Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

HAL Id: hal-00859700

https://hal.archives-ouvertes.fr/hal-00859700

Submitted on 9 Sep 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

François Bouchut, Robert Eymard, Alain Prignet

To cite this version:

François Bouchut, Robert Eymard, Alain Prignet. Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems. Journal of Evolution Equations, Springer Verlag, 2014, pp.14(3):635-669, 2014. �10.1007/s00028-014-0231-9�. �hal-00859700�

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

F. Bouchut, R. Eymard and A. Prignet^∗ September 9, 2013

Abstract

We study approximations by conforming methods of the solution to the variational in- equality h∂tu, v−ui+ψ(v)−ψ(u) ≥ hf, v−ui, which arises in the context of inviscid incompressible Bingham fluid flows and of the total variation flow problem. We propose a general framework involving total variation functionals, that enables to prove convergence of space, or time-space approximations, for steady or transient problems. We consider time implicit, or time implicit regularized (linearized or not) algorithms, and prove their conver- gence for general total variation functionals. Comparison with analytical solutions show the accuracy of the methods.

Keywords. Total variation flow, Bingham fluids, conforming approximations, regularization method, convergence

1 Introduction

A series of practical physical and engineering problems involve the flows of the so-called incom- pressible “Bingham fluids”. For such flows within a domain Ω ⊂R^N, N = 2 or 3, the relation between the stress tensorσ(t, x) (seen as aN×N matrix), the pressurep(t, x) and the velocity u(t, x)∈R^N is given by (see for example [12] and references therein)

σ=−pI_N + κ

|Du|+ 2ν

Du, (1.1)

where κ >0 andν >0 are given physical coefficients (ν is called the viscosity of the fluid), I_N is the N×N identity matrix, and where

(Du)ij = 1

2(∂iuj+∂jui), i, j = 1, . . . , N, (1.2) denoting by∂_i the partial derivative with respect to the i-th coordinate of a point x∈Ω, and

|Du|² =

N

X

i,j=1

(Du)²_ij. (1.3)

∗Universit´e Paris-Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050), CNRS, UPEMLV, UPEC, F-77454, Marne-la-Vall´ee, France(francois.bouchut, robert.eymard, [email protected])

Since the case of Bingham fluids with negligible viscosity arises in practice [23], our motiva- tion is to provide here numerical methods which remain available in the case when ν is small.

Therefore, this paper is focused on approximate methods allowing for the limit caseν = 0. The incompressibility condition for the fluid reads

divu=

N

X

i=1

∂_iu_i = 0. (1.4)

Assuming a constant density ρ = 1 for the fluid and neglecting the nonlinear convection term, the momentum conservation equation is given by

∂tui−

N

X

j=1

∂jσij =fi, i= 1, . . . , N, (1.5) wheref has values inR^N. An initial condition

u(0,·) =u⁰ (1.6)

is considered, as well as boundary conditions. For simplicity, we assume here homogeneous Neumann boundary conditions, which can be written

N

X

j=1

σijnj = 0 on ∂Ω, i= 1, . . . , N, (1.7) wheren is the outward normal unit vector on the boundary∂Ω of Ω.

In order to provide a formal variational formulation, let us define a set E of regular functions v : Ω → R^N such that divv = 0. We then multiply (1.5) by vi, sum over i= 1, . . . , N and integrate on Ω. We get, after an integration by parts in space accounting for the homogeneous Neumann boundary conditions and for the relation divv= 0,

∀v∈E, Z

Ω

∂_tu·v+ κ

|Du|+ 2ν

Du:Dv

dx= Z

Ω

f ·vdx, (1.8) where we denote by

Du:Dv=

N

X

i,j=1

(Du)_ij(Dv)_ij. (1.9)

In (1.8) and (1.5), the ratio Du/|Du|is not always defined. The physical understanding of this term indicates that it should be interpreted whenDu = 0 as “any trace-free symmetric matrix with norm less or equal to one”. In the mathematical language such quantity is called “multi- valued”. This fundamental difficulty makes the approximation of the problem (1.8) already a complex challenge forν >0, and has motivated a large literature, see [12, 15, 16] and references therein.

Since our motivation is to provide discretization methods also available in the caseν = 0, a first step is to rewrite (1.8) under a form which provides a well-posed continuous formulation. We then follow [12], writing a variational inequality giving a rigorous sense to this problem. Let us define, for u, v∈E andx∈Ω,

A(u, v)(x) =Du(x) :Dv(x) and a(u)(x) = A(u, u)(x)1/2

. (1.10)

Then from the Cauchy-Schwarz inequality one has

A(u, v) +A(u, u) =A(u, u+v)≤a(u)a(u+v), which gives

1

a(u)A(u, v)≤a(u+v)−a(u).

Let us also note that

2A(u, v)≤A(u+v, u+v)−A(u, u).

Then, denoting

ψ(v) = Z

Ω

(κ a(v) +ν A(v, v)) dx, (1.11)

the formulation (1.8) implies

∀v∈E, Z

Ω

∂_tu·vdx+ψ(u+v)−ψ(u)≥ Z

Ω

f·vdx. (1.12)

Reciprocally, letting v=θw in the previous inequality, we note that A(u+θw, u+θw)−A(u, u) =θ 2A(u, w) +θA(w, w)

, and

a(u+θw)−a(u) =θ2A(u, w) +θA(w, w) a(u+θw) +a(u) .

We then formally recover (1.8) from (1.12) by dividing by θ, lettingθ >0 tend to 0 and letting θ <0 tend to 0. We conclude that (1.12) is more general than (1.8) since it can be written in cases whena(u)(x) = 0 occurs for somex∈Ω. The formulation (1.12) can indeed be understood as saying that the linear formv7→R

(f−∂_tu)·v dx belongs to the subdifferential of the convex functional ψ atu. The term in (1.8) appears in fact as the formal differential of the functional ψ.

