T ],

(33)

‖{uhe}‖V ≤ c2(αh,h◦)‖{u0he}‖H + c3(αh,h◦)‖f∗ + ΛTCg‖V∗, (34)

∥∥∥∥{duhe

dt

}∥∥∥∥V∗≤ mA+∇(G◦Λ)c2(αh,h◦)‖u0‖H

+ (mA+∇(G◦Λ)c3(αh,h◦) + 1)‖f∗ + ΛTCg‖V∗,(35)

692 G. Alduncin

‖{p∗h∗e}‖Y∗ ≤ mΛc2(αh∗ ,h◦)

βC∗h∗

‖{u0he}‖H +

mΛc3(αh,h◦)

βC∗h∗

‖f∗+ΛTCg‖V∗+1

βC∗h∗

‖g‖Y ,

(36)and

‖{λ∗h◦

e}‖B∗

{Ωe}≤ 1

βπTΓh,h◦

(mΛT mΛ

βC∗h∗

+ mA+∇(G◦Λ) + mA

)c2(αh∗,h◦)‖{u0he

}‖H

+1

βπTΓh,h◦

((mΛT mΛ

βC∗h∗

+ mA+∇(G◦Λ) + mA

)c3(αh∗ ,h◦) + 1

)‖f∗ + ΛTCg‖V∗

+1

βπTΓh,h◦

‖f∗‖V∗ +mΛT

βπTΓh,h◦

βC∗h∗

‖g‖Y ,

(37)where αh,h◦ = min {αhe,h◦

e: e = 1, 2, ..., E}. Consequently, {uhe} ∈ L∞(0, T ;

H) and the mapping (u0, f∗ + ΛT Cg) ∈ H × V∗ �→ ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈

W{he} × Y∗{h∗

e} ×B∗{h◦

e} is continuous.

Remark 4.7 By the usual monotone weak compactness arguments in re-flexive Banach spaces, the above boundedness results of Theorem 4.6 guaran-tee the weak convergence of the sequence ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) to the solution

({ue}, {p∗e}, {λ∗e}) of problem (MH) in W ×Y∗ × B∗ as {he, h

∗e, h

◦e} ↓ 0.

Remark 4.8 In the numerical resolution of semi-discrete problem (MHh,h∗ ,h◦), upon the application of time marching schemes for full discretization(cf. [5]), the well-posedness of its local primal stationary mixed systems, fore = 1, 2, ..., E,

(M he,h∗e ,h◦

e)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find (uhe , p

∗h∗

e) ∈ Vhe × Y ∗

h∗e

:

−ΛThe,h∗

ep∗h∗

e− πT

Γhe,h◦eλ∗

h◦e

= Aheuhe − f∗he

, in V ∗he

,


h∗ep∗h∗

e+ gh∗

e, in Yh∗

e,

has to be guaranteed. In this manner, the macro-hybrid variational structureof the problem permits the definition of parallel algorithms implementable interms of finite element schemes. As observed in Remark 2.6, a dual compositionduality principle can be established for qualitative analysis via the numericaldual compatibility condition

(CΛThe,h∗e

)

{ΛT

he,h∗e∈ L(Y ∗

h∗e, V ∗

he) is surjective

⇐⇒ Λhe,h∗e∈ L(Vhe , Yh∗

e) is injective,


which is equivalent to the transpose condition

(CTΛT

he,h∗e)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Λhe,h∗e∈ L(Vhe , Yh∗

e) is bounded below; i.e., ∃ βΛhe,h∗e

> 0 :

‖Λhe,h∗evhe‖Yh∗

e= sup

q∗h∗

e∈Y ∗

h∗e\{0∗

h∗e}

〈Λhe,h∗evhe, q

∗h∗

e〉Y ∗

h∗e

‖q∗h∗e‖Y ∗

h∗e

≥ βΛhe,h∗e‖vhe‖Vhe

,

∀vhe ∈ Vhe .

Again, note that the equivalence in condition (CΛThe,h∗e

) is due to the closeness

of the finite dimensional range R(Λhe,h∗e). Furthermore, condition (CΛT

he,h∗e),

in fact, corresponds to the primal mixed Ladysenskaja-Babuska-Brezzi inf-supcondition [7] (cf. Remark 4.2)

(LBBΛThe,h∗e

) infvhe∈Vhe\{0he}

supq∗h∗

e∈Y ∗

h∗e\{0∗

h∗e}

〈vhe , ΛThe,h∗

eq∗h∗

e〉Y ∗

h∗e

‖vhe‖Vhe‖q∗h∗

e‖Y ∗

h∗e

≥ βΛhe,h∗e.

Then, by dualization and compositional dualization (cf. Remark 2.6), it canbe concluded that under condition (CΛT

he,h∗e) the discrete primal stationary

mixed problem (M he,h∗e ,h◦

e) is uniquely solvable if, and only if, its discrete dual

stationary problem

(Dhe,h∗e ,h◦

e)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find p∗h∗

e∈ Y ∗

h∗e

:

0 ∈ C∗h∗

ep∗h∗

e+ ∂(F ∗

he◦ (−ΛT

he,h∗e))(p∗h∗

e+ r∗

f∗he

+πTΓhe,h◦e

λ∗h◦

e

) + gh∗e,

in Yh∗e,

is uniquely solvable. Here F ∗he

: V ∗he

→ �∪{+∞} is the conjugate of the primalquadratic potential Fhe , and r∗

f∗he

+πTΓhe,h◦e

λ∗h◦

e

∈ Y ∗h∗

eis a fixed −ΛT

he,h∗e-preimage

of function f∗he

+ πTΓhe,h◦e

λ∗h◦

e.

4.2 Semi-discrete dual composition duality principle and

well-posedness

Next, for the semi-discrete dual problem (MH∗h,h∗,h◦), we introduce the clas-

sical dual compatibility condition

(C [F ∗he |V ∗

he

,−ΛThe,h∗e

])

⎧⎨⎩ intD(F ∗he |V ∗

he

) ∩R(−ΛThe,h∗

e) �= ∅, e = 1, 2, ..., E,

⇐⇒ Ahe = ∇Fhe �= 0, e = 1, 2, ..., E,

with the following compositional result.

694 G. Alduncin

Lemma 4.3∗ Let condition (C [F ∗he |V ∗

he

,−ΛThe,h∗e

]) be satisfied. Then the discrete

compositional operator equalities

∇(F ∗he |V ∗

he

◦ −ΛThe,h∗

e) = −Λhe,h∗

e∇F ∗

he |V ∗he

◦ −ΛThe,h∗

e, e = 1, 2, ..., E, (32∗)

are guaranteed.

Here, [F ∗he

]|V ∗{he}

= ([Fhe] + IQ∗h◦ ◦ [πhe,h◦

e])∗ is the restriction of the conjugate of

primal superpotential [Fhe] : V h,h◦ → � to V ∗{he} ⊂ V ∗

h,h◦ . Then, introducingthe condition for the local potential discrete primal operators

(C [Ahe ]) [Ahe] = [∇Fhe] ∈ L(V h,h◦ , V ∗h,h◦) is positive definite,

and applying Lemmas 4.1 and 4.3∗, the semi-discrete dual composition dualityprinciple can be established (cf. Theorems 2.2∗, 3.3∗ and 3.5∗).

Theorem 4.4∗ Let discrete compatibility conditions (C [F ∗he |V ∗

he

,−ΛThe,h∗e

]) and

(C [πΓhe,h◦e]) be satisfied, as well as condition (C [Ahe ]). Then the semi-discrete

dual evolution macro-hybrid mixed problem (MH∗h,h∗ ,h◦) has a unique solution

if, and only if, the semi-discrete macro-hybridized dual evolution problem

(DMHh,h∗,h◦ )

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

Find {p∗h∗e} ∈ X ∗

{h∗e} :

{0h∗e} =

{dph∗

e

dt

}+ {C∗

h∗ep∗h∗

e}

+ ∇([F ∗he

]|V ∗{he}

◦ [−ΛThe,h∗

e])({p∗h∗

e+ r∗f∗

he}) + {gh∗

e}, in Y{h∗

e},

{p∗h∗e(0)} = {p∗0h∗

e},

has a unique solution, where {r∗f∗he} ∈ Y∗

{h∗e} is a fixed [−ΛT

he,h∗e]-preimage of

function {f∗he}: {−ΛT

he,h∗e

r∗f∗he} = {f∗

he}. That is, if ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈

V{he}×X ∗{h∗

e}×B∗{h◦

e} is the solution of problem (MH∗h,h∗,h◦) then dual function

{p∗h∗e} is the solution of problem (DMHh,h∗,h◦ ) and, conversely, if {p∗h∗

e} ∈ X ∗

{h∗e}

is the solution of problem (DMHh,h∗,h◦ ) then there are unique primal function{uhe} = ∇[F ∗

he]|V ∗

{he}({−ΛT

he,h∗e

p∗h∗e

+ f∗he}) ∈ V{he} and dual function {λ∗

h◦e} ∈

∂IQh◦ ({πhe,h◦euhe}) ⊂ B∗

{h◦e} such that ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) is the solution of

problem (MH∗h,h∗ ,h◦).

Remark 4.5∗. We observe that the discrete dual condition (C [F ∗he |V ∗

he

,−ΛThe,h∗e

])

is in fact implied by the continuous local condition (C [F ∗|V ∗(Ωe)

,−ΛTe ]), equivalent

to the global condition (CF∗,−ΛT ) (cf. Remark 3.4∗).

Moreover, under the discrete dual coercivity condition


(C [C∗h∗

e]+∇([F∗

he]|V ∗

{he}◦[−ΛT

he,h∗e]))

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[C∗h∗

e] + ∇([F ∗

he]|V ∗

{he}◦ [−ΛT

he,h∗e]) ∈ L(Y ∗

{h∗e},

Y {h∗e}) is Y ∗

{h∗e} − coercive; i.e., ∃ α∗

t h∗ > 0 :

〈[C∗h∗

e] + ∇([F ∗

he]|V ∗

{he}◦ [−ΛT

he,h∗e])y∗

{h∗e},

y∗{h∗

e}〉Y ∗{h∗

e}

≥ α∗h∗‖y∗

{h∗e}‖2

Y ∗{h∗

e}, ∀y∗

{h∗e} ∈ Y ∗

{h∗e},

⇐⇒〈[C∗

h∗e] + ∇([F ∗

he]|V ∗

{he}◦ [−ΛT

he,h∗e])y∗

{h∗e},

y∗{h∗

e}〉1/2Y{h∗

e}is a norm in Y ∗

{h∗e},

the well-posedness of problem (MH∗h,h∗ ,h◦) can be concluded (cf. Theorems

2.3∗ and 3.6∗). Let βAhdenote the positive definiteness constant of discrete

primal operator [Ahe] = [∇Fhe].