Note that all the theoretical background of [12] relies on the viscous term, and strongly depends on the assumption ν >0, while here we want to also handle the case ν = 0. We would like to extend the definition of ψ(v) to all functionsv∈L²(Ω)^N, thus we write forv∈E

Z

Ω

|Dv|dx= sup

ϕ∈(C_c¹(Ω))^N2,kϕk_L∞(Ω)≤1

Z

Ω

Dv:ϕdx, (1.13)

Z

Ω

|Dv|²dx 1/2

= sup

ϕ∈(C_c¹(Ω))^N2,kϕk_L2(Ω)≤1

Z

Ω

Dv:ϕdx, (1.14)

whereC_c¹(Ω) is the set ofC¹(Ω) functions with compact support in Ω. Integrating by parts, we thus extend the definition of ψ(v) to all functionsv∈L²(Ω)^N by defining

ψ(v) =κ sup

w∈VN,N

hv, wi+ν sup

w∈WN,N

hv, wi

!2

∈[0,∞], (1.15)

whereh·,·idenotes theL²(Ω)^N scalar product, and where VN,N =

w∈L²(Ω)^N,∃ϕ∈ C_c¹(Ω)N²

, wi = 1 2

N

X

j=1

∂j(ϕij +ϕji),

N

X

i,j=1

ϕ²_ij ≤1 in Ω

, (1.16) W_N,N =

w∈L²(Ω)^N,∃ϕ∈ C_c¹(Ω)N²

, w_i = 1 2

N

X

j=1

∂_j(ϕ_ij +ϕ_ji), Z

Ω N

X

i,j=1

ϕ²_ijdx≤1

. (1.17) We consider the space H of all functions v∈L²(Ω)^N such that divv= 0. Then the functional ψ may be defined by (1.15) as a mapping fromH to [0,∞], and we may define the set B of all functions v ∈ H such that ψ(v) < ∞. Indeed, the finiteness of the first term in (1.15) means that Dv is a finite measure on Ω, while the finiteness of the second term (ifν >0) means that Dv∈L²(Ω).

The problem is then to find, for a given T >0, a function u such that u∈L²(0, T;H),

Z T 0

ψ(u(t))dt <∞, ∂_tu∈L²(0, T;H), u(0) =u⁰ and

∀v∈L²(0, T;H), Z T

0

h∂_tu(t), v(t)−u(t)i+ψ(v(t))−ψ(u(t))

dt

≥ Z T

0

hf(t), v(t)−u(t)idt.

(1.18)

For such nonlinear monotone variational inequalities in a Hilbert space, involving a convex lower semi-continuous functional (which is precisely the case for the function ψ defined above and valued in ]− ∞,∞]), the theory of [6] applies, giving the existence and uniqueness of the solution.

This theory has been used in several works, applied to the total variation flow, and we refer to [22] for a general exposition on the subject. The total variation flow problem is a scalar problem which consists in looking for the solution u : Ω→Rto the problem

∂_tu−div ∇u

|∇u|

=f, (1.19)

with Neumann boundary condition (∇u/|∇u|)·n= 0 on∂Ω. Indeed, this problem may also be written under the form of inequality (1.18), denoting by H the space L²(Ω), and introducing the BV seminorm defined by

ψ(v) = sup

w∈V_N

hv, wi, (1.20)

whereh·,·idenotes theL²(Ω) scalar product and where VN =

(

w∈L²(Ω),∃ϕ∈ C_c¹(Ω)N

, w= divϕ,

N

X

i=1

ϕ²_i ≤1 in Ω )

. (1.21)

The total variation problem is however more “regular” than the inviscid Bingham problem, since it is also monotone inL¹(Ω), as proved in [1]. The theory of monotone problems in Banach spaces is provided in [9, 2, 3]. This L¹ monotonicity (implying the almost everywhere monotonicity,

via the well-known Crandall-Tartar lemma [10]) enables to use the Kruzkov entropies, as for hyperbolic scalar conservation laws, and therefore to include a transport term. This is done in [4], including convergence results for numerical approximations.

The Bingham problem does not have this L¹ structure. Nevertheless, there is a very strong result in [20], that states that in two-dimensions, the Bingham evolution problem has a smooth solution, without viscosity, and including advection.

It is worth noticing that the form of inequality (1.18) includes both linear problems and nonlinear problems such as the p−Laplacian, which shows that it is quite general. Therefore it presents some interest to give general lines for its approximation.

This paper is devoted to the approximation by conforming methods of the solution to (1.18) in the general case of a convex lower semi-continuous (l.s.c. for short) functional on a Hilbert space.

It is then applied to the framework of general total variation functionals, which includes the particular cases of the inviscid Bingham problem and of the total variation flow problem. In this situation we use a regularization procedure and time implicit or linearized implicit integration.

Our results generalize the ones of [13, 14], obtained for the total variation flow, and for more regular data. Our analysis of the linearized implicit scheme, which is the one applied in practice for Bingham flows (see [19, 21]), seems to be new.

The paper is organized as follows. In Section 2, we first recall some properties of the steady problem, following [6], and we propose sufficient conditions for a convergent approximation in a finite dimensional subspace, provided that some interpolation conditions be satisfied (Subsection 2.1). We then provide in Subsection 2.2 the analysis of the regularization method, applied to general total variation functionals. We turn to the transient problem in Section 3: we first state some basic properties in Subsection 3.1, and we then prove the convergence of a time-space conforming approximation of fully implicit type in Subsection 3.2. We provide in Subsection 3.3 the proof of convergence for the implicit regularization method for general total variation functionals, and treat the linearized algorithm in Subsection 3.4. We finally propose in Section 4 the study of numerical convergence in the particular case of the total variation flow. In a first subsection, we show that the problem to be solved in a finite dimensional space is itself approximated in the steady case. In a second subsection we consider the transient case, for which we show the convergence of the implicit method, where the use of a linearization is studied in the case of the regularized problem. Our results extend the ones of [13, 14], since they apply without further hypotheses on the regularity of the continuous solution. A short conclusion is finally given in Section 5.

2 Approximation of the steady problem

2.1 General framework

As stated in the introduction of this paper, we focus on the approximation of a steady version of the problem (1.18), using the framework of [6].

LetH be a Hilbert space, with scalar producth·,·iand normk · k. Letψ : H→]− ∞,∞] be a convex, lower semi-continuous function such that the set B ={v∈H, ψ(v) <∞}(the domain of ψ) is not empty. Classical results in this situation can be found for example in [7].

We first recall the following standard lemma in convex analysis.

Lemma 2.1 A functional ψ is convex, lower semi-continuous with non-empty domain B, and first order positively homogeneous (i.e. for all λ∈ R, v ∈ B, ψ(λv) = |λ| ψ(v)) if and only if there exists a non empty setV ⊂H such that

∀w∈V, −w∈V, (2.1)

and

∀v∈H, ψ(v) = sup

w∈V

hw, vi. (2.2)

In this case, B is a subspace of H, andψ satisfies ψ(u+v)≤ψ(u) +ψ(v).