Theorem 4.6∗. Let continuous condition (CF∗ ,−ΛT ), and discrete condi-tions (C [πΓhe,h◦e

]), (C [Ahe ]) and (C [C∗h∗

e]+∇([F∗

he]|V ∗

{he}◦[−ΛT

he,h∗e])) be fulfilled. Then

semi-discrete dual evolution macro-hybrid mixed problem (MH∗h,h∗,h◦) has a

unique solution ({uhe}, {p∗h∗e}, {λ∗

h◦e}) ∈ V{he} × X ∗

{h∗e} × B∗

{h◦e} such that

‖{p∗h∗e(τ )}‖Z∗ ≤ ‖{p∗0h∗

e}‖Z∗ + c∗1

{∫ τ

0‖g(t) − ΛA∗f∗(t)‖2

Y dt}1/2

, ∀τ ∈ [0, T ],

(33∗)

‖{p∗h∗e}‖Y∗ ≤ c∗2(α

∗h∗)‖{p∗0h∗

e}‖Z∗ + c∗3(α

∗h∗)‖g − ΛA∗f∗‖Y , (34∗)

∥∥∥∥{dp∗h∗e

dt

}∥∥∥∥Y ≤ mC∗+∇(F∗◦(−ΛT ))c∗2(α

∗h∗)‖{p∗0h∗

e}‖Z∗

+ (mC∗+∇(F∗◦(−ΛT ))c∗3(α

∗h∗) + 1)‖g − ΛA∗f∗‖Y ,

(35∗)

‖{uhe}‖V ≤ m−ΛT c∗2(α∗h∗)

βAh

‖{p∗0h∗e}‖Z∗ +

m−ΛT c∗3(α∗h∗)

βAh

‖g − ΛA∗f∗‖Y

+1

βAh

‖f∗‖V∗,(36∗)

and

‖{λ∗h◦

e}‖B∗

{Ωe}≤ 1

βπTΓh,h◦

(mΛT +

mAm−ΛT

βAh

)c∗2(α

∗h∗)‖p∗0‖Z∗

+1

βπTΓh,h◦

(mΛT +

mAm−ΛT

βAh

)c∗3(α

∗h∗)‖g − ΛA∗f∗‖V∗ +

mA

βπTΓh,h◦

βAh

‖f∗‖V∗ .

696 G. Alduncin

(37∗)

Consequently, {p∗h∗e} ∈ L∞(0, T ; Z∗) and the mapping (p∗0, g − ΛA∗f∗) ∈ Z∗ ×

Y �→ ({uhe}, {p∗h∗e}, {λ∗

h◦e}) ∈ V{he} × X ∗

{h∗e} ×B∗

{h◦e} is continuous.

Remark 4.7∗. As in Remark 4.7, we note that due to the boundednessresults of Theorem 4.6∗ the sequence ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) converges weakly to

the solution ({ue}, {p∗e}, {λ∗e}) of dual problem (MH)∗ in V × X ∗ × B∗ as

{he, h∗e, h

◦e} ↓ 0.

Remark 4.8∗. Relative to fully discrete versions of semi-discrete problem(MH∗

h,h∗,h◦) (cf. [5]), in contrast with the primal case of Remark 4.8, thewell-posedness of the local stationary mixed systems, for e = 1, 2, ..., E,

(M ∗he,h∗

e ,h◦e)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find (uhe , p

∗h∗

e) ∈ Vhe × Y ∗

h∗e

:

−ΛThe,h∗

ep∗h∗

e− πT

Γhe,h◦eλ∗

h◦e

= Aheuhe − f∗he

, in V ∗he

,


h∗ep∗h∗

e+ gh∗

e, in Yh∗

e,

has to be guaranteed, but regarded now in a dual sense. Thus, for composi-tional dualization, the numerical primal compatibility condition

(CΛhe,h∗e)

{Λhe,h∗

e∈ L(Vhe , Yh∗

e) is surjective

⇐⇒ ΛThe,h∗

e∈ L(Y ∗

h∗e, V ∗

he) is injective,

is imposed, being equivalent to the transpose condition

(CTΛhe,h∗e

)

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ΛThe,h∗

e∈ L(Y ∗

h∗e, V ∗

he) is bounded below; i.e., ∃ β∗

ΛThe,h∗e

> 0 :

‖ΛThe,h∗

eq∗h∗

e‖V ∗

he= sup

vhe∈Vhe\{0he}

〈ΛThe,h∗

eq∗h∗

e, vhe〉Vhe

‖vhe‖Vhe

≥ β∗ΛT

he,h∗e‖q∗h∗

e‖Y ∗

h∗e

,

∀q∗h∗e∈ Y ∗

h∗e,

and which in turn corresponds to the dual mixed Ladysenskaja-Babuska-Brezziinf-sup condition [7]

(LBBΛhe,h∗e) inf

q∗h∗

e∈Y ∗

h∗e\{0∗

h∗e}

supvhe∈Vhe\{0he}

〈q∗h∗e, Λhe,h∗

evhe〉Yh∗

e

‖q∗h∗e‖Y ∗

h∗e‖vhe‖Vhe

≥ β∗ΛT

he,h∗e.

Then, a dual composition duality principle can be established under condi-tion (CΛhe,h∗e

) (cf. Remark 2.6∗). That is, the discrete dual stationary mixed

problem (M∗he,h∗

e ,h◦e) has a unique solution if, and only if, its discrete primal

stationary problem


(P he,h∗e ,h◦

e)

⎧⎪⎪⎪⎨⎪⎪⎪⎩Find uhe ∈ Vhe :

0V ∗he

∈ Aheuhe + ∂(Gh∗e◦ Λhe,h∗

e)(uhe − rgh∗

e) − f∗

he+ πT

Γhe,h◦eλ∗

h◦e,

in V ∗he

,

has a unique solution, where Gh∗e

: Yh∗e→ �∪{+∞} is the conjugate of the dual

quadratic potential G∗h∗

e, and rgh∗

e∈ Vhe is a fixed Λhe,h∗

e-preimage of function

gh∗e.

5 Strong Convergence Analysis

We next continue to construct a priori error estimates from which the strongconvergence of the primal and dual semi-discrete processes follow. Toward thisend, we shall assume the discrete conforming condition

(CVh,h◦ ) Qh◦ ⊂ Q ⇐⇒ V h,h◦ ⊂ V (Ω).