Proof. It is given in [8] or [18, theorem 3.1.1], but for completeness we give it shortly. Since the “if” part is obvious, considerψ convex, lower semi-continuous with non-empty domain, and first order positively homogeneous. Then applying the homogeneity property to some v₀ ∈ B andλ= 0 yields thatψ(0) = 0. We deduce thatψ(λv) =|λ|ψ(v) for allλ∈Randv ∈H, with the convention that 0× ∞ = 0. In particular, ψ(−v) = ψ(v). Applying the Fenchel-Moreau theorem we have thatψis the supremum of all affine functions upper dominated byψ. Consider such an affine functionv7→µ+hw, viwith µ∈R and w∈H. We have

ψ(v)≥µ+hw, vi for all v∈H. (2.3) Applying this inequality to λv, using the homogeneity and lettingλ→ ∞ yields that

ψ(v)≥ hw, vi for all v∈H. (2.4) Applying this to −v gives then that ψ(v) ≥ |hw, vi| ≥ 0. Since (2.3) applied to v = 0 gives µ≤0, we deduce that the linear function v7→ hw, vi, which is upper dominated byψ by (2.4), is greater than the affine functionv7→µ+hw, vi. Therefore,ψis also the supremum of all linear functions upper dominated byψ. In other words, (2.2) holds with

V ={w∈H, ∀v∈H hw, vi ≤ψ(v)}. (2.5)

It is easy to check finally that 0∈V and that (2.1) holds.

Let us notice that the examples provided in the introduction of this paper (without viscosity) fall into the class given by Lemma 2.1.

We recall that, according to the convexity ofψ, the property of lower semi-continuity also holds for the weak topology ofH, which implies that, for any sequence (v_n)n∈Nof elements of B that weakly converges to v ∈ H, and such that there exists C ∈ R with ψ(vn) ≤ C for all n ∈ N, thenψ(v)≤lim inf

n ψ(vn), which implies thatv∈B.

Let α > 0 andf ∈H be given. The aim of this section is to approximate the solution to the following problem: find u such that

u∈B,

∀v∈B, αhu, v−ui+ψ(v)−ψ(u)≥ hf, v−ui. (2.6) Note that (2.6) may also be written as

αu+∂ψ(u)3f, (2.7)

with introducing the subdifferential of the function ψatu,

∂ψ(u) ={w∈H, ∀v∈H, ψ(v)≥ψ(u) +hw, v−ui}. (2.8) Then according to [6] we have the following result.

Lemma 2.2 There is existence and uniqueness of the solution u to the problem (2.6), which moreover satisfies

u= argmin

v∈B

J(v), (2.9)

where J : B →R is defined by J(v) = α

2kvk²+ψ(v)− hf, vi, ∀v∈B. (2.10) Corollary 2.3 There exists u0 ∈B such that ∂ψ(u0) is not empty.

Proof. Take α = 1 and f = 0. Then the solution u obtained by Lemma 2.2 satisfies (2.7),

which implies that ∂ψ(u) is not empty.

Remark 2.4 In the case whenψ is first order positively homogeneous (the case of Lemma 2.1), thenu∈B is solution to the problem (2.6) if and only if

u∈H

∀v∈H, αhu, vi+ψ(v)≥ hf, vi, (2.11) and

αkuk²+ψ(u) =hf, ui. (2.12)

It indeed suffices to let v = 0 and v = 2u in (2.6) for obtaining (2.12), which also shows that u∈B. This characterization is used in the examples for exhibiting an analytical solution.

Remark 2.5 If we have two solutions u₁ andu₂ to (2.6) associated to two different right-hand sides f₁ and f₂, then one hasku₂−u₁k ≤ kf₂−f₁k/α. This is obtained by taking v=u₂ in the formulation (2.6) for u1, taking v=u1 in the formulation (2.6) for u2, and adding the results.

This contraction property is in the heart of the theory of monotone operators (here ∂ψ is the monotone operator in (2.7)).

Let us now introduce a reduction argument. Take u0 ∈B such that ∂ψ(u0) is not empty, and pick some w ∈ ∂ψ(u₀). Then one has ψ(v) ≥ ψ(u₀) +hw, v−u₀i for all v ∈ H. Therefore, setting for v∈H

ψ(v) =e ψ(v+u0)−ψ(u0)− hw, vi, (2.13) the functional ψe : H →]− ∞,∞] is convex and lower semi-continuous, and satisfies ψe ≥ 0, and ψ(0) = 0 (implying 0e ∈Be={v∈H,ψ(v)e <∞}). For a∈R(chosen later) and for v∈H, define then

J(v) =e J(v+u₀)−a= α

2kv+u₀k²+ψ(v+u₀)− hf, v+u₀i −a.

Then the minimum of Jeis obtained at the point ue = u−u0, where u is the solution to the problem (2.6). Settingfe=f−w−αu0, we have

Je(v) = α

2kvk²+ψ(v)e − hf , vie +ψ(u0) + Dα

2u0−f, u0

E

−a, thus choosinga=ψ(u0) +h^α₂u0−f, u0i, the problem is to find the minimum of

J(v) =e α

2kvk²+ψ(v)e − hf , vi.e Therefore, eu is the solution to the problem

ue∈B,e

∀v∈B,e αhu, ve −eui+ψ(v)e −ψ(eu)e ≥ hf , ve −ui.e (2.14) Indeed, (2.14) can also be deduced directly from (2.6). We conclude that we may assume, without loss of generality, that ψ≥0 andψ(0) = 0. This is done in the following Hypothesis.

Hypothesis 2.6 We consider the following assumptions:

1. H is a Hilbert space,

2. ψ : H→]− ∞,∞]is convex and lower semi-continuous, 3. ψ(v)≥0 for all v∈H, and ψ(0) = 0.

Note that the third assumption ensures that if f = 0, the solution to (2.6) isu= 0.

Assuming Hypothesis 2.6 (that implies 0∈B={v∈H, ψ(v)<∞}), letHb be a closed subspace of H and let Bb =Hb∩B. Note that 0 ∈Bb (in the examples, B is a subspace ofH and Hb is a finite dimensional subspace of H). We define the approximate problem as: findubsuch that

bu∈B,b

∀v∈B,b αhu, vb −uib +ψ(v)−ψ(u)b ≥ hf, v−ui.b (2.15) It is then immediate to get the existence and uniqueness ofubsolution to (2.15).

Lemma 2.7 Under Hypothesis 2.6, let α >0 and f ∈H be given. Let Hb be a closed subspace of H and let Bb =Hb ∩B. Then there exists one and only one solution to the problem (2.15), which moreover satisfies

ub= argmin

v∈Bb

J(v), (2.16)

where J is defined by (2.10).

Proof. It suffices to consider f, the orthogonal projection ofb f on H. Then (2.15) is identicalb to (2.6) replacing ψ by its restriction to Hb and f by fb, since hf , vib = hf, vi for all v ∈ H.b Therefore, Lemma 2.2 gives the existence and uniqueness of the solution to (2.15).

The scheme (2.15) leads to the resolution of a convex minimization problem in a finite dimen- sional space. There are many numerical methods well-suited to that (gradient or conjugate gra- dient methods for example). We analyze a different type of method, the regularization method, in the particular case of total variation functionals in Subsection 2.2, based on the particular form of the function ψ. Let us however give the following error estimate result for the scheme (2.15) in the general case.

Lemma 2.8 Under Hypothesis 2.6, letα >0 and f ∈H be given and letu∈B be the solution to the problem (2.6). Let Hb be a closed subspace of H, let Bb = Hb ∩B and let ub ∈ Bb be the solution to (2.15). Then

ku−buk ≤2 Rb_u,f α

!1/2

, (2.17)

and

ψ(u)−ψ(u)b

≤6kfk Rb_u,f α

!1/2

, (2.18)

with

Rb_u,f = inf

v∈Bb

kfk kv−uk+ ψ(v)−ψ(u)+

, (2.19)

where we denote for all x∈R, x⁺= max(0, x).

Proof. Let us first observe that ubsatisfies, setting v= 0 in (2.15) and using thatψ(0) = 0, αkukb ²+ψ(u)b ≤ hf,bui,

which gives, according to the Young inequality, α

2kukb ²+ψ(bu)≤ 1

2αkfk². (2.20)

Since ψ≥0 this shows that

kuk ≤b 1

αkfk. (2.21)

We similarly write, from taking v= 0 in (2.6), kuk ≤ 1

αkfk. (2.22)

From (2.15), we get for anyv∈Bb

αhbu, u−bui+ψ(u)−ψ(u) +b R(v)≥ hf, u−ui,b with

R(v) =αhu, vb −ui+ψ(v)−ψ(u) +hf, u−vi. (2.23) Hence, taking the infimum, we get

αhbu, u−bui+ψ(u)−ψ(u) + infb

v∈Bb

R(v)≥ hf, u−ui,b (2.24) while from (2.23) and (2.19) we have

inf

v∈Bb

R(v)≤2Rbu,f. (2.25)

We have also, lettingv= 0 in (2.23) and using that ψ(0) = 0, ψ≥0 and (2.21)-(2.22), inf

v∈Bb

R(v)≤R(0)≤ 2

αkfk². (2.26)

Taking v=bu in (2.6), we get

αhu,ub−ui+ψ(bu)−ψ(u)≥ hf,ub−ui. (2.27) Adding the inequalities (2.24) and (2.27) yields

αku−buk² ≤ inf

v∈Bb

R(v), (2.28)

which provides (2.17), using (2.25). We deduce also that the right-hand side of (2.28) is non- negative. We then write, again using (2.24) and (2.21),

ψ(bu)−ψ(u)≤2kfkku−ukb + inf

v∈Bb

R(v),

and similarly from (2.27)

ψ(u)−ψ(bu)≤αhu,ub−ui+hf, u−bui ≤2kfkku−uk.b We deduce, with (2.28) and (2.26), that

ψ(u)−ψ(u)b

≤2kfkku−buk+ inf

v∈Bb

R(v)

≤r inf

v∈Bb

R(v) 2kfk

√α +r inf

v∈Bb

R(v)

!

≤ 4kfk

√α r

inf

v∈Bb

R(v),

which leads to (2.18) using (2.25).

We deduce the following convergence result for the approximation method.

Corollary 2.9 Under Hypothesis 2.6, let (Hbn)n∈N be a sequence of closed subspaces of H, and let, for all n∈N, Bb_n=Hb_n∩B. We assume that

n→∞lim inf

w∈Bbn

kw−vk+ ψ(w)−ψ(v)+

= 0, ∀v∈B. (2.29)

Let, for alln∈N,ubn∈Bbnbe the unique solution ubto (2.15) withBb=Bbn. Then, ubn converges in H to the unique solution u∈B to the problem (2.6) asntends to ∞ and ψ(ubn) converges to ψ(u).

2.2 Total variation functionals

In this subsection we apply the framework of the previous section to the case of functionalsψof total variation type, generalizing (1.20), or (1.15) withν = 0, in the introduction of this paper.

Hypothesis 2.10 We assume Hypothesis 2.6. Moreover, we assume that there exists a subspace H1 ⊂ H, an open set Ω ⊂ R^N with N ≥ 1, a nonnegative Borel measure µ on Ω such that

µ(Ω)<∞, and a symmetric bilinear mappingA : H1×H1 →L¹_µ(Ω)such thatA(u, u)(x)≥0 for µ a.e. x∈Ω, for all u∈H1, and

∀u∈H₁, ψ(u) = Z

Ω

a(u)(x)dµ, (2.30)

denoting by a(u)(x) = (A(u, u)(x))^1/2 for µ a.e. x ∈ Ω. In particular, ψ is finite on H1, i.e.

H₁⊂B.

The two examples we are interested in obviously satisfy these conditions, with the following choices.

Example 2.11 (Total variation flow) The Hilbert space is taken H = L²(Ω), with Ω a bounded open set inR^N,ψis as in (1.20), (1.21),B =L²(Ω)∩BV(Ω),µis the Lebesgue measure onΩ, H₁ is any space such that C^∞(Ω)⊂H₁ ⊂H¹(Ω), andA(u, v)(x) =∇u(x)· ∇v(x).

Example 2.12 (Inviscid Bingham flow) The Hilbert space is takenH ={u∈L²(Ω)^N, divu= 0} with Ω a bounded open set in R^N, ψ is as in (1.15), (1.16) with ν = 0, B = {u ∈ L²(Ω)^N, divu = 0, Du ∈ M(Ω)}, where Du = (∇u + (∇u)^t)/2, M(Ω) is the space of fi- nite measures over Ω, µ is κ times the Lebesgue measure on Ω, H1 is any space such that {u∈C^∞(Ω)^N,divu= 0} ⊂H1⊂ {u∈H¹(Ω)^N,divu= 0}, andA(u, v)(x) =Du(x) :Dv(x).

We next analyze the algorithm byregularizationfor computing an approximate solution to (2.16) (which converges to the continuous solution according to Corollary 2.9).