That is, the family of discrete primal spaces V h,h◦ , defined in (30), conforms tointernal approximations of the primal space V (Ω). Let the evolution discreteprimal space be denoted by Vh,h◦ = L2(0, T ; V h,h◦).

5.1 Primal strong convergence

Theorem 5.1 Let the continuous and discrete conditions of Theorems 3.6and 4.6, (CG,Λ), (CC∗), (CA+∇(G◦Λ)), (C [πΓe ]) and (C [πΓhe,h◦e

]), (C [Ahe+∇(Gh∗e

◦Λhe,h∗e )]) be satisfied, as well as conforming condition (CVh,h◦ ). Let ({ue}, {p∗e},{λ∗

e}) ∈ W{Ωe}×Y∗{Ωe}×B∗

{Γe} be the solution of primal evolution macro-hybridmixed problem (MH), and ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈ W{he} ×Y∗

{h∗e}×B∗

{h◦e} the

solution of the corresponding semi-discrete problem (MHh,h∗,h◦). Then, as{he, h

∗e, h

◦e} ↓ 0,

{uhe} −→ {ue} strongly in L∞(0, T ; H),

{uhe} −→ {ue} strongly in V ,

{p∗h∗e} −→ {p∗e} strongly in Y∗,

{λ∗h◦

e} −→ {λ∗

e} strongly in B∗.

(38)

Moreover, the following primal error estimates hold, ∀{whe} ∈ Vh,h◦ ∩W:

‖{ue(τ )} − {uhe(τ )}‖H ≤ ‖{u0he} − {whe(0)}‖H + ‖{ue(τ )} − {whe(τ )}‖H

+ c1‖{ue} − {whe}‖V + c2

∥∥∥∥{due

dt

}−

{dwhe

dt

}∥∥∥∥V∗, ∀τ ∈ [0, T ],

(39)

698 G. Alduncin

‖{ue} − {uhe}‖V ≤ c3‖{u0he} − {whe(0)}‖H + c4‖{ue} − {whe}‖V

+ c5

∥∥∥∥{due

dt

}−

{dwhe

dt

}∥∥∥∥V∗,

(40)

‖{p∗e} − {p∗h∗e}‖Y∗ ≤ mΛ

βC∗

[c3‖{u0he

} − {whe(0)}‖H + c4‖{ue} − {whe}‖V

+ c5

∥∥∥∥{due

dt

}−

{dwhe

dt

}∥∥∥∥V∗

],

(41)

‖{λ∗e − λ∗

h◦e}‖B∗

≤ 1

βπTΓ

(mA +

mΛT mΛ

βC∗

)[c3‖{u0he

} − {whe(0)}‖H + c4‖{ue} − {whe}‖V]

+1

βπTΓ

((mA +

mΛT mΛ

βC∗

)c5 + 1

)∥∥∥∥{due

dt

}−{

dwhe

dt

}∥∥∥∥V∗,

(42)where c1 = c1(mA+∇(G◦Λ)), c2, c3 = c3(α), c4 = c4(α, mA+∇(G◦Λ)) and c5 =c5(α) are strictly positive constants.

Proof Let us denote the primal operator by A = A+∇(G◦Λ) ∈ L(V (Ω), V ∗(Ω)).From integration by parts formula (7) and coercivity condition (CA+∇(G◦Λ)),it follows that∫ τ

0

⟨dv

dt(t) − dw

dt(t), v(t)− w(t)

⟩V

dt +∫ τ

0〈Av(t) − Aw(t), v(t) − w(t)〉V dt

≥ 1

2‖v(τ )− w(τ )‖2

H − 1

2‖v(0) −w(0)‖2

H + α∫ τ

0‖v(t)− w(t)‖2

V dt,

∀τ ∈ [0, T ], ∀v, w ∈ W,

and, under conforming condition (CVh,h◦ ), from the equations of macro-hybridprimal problems (PMH) and (PMHh,h∗,h◦ ) the following orthogonality formulaholds, ∫ τ

0

⟨{due

dt

}(t) −

{duhe

dt

}(t), {vhe}(t)

⟩V

dt

+∫ τ

0〈Aeue}(t) − {Aeuhe}(t), {vhe}(t)〉V dt = 0,

∀τ ∈ [0, T ], ∀{vhe} ∈ Vh,h◦ ⊂ V .

(43)

Hence, from these two results and applying Young’s inequality, with constantb > 0, we have, ∀τ ∈ [0, T ] and ∀{whe} ∈ Vh,h◦ ∩W,


1

2‖{uhe}(τ ) − {whe}(τ )‖2

H − 1

2‖{u0he} − {whe}(0)‖2

H

+α∫ τ

0‖{uhe}(t) − {whe}(t)‖2

V dt

≤∫ τ

0

⟨{due

dt

}(t) −

{dwhe

dt

}(t), {uhe}(t) − {whe}(t)

⟩V

dt

+∫ τ

0〈{Aeue}(t) − {Aewhe}(t), {uhe}(t) − {whe}(t)〉V dt

≤ 1

2b2

[ ∫ τ

0

∥∥∥∥{due

dt

}(t) −

{dwhe

dt

}(t)

∥∥∥∥2

V ∗dt +

∫ τ

0‖{Aeue}(t)

−{Aewhe}(t)‖2V ∗ dt

]+

b2

2

∫ τ

0‖{uhe}(t) − {whe}(t)‖2

V dt.