Lemma 2.13 Under Hypothesis 2.10, let α > 0 and f ∈ H. Let Hb be a finite dimensional subspace of H1. Then, forε >0, there exists one and only one function buε∈Hb solution to

ubε∈H,b

∀v∈H, αhb bu_ε, vi+ Z

Ω

A(ubε, v)(x)

ε+a(ub_ε)(x)dµ=hf, vi. (2.31) Moreover, denoting by bu∈Hb the unique solution to (2.16), we have

kubε−uk ≤b

rµ(Ω)ε

α , (2.32)

and

|ψ(buε)−ψ(bu)| ≤2kfk

rµ(Ω)ε

α + 3µ(Ω)ε. (2.33)

The estimates (2.32), (2.33) provide, with Corollary 2.9, the convergence of (ub_ε)_n to u under the conditions (2.29) andε→0. Note that sinceHbn⊂H1, it is necessary for having (2.29) that the following condition holds:

∀v ∈B, inf

w∈H₁

kw−vk+ ψ(w)−ψ(v)+

= 0. (2.34)

This condition is proved to hold true for the total variation flow and for the Bingham flow in the appendix. Then to recover (2.29) from (2.34), it suffices to require that

n→∞lim inf

w∈Hbn

kw−vk+ ψ(w)−ψ(v)+

= 0, ∀v∈H₁, (2.35)

which is easily fulfilled.

Proof of Lemma 2.13. Let us begin with the proof of uniqueness. Consider two elements u1

and u₂ ofHb satisfying (2.31). Subtracting (2.31) withu₁ and u₂, and takingv=u₁−u₂ leads to

αku₁−u2k²+ Z

Ω

A(u1, u1−u2)(x)

ε+a(u₁)(x) −A(u2, u1−u2)(x) ε+a(u₂)(x)

dµ= 0. (2.36) We have according to the Cauchy-Schwarz inequalityA(u1, u2)(x)≤a(u1)(x)a(u2)(x) forµa.e.

x∈Ω. Sinces→s/(ε+s) is strictly increasing from [0,∞) to [0,1), omitting for simplicity the argumentx we get

0≤

a(u1)

ε+a(u₁) − a(u2) ε+a(u₂)

a(u1)−a(u2)

≤

A(u₁, u₁−u₂)

ε+a(u₁) −A(u₂, u₁−u₂) ε+a(u₂)

. Using this information in (2.36) we obtain

αku₁−u₂k² ≤0,

which concludes the proof of uniqueness. Turning to the existence proof, we consider the operator T :Hb →Hb defined for u∈Hb by

∀v∈H,b hT(u), vi= Z

Ω

A(u, v)(x)

ε+a(u)(x)dµ. (2.37)

SinceAis bilinear andHb is finite dimensional, the restriction ofAtoHb×Hb is continuous, with values inL¹_µ(Ω). Therefore, the operatorT is continuous fromHb to itself. Then, we have that any solution to (2.31), in which a factorλ∈[0,1] is introduced in front of the integral, satisfies the estimate

α

2kub_εk²+λ Z

Ω

a(ub_ε)(x)²

ε+a(bu_ε)(x)dµ≤ 1

2αkfk². (2.38)

Since, for λ = 0, the problem is a finite dimensional invertible linear problem, we get by a standard topological degree argument that there exists at least one solution to the problem for λ= 1. Let us now turn to the proof of (2.32). We remark that, foru, v∈H1 andµ a.e. x∈Ω, omitting for simplicity the argumentx, applying the Cauchy-Schwarz inequality and (5.6),

A(u, v−u)

ε+a(u) ≤ a(u)a(v)−a(u)²

ε+a(u) ≤ε+a(v)−a(u). (2.39) Using (2.31) where we replacev by v−ub_ε, we get

∀v∈H,b

αhubε, v−ubεi+µ(Ω)ε+ψ(v)−ψ(buε)≥ hf, v−buεi. (2.40) Lettingv =bu (the solution to (2.16)) in (2.40), we obtain

ψ(buε)−ψ(bu)≤ hαubε−f,ub−ubεi+µ(Ω)ε. (2.41) Then, taking v=ubε in (2.15) we get

ψ(bu)−ψ(ubε)≤ hαub−f,buε−bui. (2.42)

Adding the two inequalities, we then get (2.32). Let us now turn to the proof of (2.33). Setting v= 0 in (2.40), we get

αkub_εk²+ψ(ub_ε)≤µ(Ω)ε+hf,bu_εi, which implies, sincehf,buεi ≤ _2α¹ kfk²+^α₂kubεk²,

α²kbuεk² ≤2αµ(Ω)ε+kfk²≤(kfk+p

2αµ(Ω)ε)². This leads with (2.41) to

ψ(ub_ε)−ψ(u)b ≤(2kfk+p

2αµ(Ω)ε)kbu_ε−buk+µ(Ω)ε.

Similarly, using (2.21) in (2.42) yields

ψ(u)b −ψ(ub_ε)≤2kfkkub_ε−buk.

Using (2.32), we finally get (2.33).

The last step is to approximate the solution ubε to the regularized problem (2.31), since it is nonlinear. For given initialu⁽⁰⁾∈Hb (and fixedε >0), we now define the sequence (u^(k))k∈Nby

u^(k+1) ∈H,b

∀v∈H,b αhu^(k+1), vi+ Z

Ω

A(u^(k+1), v)(x)

ε+a(u^(k))(x) dµ=hf, vi. (2.43) We have the following result.

Lemma 2.14 Under Hypothesis 2.10, let α > 0 and f ∈ H. Let Hb be a finite dimensional subspace of H₁, and let u⁽⁰⁾ ∈ Hb and ε > 0 be given. Then there exist a unique sequence (u^(k))k∈N defined by (2.43). Moreover, as k→ ∞, u^(k) converges to buε, the unique solution to (2.31), and ψ(u^(k)) converges to ψ(ub_ε).

Proof. The estimate α

2ku^(k+1)k²+ Z

Ω

a(u^(k+1))(x)²

ε+a(u^(k))(x)dµ≤ 1

2αkfk², (2.44)

obtained by letting v =u^(k+1) in (2.43) and applying the Young inequality, shows that with a null right-hand side, the square linear system to be solved has the unique solution 0. Hence it is invertible, showing the existence and uniqueness of the sequence (u^(k))k∈N.

We then let v=u^(k+1)−u^(k) in (2.43). Since the Cauchy-Schwarz inequality implies, µa.e. in Ω,

a(u^(k+1)) ε+a(u^(k))

a(u^(k+1))−a(u^(k))

≤ A(u^(k+1), u^(k+1)−u^(k)) ε+a(u^(k)) , we get from (5.4) proved in Lemma 5.1 that

α 2

ku^(k+1)k²+ku^(k+1)−u^(k)k²− ku^(k)k² +

Z

Ω

Fε

a(u^(k+1))(x)

−Fε

a(u^(k))(x)

+ a(u^(k+1))(x)−a(u^(k))(x)2

2(ε+a(u^(k))(x))

! dµ

≤ hf, u^(k+1)−u^(k)i.

(2.45)

Therefore, summing the above inequality for k = 0, . . . , m and applying the Young inequality to the right-hand side, we get

α 2

ku^(m+1)k²+

m

X

k=0

ku^(k+1)−u^(k)k²− ku⁽⁰⁾k²

+ Z

Ω

Fε

a(u^(m+1))(x)

−Fε

a(u⁽⁰⁾)(x)

+1 2

m

X

k=0

a(u^(k+1))(x)−a(u^(k))(x)2

ε+a(u^(k))(x)

! dµ

≤ 1

αkfk²+α

4ku^(m+1)k²− hf, u⁽⁰⁾i.

(2.46)

This proves on one hand that ^α₄ku^(m+1)k²+R

ΩF_ε(a(u^(m+1))(x))dµ remains bounded indepen- dently of m, and using (5.3) proved in Lemma 5.1, we get thatψ(u^(m+1)) remains bounded.

This proves on the other hand that the two series in the left-hand side of the above inequality converge, and therefore that

ku^(k+1)−u^(k)k →0, ask→ ∞. (2.47)

We next observe thatv7→ ka(v)k_L2

µ(Ω) is a semi-norm onH1. SinceHb has finite dimension, this implies that there exist a constantMcsuch that

∀v∈H,b ka(v)k_L2

µ(Ω)≤Mkvk.c (2.48)

Writing

ka(u^(k+1))−a(u^(k))k_L2

µ(Ω)≤ ka(u^(k+1)−u^(k))k_L2

µ(Ω)≤Mcku^(k+1)−u^(k)k, (2.49) we get that a(u^(k+1))−a(u^(k)) → 0 in L²_µ(Ω). Using again that the dimension of Hb is finite, we deduce from (2.44) that there exists a subsequence of (u^(k))k∈N, again denoted (u^(k))k∈N, strongly convergent in the finite dimensional vector space to some elementu∈H. This impliesb that (a(u^(k)))k∈N converges in L²_µ(Ω) toa(u), and that ψ(u^(k)) tends to ψ(u). Using (2.47), we may then pass to the limit in (2.43) for this extracted subsequence, and we get that the limitu satisfies (2.31). Since the solution to (2.31) is unique, we get that the whole sequence (u^(k))k∈N

converges to this solution.

Lemma 2.15 With the assumptions of Lemma 2.14, assume further that

∀v∈H,b ka(v)k_L∞

µ(Ω)<∞. (2.50)

Then the convergence of u^(k) to buε as k→ ∞ is asymptotically geometric with ratio arbitrarily close to

ka(ub_ε)k_L∞ µ(Ω)

ε+ka(ubε)k_L^∞

µ(Ω)

<1. (2.51)

Proof. With the assumption (2.50), v 7→ ka(v)k_L^∞

µ(Ω) is a semi-norm on H, which is finiteb dimensional. Thus there exists a constant Gb such that

∀v∈H,b ka(v)k_L^∞

µ(Ω)≤Gkvk.b (2.52)

It implies thata(u^(k))→a(buε) inL^∞_µ(Ω). Then, take successively v=u^(k+2)−u^(k+1) in (2.43), and in (2.43) withk incremented of 1. The difference yields

αku^(k+2)−u^(k+1)k²+ Z

Ω

A(u^(k+2), u^(k+2)−u^(k+1))(x) ε+a(u^(k+1))(x) dµ

− Z

Ω

A(u^(k+1), u^(k+2)−u^(k+1))(x)

ε+a(u^(k))(x) dµ= 0.

(2.53)

We deduce omitting thex that αku^(k+2)−u^(k+1)k²+ Z

Ω

a(u^(k+2)−u^(k+1))² ε+a(u^(k+1)) dµ

= Z

Ω

A(u^(k+1), u^(k+2)−u^(k+1))

1

ε+a(u^(k)) − 1 ε+a(u^(k+1))

dµ

≤ Z

Ω

a(u^(k+1))a(u^(k+2)−u^(k+1))

a(u^(k+1))−a(u^(k)) (ε+a(u^(k)))(ε+a(u^(k+1)))dµ

≤ 1 2

Z

Ω

a(u^(k+2)−u^(k+1))²

ε+a(u^(k+1)) +a(u^(k+1))²

a(u^(k+1))−a(u^(k))

2

(ε+a(u^(k)))²(ε+a(u^(k+1)))

! dµ,

(2.54)

and therefore that

2αku^(k+2)−u^(k+1)k²+ Z

Ω

a(u^(k+2)−u^(k+1))² ε+a(u^(k+1)) dµ

≤ Z

Ω

a(u^(k+1))² a(u^(k+1)−u^(k))²

(ε+a(u^(k)))²(ε+a(u^(k+1)))dµ.

(2.55)

Now, since a(u^(k))→a(buε) in L^∞_µ(Ω), for givenη >0 there exist somek0 such that

∀k≥k0, µ a.e.in Ω, a(u^(k+1))²

(ε+a(u^(k)))(ε+a(u^(k+1))) ≤(r+η)², (2.56) wherer is the left-hand side of (2.51). Then, fork≥k₀ one has

2αku^(k+2)−u^(k+1)k²+ Z

Ω

a(u^(k+2)−u^(k+1))² ε+a(u^(k+1)) dµ

≤(r+η)² Z

Ω

a(u^(k+1)−u^(k))² ε+a(u^(k)) dµ,

(2.57)

which implies that for k≥k₀ Z

Ω

a(u^(k+1)−u^(k))²

ε+a(u^(k)) dµ≤(r+η)^2(k−k0) Z

Ω

a(u^(k0+1)−u^(k0))²

ε+a(u^(k0)) dµ. (2.58)

Plugging this in (2.57) we deduce that for k≥k0,

ku^(k+2)−u^(k+1)k ≤C(r+η)^k−k0, (2.59)

whereC does not depend onk. Ifr+η <1 we finally write fork≥k₀ ku^(k)−bu_εk ≤

∞

X

k⁰=k

ku^(k0+1)−u^(k0)k ≤C(r+η)^k−k0, (2.60)

which proves the claim.