Therefore, choosing b > 0 such that α = α− b2/2 > 0, we obtain the followingbasic estimates ∀{whe} ∈ Vh,h◦ ∩W:

‖{ue}(τ ) − {uhe}(τ )‖H ≤ ‖{ue}(τ ) − {whe}(τ )‖H + ‖{u0he} − {whe}(0)‖H

+1

b

[∥∥∥∥{due

dt

}−

{dwhe

dt

}∥∥∥∥V∗+ ‖{Aeue} − {Aewhe}‖V∗

], ∀τ ∈ [0, T ],

(44)

‖{ue} − {uhe}‖V ≤ ‖{ue} − {whe}‖V +(

1

2α

)1/2

‖{u0he} − {whe}(0)‖H

+(

1

2αb2

)1/2[∥∥∥∥{due

dt

}−

{dwhe

dt

}∥∥∥∥V∗+ ‖{Aeue} − {Aewhe}‖V∗

].

(45)Now, taking into account that Vh,h◦ ∩ W is dense in W, as {h, h◦} ↓ 0, aswell as convergent discrete initial condition (28)1, the strong convergence of{uhe} to {ue} in L∞(0, T ; H) and V follows from these estimates, (44) and(45), due to the continuity of the primal operator A ∈ L(V ,V∗) as well as thecontinuity of the embedding W ⊂ C([0, T ]; H). Further, estimates (39) and(40) are concluded from the boundedness of the operator A, with subdifferen-tial ∂F ∗

he: V ∗

he→ 2Vhe . Then, estimates (41) and (42) follow respectively from

the dual and primal equations of the semi-discrete and continuous problems,(MHh,h∗,h◦) and (MH), utilizing property (CC∗) as well as the Λ-, A- andΛT -boundedness, and the πT

Γ -lower boundedness; the dual strong convergence(38)3,4 then holds true.

5.2 Dual strong convergence

Similarly, the strong convergence of the dual semi-discrete problem (MH∗h,h∗,h◦)

to the dual evolution macro-hybrid mixed problem (MH∗) can be demon-

700 G. Alduncin

strated, assuming in addition the primal coupling condition

(CΛ) Λ ∈ L(V, Y ) is bounded below,

with corresponding bounded below constant βΛ > 0. Note that in this dualcase the discrete conforming condition (CVh,h◦ ) is still a required conditionfor making consistent the continuous and discrete dual potential restrictionsF ∗|V ∗({Ωe}) and F ∗

he]|V ∗

{he}of dual problems (DMH) and (DMHh,h∗,h◦ ), since then

the restricted domains V ∗({Ωe}) and V ∗{he} are both subspaces of the dual V ∗

h,h◦

of the discrete primal space V h,h◦ ⊂ V (Ω).

Theorem 5.1∗ Let the continuous and discrete conditions of Theorems3.6∗ and 4.6∗, (CF∗,−ΛT ), (CA), (CC∗+∇(F∗◦(−ΛT ))), (C [πΓe ]) and (C [πΓhe,h◦e

]),

(C [C∗h∗

e]+∇([F∗

he]|V ∗

{he}◦[−ΛT

he,h∗e])) be fulfilled, as well as conforming condition (C

Vh,h◦ ) and bounded below condition (CΛ). Let ({ue}, {p∗e}, {λ∗e}) ∈ V{Ωe} ×

X ∗{Ωe} × B∗

{Γe} be the solution of dual evolution macro-hybrid mixed problem(MH∗), and ({uhe}, {p∗h∗

e}, {λ∗

h◦e}) ∈ V{he}×X ∗

{h∗e}×B∗

{h◦e} the solution of the

corresponding semi-discrete problem (MH∗h,h∗ ,h◦). Then, as {he, h

∗e, h

◦e} ↓ 0,

{p∗h∗e} −→ {p∗e} strongly in L∞(0, T ; Z∗),

{p∗h∗e} −→ {p∗e} strongly in Y∗,

{uhe} −→ {ue} strongly in V ,

{λ∗h◦

e} −→ {λ∗

e} strongly in B∗.

(38∗)

Moreover, the following dual error estimates hold, ∀{s∗h∗e} ∈ X ∗:

‖{p∗e(τ )} − {p∗h∗e(τ )}‖Z∗ ≤ ‖{p∗0h∗

e} − {s∗h∗

e(0)}‖Z∗ + ‖{p∗e(τ )} − {s∗h∗

e(τ )}‖Z∗

+ c∗1‖{p∗e} − {s∗h∗e}‖Y∗ + c∗2

∥∥∥∥{dp∗edt

}−

{ds∗h∗e

dt

}∥∥∥∥Y , ∀τ ∈ [0, T ],

(39∗)

‖{p∗e} − {p∗h∗e}‖Y∗ ≤ c∗3‖{p∗0h∗

e} − {s∗h∗

e(0)}‖Z∗

+ c∗4‖{p∗e} − {s∗h∗e}‖Y∗ + c∗5


}−

{dsh∗

e

dt

}∥∥∥∥Y ,(40∗)

‖{ue} − {uhe}‖V ≤ mC∗

βΛ

[c∗3‖{p∗0h∗

e} − {s∗h∗

e(0)}‖Z∗

+ c∗4‖{p∗e} − {s∗h∗e}‖Y∗

]+

1

βΛ

(mC∗ c∗5 + 1

)∥∥∥∥{dp∗edt

}−{ds∗h∗

e

dt

}∥∥∥∥Y ,

(41∗)


‖{λ∗e − λ∗

h◦e}‖B∗ ≤ 1

βπTΓ

(mΛT +

mA mC∗

βΛ

)[c∗3‖{p∗0h∗

e} − {sh∗

e(0)}‖Z∗

+c∗4‖{p∗e} − {s∗h∗e}‖Y∗

]+

1

βπTΓ

(mΛT c∗5 +

mA

βΛ

(mC∗ c∗5 + 1))


}−

{dsh∗

e

dt

}∥∥∥Y ,

(42∗)

where c∗1 = c∗1(mC∗+∇(F∗◦(−ΛT ))), c∗2, c∗3 = c∗3(α∗), c∗4 = c∗4(α

∗, mC∗+∇(F∗◦(−ΛT )))and c∗5 = c∗5(α

∗) are strictly positive constants.

It is important to observe that the primal and dual a priori error esti-mates (39)-(42) and (39∗)-(42∗) may not be suitable for determining rates ofconvergence of corresponding finite element implementations, because of thepresence of the time derivative error terms ‖{due/dt} − {dwhe/dt}‖V∗ and‖{dp∗e/dt} − {dsh∗

e/dt}‖Y , respectively. Appropriate estimates will be derived

below, once some regularity in time is established.

6 Regularity in Time Analysis

In this final section, taking advantage of the potentiality and coercivity proper-ties of the primal and dual evolution macro-hybrid mixed problems, we performa regularity in time analysis that permits to construct a priori error estimates,which may provide rates of convergence of finite element implementations.

6.1 Primal regularity in time

Let ({ue}, {p∗e}, {λ∗e}) ∈ W{Ωe} × Y∗

{Ωe} × B∗{Γe} be the solution of primal

evolution problem (MH), and ({uhe}, {p∗h∗e}, {λ∗

h◦e}) ∈ W{he} ×Y∗

{h∗e} ×B∗

{h◦e}

the solution of the corresponding semi-discrete problem (MHh,h∗ ,h◦).

Theorem 6.1 Let the conditions of Theorem 5.1 be fulfilled. Then, forprimal data (f∗ + ΛTCg, u0) ∈ V∗ × H as regular as

f∗ + ΛT Cg ∈ L2(0, T ; H) and u0 ∈ V, (46)

and assuming V -convergence of the discrete initial data,

lim{he}↓0

‖{u0e} − {u0he}‖V = 0, (47)

the primal component solution {ue} ∈ L∞(0, T ; H) ∩ W{Ωe} of the primalevolution macro-hybrid mixed problem (MH) satisfies the following regularityin time:

702 G. Alduncin

{ue} ∈ L∞(0, T ; V ({Ωe})),{uhe} →∗ {ue} weakly star in L∞(0, T ; V ({Ωe})),

(48)

and

{due

dt

}∈ L2(0, T ; H({Ωe})),{

duhe

dt

}→

{due

dt

}weakly in L2(0, T ; H({Ωe})).

(49)

Moreover, the following primal error estimates hold: ∀{whe} ∈ Vh,h∗ ⊂ V,

‖{ue(τ )}−{uhe(τ )}‖H ≤ ‖{u0e}−{u0he}‖H +c1‖{ue}−{whe}‖1/2

V , ∀τ ∈ [0, T ],(50)

‖{ue} − {uhe}‖V ≤(

1

2α

)1/2

‖{u0e} − {u0he}‖H + c2‖{ue} − {whe}‖1/2

V , (51)

‖{p∗e} − {p∗h∗e}‖Y∗

≤ mΛ

βC∗

[(1

2α

)1/2

‖{u0e} − {u0he}‖H + c2‖{ue} − {whe}‖1/2

V

],

(52)

‖{λ∗e − λ∗

h◦e}‖B∗ ≤ 1

βπTΓ

(mA +

mΛT mΛ

βC∗

)[(

1

2α

)1/2

‖{u0e} − {u0he}‖H + c2‖{ue} − {whe}‖1/2

V

],

(53)

where c1 = c1(mA+∇(G◦Λ), α, βJ(μ0), f∗ + ΛT Cg) and c2 = c2(mA+∇(G◦Λ), α,

βJ(μ0), f∗ +ΛT Cg) are strictly positive constants, μ0 being a bound for {u0he

}.

In order to established this regularity result, we first observe that at thesemi-discrete level, the primal solution {uhe} ∈ W{he} of problem (PMHh,h∗,h◦ ),with internal boundary traces {πhe,h◦

euhe} ∈ Qh◦ , is indeed an absolutely con-

tinuous vector function [10], whose strong time derivative that exists a.e. on(0, T ), clearly belongs to L2(0, T ; V {he}) since in this regular case f∗+ΛT Cg ∈L2(0, T ; H). Then the primal semi-discrete solution is such that

{uhe} ∈ CA(0, T ; V {he}) ⊂ L∞(0, T ; V {he}),{duhe

dt

}∈ L2(0, T ; V {he}).

(54)


Further, considering the potentiality of the primal discrete operator [Ahe,h∗e] =

[Ahe + ∇(Ghe ◦ Λhe,h∗e)] ∈ L(V h,h◦ , V ∗

h,h◦), we have the relation

d

dtJ

A(uhe(t)) =

⟨{Ahe,h∗

euhe}(t),

{duhe

dt

}(t)

⟩V, for a.e. t ∈ [0, T ], (55)

where JA

: V → � is the primal Frechet differentiable potential of the primal

operator A = A + ∇(G ◦ Λ) ∈ L(V, V ∗), defined uniquely modulo a constantby J

A(v) =

∫ 10 〈A(sv), v〉V ds, v ∈ V , which due to the coercivity condition

(CA+∇(G◦Λ)) and boundedness of A, turns out to be bounded in the sense (see,in the setting of general monotone operators, [17])

α

2‖v‖2

V ≤ JA(v) ≤ βJ

A(‖v‖V ) = mA+∇(G◦Λ)

∫ 1

0(s‖v‖V ) ds‖v‖V , ∀v ∈ V. (56)

Lemma 6.2 Under the conditions of Theorem 6.1, the sequences of pri-mal semi-discrete approximations {uhe} and {duhe/dt} of (54) are boundedin L∞(0, T ; V ) and L2(0, T ; H), respectively. In fact, the following a prioriestimates hold true:

‖{uhe(τ )}‖V ≤{

2

αβJ

A(‖{u0he

}‖V )}1/2

+{

1

α

∫ τ

0‖f∗(t) + ΛT Cg(t)‖2

H dt}1/2

, ∀τ ∈ [0, T ],

(57)