3 Transient problem

3.1 Continuous framework

Assuming Hypothesis 2.6 (we again notice that the assumptions ψ ≥0 and ψ(0) = 0 bring no restriction to generality in Problem (3.3) below), let T > 0 be given. The space L²(0, T;H) is defined as the Hilbert space of all measurable, almost everywhere defined functions u (in the so-called “Bochner integral” sense, recalled for example in [11]) from (0, T) to H such that kuk²_L2(0,T;H) :=RT

0 ku(t)k²dt <∞. The setBT is defined as B_T =

u∈L²(0, T;H);

Z T 0

ψ(u(t))dt <∞

. (3.1)

For anyu∈L¹_loc(0, T;H), we denote by ∂_tu∈L¹_loc(0, T;H) the time weak derivative of u when it exists, that is∂tu∈L¹_loc(0, T;H) is such that

∀ϕ∈C_c¹(]0, T[), Z T

0

ϕ⁰(t)u(t)dt=− Z T

0

ϕ(t)∂_tu(t)dt.

We recall that C^∞([0, T];H) is dense in the Hilbert spaces L²(0, T;H) and in H¹(0, T;H) :=

{u ∈L²(0, T;H);∂tu ∈L²(0, T;H)}, and that H¹(0, T;H) ⊂C⁰([0, T];H) (see [11]). We also recall that, for allu, v∈H¹(0, T;H), we have for anyt₁, t₂∈[0, T]

Z t2

t1

h∂_tu(t), v(t)idt+ Z t2

t1

hu(t), ∂_tv(t)idt=hu(t₂), v(t₂)i − hu(t₁), v(t₁)i. (3.2) Letf ∈L²(0, T;H) and u⁰ ∈B be given. We look in this section for a functionu such that

u∈H¹(0, T;H)∩BT, u(0) =u⁰,

Z _T

0

h∂_tu(t), v(t)−u(t)idt+ Z _T

0

ψ(v(t))−ψ(u(t)) dt

≥ Z T

0

hf(t), v(t)−u(t)idt, ∀v ∈BT.

(3.3)

Let us remark that the inequality (3.3) implies Z _t2

t1

h∂_tu(t), v(t)−u(t)idt+ Z _t2

t1

ψ(v(t))−ψ(u(t)) dt

≥ Z t2

t1

hf(t), v(t)−u(t)idt, ∀v∈B_t1,t2, ∀t₁ < t₂ ∈[0, T],

(3.4)

where B_t1,t2 denotes the set of all v ∈ L²(t₁, t₂;H) such that Rt2

t1 ψ(v(t))dt < ∞. It indeed suffices, for a given ˜v ∈Bt1,t2, to define v(t) = ˜v(t) for a.e. t ∈]t₁, t2[ and v(t) = u(t) for a.e.

t∈]0, t₁[∪]t₂, T[, and then use thisv ∈BT as test function in (3.3). Then, since the restriction of v∈B_T to any interval ]t₁, t₂[ belongs toB_t1,t2, we conclude that the inequality (3.3) is equivalent to

h∂_tu(t), v(t)−u(t)i+ψ(v(t))−ψ(u(t))

≥ hf(t), v(t)−u(t)i, for a.e. t∈]0, T[, ∀v∈BT. (3.5) One can derive even a stronger formulation. Let E ⊂]0, T[ a set such that meas(]0, T[\E) = 0 and for all t∈E, ast₁, t₂→t with 0< t₁ < t < t₂ < T,

Z t2

t1

ψ(u(s))ds→ψ(u(t)), Z t2

t1

h∂_tu(s)−f(s), u(s)ids→ h∂_tu(t)−f(t), u(t)i, Z t2

t1

∂_tu(s)−f(s)

ds * ∂_tu(t)−f(t) weakly inH,

where the bar integral denotes the average, i.e. the integral normalized by the volume. Dividing (3.4) by t₂−t₁, passing to the limit ast₁, t₂ →t for all t∈E and taking v equal to a constant element ofB, we get

h∂_tu(t), v−u(t)i+ψ(v)−ψ(u(t))

≥ hf(t), v−u(t)i, ∀v∈B, for a.e. t∈]0, T[. (3.6) This strong formulation is therefore equivalent to (3.5) and to (3.3). It may also be written as

∂tu(t) +∂ψ(u(t))3f(t) for a.e. t∈]0, T[, (3.7) with the subdifferential defined in (2.8). However, in this paper we shall not use this formulation.

The study of existence and uniqueness for Problem (3.3) is given in [22, Theorem 20], following [6], in the case when f = 0, using the semi-group approach and the Yosida regularization.

Since we focus on the approximation of this problem, we indeed recover the existence through the convergence of a semi-discrete (in time) approximation. A convergence result is however established for general time-space approximations.

The following lemma is classical. We recall its short proof.

Lemma 3.1 Under Hypothesis 2.6, let T > 0 be given, and f ∈ L²(0, T;H). If u1 and u2

are two solutions to Problem (3.3), with possibly different initial data, then ku₂(t)−u1(t)k is nonincreasing in [0, T]. In particular, there exists at most one solution to Problem (3.3) with a given initial data u⁰ ∈B.

Proof. Let u1 and u2 be two solutions to Problem (3.3), with possibly different intial data.

Choosing, for given t1 < t2 ∈ [0, T], v = u2 (respectively v = u1) in (3.4) with u = u1

(respectively u=u₂), and adding the two obtained inequalities, we get Z t2

t1

h∂_tu₁(t)−∂_tu₂(t), u₂(t)−u₁(t)idt≥0.

Taking into account (3.2), it yields 1

2ku₂(t2)−u1(t2)k² ≤ 1

2ku₂(t1)−u1(t1)k²,

which proves the claim.

Lemma 3.2 Under Hypothesis 2.6, let T >0 be given, f ∈L²(0, T;H) and u⁰ ∈B. Then any solution u to Problem (3.3) verifies the a priori estimates

ku(t)k ≤ ku⁰k+ 2√

tkfk_L2(0,T;H), for all t∈[0, T], (3.8) Z T

0

ψ(u(t))dt≤ 1 2

ku⁰k+ 2√

Tkfk_L2(0,T;H)

2

. (3.9)

Proof. We take v= 0 in (3.4), which gives for allt₁ < t₂ ∈[0, T] 1

2ku(t₂)k²+ Z t2

t1

ψ(u(t))dt≤ 1

2ku(t₁)k²+ Z t2

t1

hf(t), u(t)idt. (3.10) In particular this implies by taking t₁= 0 that for allt∈[0, T],

ku(t)k² ≤ ku⁰k²+ 2kfk_L2(0,T;H)

Z t 0

ku(s)k²ds 1/2

. (3.11)

Definingϕ(t) =Rt

0ku(s)k²ds, it satisfies the differential inequality ϕ⁰(t)≤ ku⁰k²+ 2kfkp ϕ(t), which implies that

d dt

ϕ(t) ku⁰k²+ 2kfkp

ϕ(t)

!