∥∥∥∥{duhe

dt

}∥∥∥∥L2(0,T ;H)

≤{2βJ

A(‖{u0he

}‖V )}1/2

+ ‖f∗ + ΛTCg‖L2(0,T ;H). (58)

Proof By virtue of (46)1 and (54), and taking as variation the time derivativevector {duhe/dt}, the variational equality of primal problem (PMHh,h∗,h◦ ) inVh,h◦ ⊂ V is expressed by

∫ τ

0

∥∥∥∥{duhe

dt

}(t)

∥∥∥∥2

Hdt +

∫ τ

0

⟨{Ahe,h∗

euhe}(t),

{duhe

dt

}(t)

⟩V

dt

=∫ τ

0

({f∗

e + ΛTe Cege}(t),

{duhe

dt

}(t)

)H

dt, ∀τ ∈ [0, T ],

where, for a.e. t ∈ [0, T ],

({f∗

e + ΛTe Cege}(t),

{duhe

dt

}(t)

)H≤ 1

2‖f∗(t) + ΛT Cg(t)‖2

H +1

2

∥∥∥∥{duhe

dt

}(t)

∥∥∥∥2

H.

704 G. Alduncin

Hence, according to (55) and (56) we can conclude the estimate

1

2

∫ τ

0

∥∥∥∥{duhe

dt

}(t)

∥∥∥∥2

Hdt +

α

2‖{uhe(τ )}‖2

V

≤ βJA(‖{u0he

(τ )}‖V ) +1

2

∫ τ

0‖{f∗

e (t) + ΛTe Cege(t)}(t)‖2

H dt, ∀τ ∈ [0, T ],

(59)from which (57) and (58) follow.

Proof of Theorem 6.1. From estimates (34) (of Theorem 4.6) and (57)(of Lemma 6.2), {uhe} converges weakly to {ue} in V and {uhe} is boundedin L∞(0, T ; V ). But, since V∗ is dense in L1(0, T ; V ∗), these conditions areequivalent to conditions (48) [18]. Similarly, conditions (49) are concludedfrom estimates (35) and (58), taking into account that the space U = {v :v ∈ V , dv/dt ∈ H = L2(0, T ; H)}, endowed with the operator norm ‖v‖U =‖v‖V+‖dv/dt‖H, is a reflexive Banach space. In order to demonstrate estimates(50) and (51), we proceed as in the proof of Theorem 5.1, obtaining fromformula (7) and the coercivity condition (CA+∇(G◦Λ)), ∀τ ∈ [0, T ],∫ τ

0

⟨{due

dt

}(t) −

{duhe

dt

}(t), {ue}(t) − {uhe}(t)

⟩V

dt

+∫ τ

0〈A{ue}(t) − A{uhe}(t), {ue}(t) − {uhe}(t)〉V dt

≥ 1

2‖{ue}(τ ) − {uhe}(τ )‖2

H − 1

2‖{u0e} − {u0he

‖2H

+ α∫ τ

0‖{ue}(t) − {uhe}(t)‖2

V dt.

Hence, applying the orthogonality condition (43) of macro-hybridized primalproblems (PMH) and (PMHh,h∗,h◦ ), under conforming condition (CVh,h◦ ), wehave the error relation

1

2‖{ue}(τ ) − {uhe}(τ )‖2

H + α∫ τ

0‖{ue}(t) − {uhe}(t)‖2

V dt

≤ 1

2‖{u0e} − {u0he

}‖2H +

[‖A{ue} − A{uhe}‖V∗ +

∥∥∥∥{due

dt

}−

{duhe

dt

}∥∥∥∥V∗

]‖{ue} − {whe}‖V , ∀{whe} ∈ Vh,h◦ ⊂ V .

(60)Thereby, because of the regularity in time results (48) and (49), necessarily[18]

‖{ue}‖L∞(0,T ;V ) ≤ lim inf{he}↓0 ‖{uhe}‖L∞(0,T ;V ),∥∥∥∥{due

dt

}∥∥∥∥L2(0,T ;H)

≤ lim inf{he}↓0∥∥∥∥{duhe

dt

}∥∥∥∥L2(0,T ;H)

,(61)


and, from Lemma 6.2,

‖{uhe}‖L∞(0,T ;V ) ≤ c1(α, βJA(μ0), f

∗ + ΛTCg),∥∥∥∥{duhe

dt

}∥∥∥∥L2(0,T ;H)

≤ c2(βJA(μ0), f

∗ + ΛTCg),(62)

where μ0 denotes a bound for ‖{u0he}‖V . Therefore, by virtue of these rela-

tions, (61) and (62), in conjunction with estimate (4) (of Theorem 2.3), theterm ‖A{ue} − A{uhe}‖V∗ + ‖{due/dt} − {duhe/dt}‖V∗ in (60) is bounded bya constant function of (mA+∇(G◦Λ), α, βJ(μ0), f

∗ + ΛTCg), and this leads toestimates (50) and (51). Finally, estimates (52) and (53) then respectivelyfollow from primal estimate (51) and the dual and primal equations of thesemi-discrete and continuous macro-hybrid mixed problems (MHh,h∗ ,h◦) and(MH).

6.2 Dual regularity in time

As a dual counterpart of the primal time regularity result of Theorem 6.1, wehave the following.

Theorem 6.1∗ Let the conditions of Theorem 5.1∗ be satisfied. Then, if thedual data (g − ΛA∗f∗, p∗0) ∈ Y × Z∗ is such that

g − ΛA∗f∗ ∈ L2(0, T ; Z∗) and p∗0 ∈ Y ∗, (46∗)

and assuming Y ∗-convergence of the discrete initial data,

lim{h∗e}↓0 ‖{p∗0e

} − {p∗0h∗e}‖Y ∗ = 0, (47∗)

the dual component solution {p∗e} ∈ L∞(0, T ; Z∗)∩X ∗{Ωe} of the dual evolution

macro-hybrid mixed problem (MH∗) has the following regularity in time:

{p∗e} ∈ L∞(0, T ; Y ∗({Ωe})),{p∗h∗

e} →∗ {p∗e} weakly star in L∞(0, T ; Y ∗({Ωe})), (48∗)

and {dp∗edt

}∈ L2(0, T ; Z∗({Ωe})),{dp∗h∗

e

dt

}→

{dp∗edt

}weakly in L2(0, T ; Z∗({Ωe})).

(49∗)

Moreover, the following dual error estimates hold: ∀{s∗h∗e} ∈ Y∗

h∗ ,

‖{p∗e(τ )}−{p∗h∗e(τ )}‖Z∗ ≤ ‖{p∗0e

}−{p∗0h∗e}‖Z∗+c∗1‖{p∗e}−{s∗h∗

e}‖1/2

Y∗ , ∀τ ∈ [0, T ],

(50∗)

706 G. Alduncin

‖{p∗e} − {p∗h∗e}‖Y∗ ≤

(1

2α∗

)1/2

‖{p∗0e} − {p∗0h∗

e}‖Z∗ + c∗2‖{p∗e} − {s∗h∗

e}‖1/2

Y∗ ,

(51∗)

‖{ue} − {uhe}‖V ≤ mC∗

βΛ

[(1

2α∗

)1/2

‖{p∗0e} − {p∗0h∗

e}‖Z∗ + c∗2‖{p∗e} − {s∗h∗

e}‖1/2

Y∗

],

(52∗)

‖{λ∗e − λ∗

h◦e‖B∗ ≤ 1

βπTΓ

(mΛT +

mA mC∗

βΛ

)[(1

2α∗

)1/2

‖{p∗0e} − {p∗0h∗

e}‖Z∗ + c∗2‖{p∗e} − {s∗h∗

e}‖1/2

Y∗

],

(53∗)

where c∗1 = c∗1(mC∗+∇(F∗◦(−ΛT )), α∗, β∗

J∗C∗

(μ∗0), g − ΛA∗f∗) and c∗2 = c∗2(mC∗

+∇(F∗◦(−ΛT )), α∗, β∗J∗

C∗(μ∗

0), g − ΛA∗f∗) are strictly positive constants, and μ∗0

is a bound for {p∗0h∗e}.

Like in the primal case, we recognize that the semi-discrete dual solution{p∗h∗

e} ∈ X ∗

{h∗e} of problem (DMHh,h∗,h◦ ) is an absolutely continuous vector

function [10], with strong time derivative belonging to L2(0, T ; Y ∗{he}) since in

this regular case g − ΛA∗f∗ ∈ L2(0, T ; Z∗); i.e.,

{p∗he} ∈ CA(0, T ; Y ∗

{h∗e}) ⊂ L∞(0, T ; Y ∗

{h∗e}),{dp∗h∗

e

dt

}∈ L2(0, T ; Y ∗

{h∗e}).

(54∗)

On the other hand, due to the potentiality of the dual discrete operator[C∗

h∗e ,he

] = [C∗h∗

e] + ∇([F ∗

he]|V ∗

{he}◦ [−ΛT

he,h∗e]) ∈ L(Y ∗

{h∗e}, Y {h∗

e}),

d

dtJ

C∗(p∗he

(t)) =⟨{C∗

h∗e ,he

p∗h∗e}(t),

{dp∗h∗e

dt

}(t)

⟩Y ∗

, for a.e. t ∈ [0, T ], (55∗)

where JC∗ : Y ∗ → � is the dual Frechet differentiable potential of the dual

operator C∗ = C∗ + ∇(F ∗ ◦ (−ΛT )) ∈ L(Y ∗, Y ), defined uniquely modulo aconstant by J

C∗(y∗) =

∫ 10 〈C∗(sy∗), y∗〉Y ∗ ds, y∗ ∈ Y ∗. Hence, under the coer-

civity condition (CC∗+∇(F∗◦(−ΛT ))) and boundedness of C∗, the dual potentialis bounded in the sense

α∗

2‖y∗‖2

Y ∗ ≤ JC∗(y

∗) ≤ βJC∗(‖y∗‖Y ∗)

= mC∗+∇(F∗◦(−ΛT ))

∫ 1

0(s‖y∗‖Y ∗) ds‖y∗‖Y ∗ , ∀y∗ ∈ Y ∗,

(56∗)

and the following result is obtained.

Lemma 6.2∗ Under the conditions of Theorem 6.1∗, the sequences of dual


semi-discrete approximations {p∗h∗e} and {dp∗h∗

e/dt} of (54∗) are bounded in

L∞(0, T ; Y ∗) and L2(0, T ; Z∗), respectively. In fact, the following a priori es-timates hold true:

‖{p∗h∗e(τ )}‖Y ∗ ≤

{2

α∗βJC∗ (‖{p∗0h∗

e}‖Y ∗)

}1/2

+{

1

α∗

∫ τ

0‖g(t) − ΛA∗f∗(t)‖2

Z∗ dt}1/2

, ∀τ ∈ [0, T ],

(57∗)

∥∥∥∥{dp∗h∗e

dt

}∥∥∥∥L2(0,T ;Z∗)

≤{2βJ

C∗ (‖{p∗0h∗e}‖Y ∗)

}1/2+ ‖g − ΛA∗f∗‖L2(0,T ;Z∗).

(58∗)

Thereby, through similar arguments as in the proof of Theorem 6.1, the validityof the dual Theorem 6.1∗ can be established.

References

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708 G. Alduncin

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Received: November 24, 2007

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Analysis of Evolution Macro-Hybrid Mixed Variational Problems

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