≤1.

Taking into account that ϕ(0) = 0, we deduce that ϕ(t) ku⁰k²+ 2kfkp

ϕ(t) ≤t, i.e.

ϕ(t)−2tkfkp

ϕ(t)−tku⁰k² ≤0, which yields

pϕ(t)≤tkfk+ t²kfk²+tku⁰k²1/2

.

Plugging this into (3.11) we obtain

ku(t)k² ≤ ku⁰k²+ 2kfk

tkfk+ t²kfk²+tku⁰k²1/2

≤ ku⁰k²+ 4tkfk²+ 2√

tku⁰kkfk

≤ ku⁰k+ 2√ tkfk2

,

(3.12)

which proves (3.8). Finally, coming back to (3.10) we easily get (3.9).

Remark 3.3 A generalization of the previous result is as follows. Under Hypothesis 2.6, let T >0 be given, f₁, f₂ ∈L²(0, T;H) and u⁰₁, u⁰₂ ∈B. Then two solutions u₁, u₂ to the Problem (3.3) with respective data verify

ku₂(t)−u₁(t)k ≤ ku⁰₂−u⁰₁k+ 2√

tkf₂−f₁k_L2(0,T;H), ∀t∈[0, T]. (3.13) This is obtained by taking as test function v =u₂ for the formulation associated to u₁, taking v =u₁ as test function for the formulation associated to u₂, adding the results and arguing as above by a Gronwall lemma.

Remark 3.4 The lemmas 3.1 and 3.2 indeed only use that u⁰ ∈ H, and not that u⁰ ∈ B.

However, in order to get an estimate on ∂_tu, the propertyψ(u⁰)<∞ is needed, as we shall see below.

3.2 Time-space implicit approximation

In this subsection we consider the space approximation (2.15), but applied to the transient prob- lem of the previous subsection. We thus consider the following approximate method. Assuming Hypothesis 2.6, let Hb be a closed subspace of H, and let Bb = Hb ∩B. We first approximate u⁰ ∈B by some

bu⁰ ∈B,b satisfying ψ(bu⁰)≤G, ψ(u⁰)≤G, (3.14) for some constant G≥0. We then take n∈N^?, we define the timestepτ =T /n, the sequence (f^k)_k=1,...,n of elements ofH and the function fn(t) by

f^k= 1 τ

Z kτ (k−1)τ

f(t)dt, ∀k= 1, . . . , n,

fn(t) =f^k, for a.e. t∈](k−1)τ, kτ[, ∀k= 1, . . . , n.

(3.15)

The sequence (bu^k)_k=1,...,n is then defined by ub^k ∈B,b

hD^ku, wb −bu^ki+ψ(w)−ψ(bu^k)≥ hf^k, w−ub^ki, ∀w∈B,b ∀k= 1, . . . , n, (3.16) where (D^ku)b _k=1,...,n is expressed in terms ofub^k and ub^k−1 by

D^kub= ub^k−ub^k−1

τ , k = 1, . . . , n. (3.17)

The existence and uniqueness of (ub^k)k=1,...,n solution to (3.16) is obvious since this problem has the same form as (2.15) with α= 1/τ andf =f^k+bu^k−1/τ. Let us denote by

u(t) =b ub^k andDtu(t) =b D^kbu, ∀t∈](k−1)τ, kτ], ∀k= 1, . . . , n,

u(0) =b ub⁰, (3.18)

henceu(t) is defined for allb t∈[0, T]. We have the following estimates on buand D_tu.b

Lemma 3.5 Under Hypothesis 2.6, let T >0, f ∈L²(0, T;H) and u⁰ ∈B be given. Let Hb be a closed subspace of H, and let Bb=Hb ∩B. Let bu⁰ ∈Bb be such that (3.14) holds. Let n∈N^?, τ = T /n, and let bu and D_tbu be defined by (3.15)-(3.18). Then, for C₂ given by (3.25), there holds

ψ(bu(t))≤C₂/2, ∀t∈[0, T], (3.19) kD_tbuk²_L2(0,T;H)≤C2, (3.20) kbu(t)k ≤ kub⁰k+ (T C2)^1/2, ∀t∈[0, T], (3.21) and

ku(tb 2)−u(tb 1)k ≤(C2)^1/2(|t₂−t1|+τ)^1/2, ∀t₁, t2∈[0, T]. (3.22) Proof. Let us first remark that, according to the Cauchy-Schwarz inequality,

kf^kk² ≤ 1 τ

Z kτ (k−1)τ

kf(t)k²dt, which leads to

n

X

k=1

τkf^kk² ≤ kfk²_L2(0,T;H). (3.23) Settingw=ub^k−1 in (3.16), we get

τkD^kbuk²+ψ(ub^k)−ψ(ub^k−1)≤ hf^k,bu^k−bu^k−1i ≤ τ

2kf^kk²+τ

2kD^kukb ², (3.24) which gives, summing for k= 1, . . . , mfor a givenm= 1, . . . , n,

ψ(bu^m) +1

2kD_tukb ²_L2(0,mτ;H)≤ψ(bu⁰) + 1

2kfk²_L2(0,T;H)

≤G+1

2kfk²_L2(0,T;H). This gives (3.19) and (3.20) with

C₂ = 2G+kfk²_L2(0,T;H). (3.25) We then write

bu^m =bu⁰+

m

X

k=1

τ D^ku,b ∀m= 1, . . . , n, which implies, using the Cauchy-Schwarz inequality and (3.20)

kub^m−ub⁰k ≤

m

X

k=1

τkD^kuk ≤b (mτkD_tukb ²_L2(0,mτ;H))^1/2 ≤(T C₂)^1/2. (3.26)

This leads to (3.21). Finally, lett1 < t2 ∈[0, T] andk1, k2 = 0, . . . , nsuch that (k1−1)τ < t1 ≤ k1τ and (k2−1)τ < t2 ≤k2τ, which implies that (k2−1−k1)τ < t2−t1. We have

ku(tb 2)−u(tb 1)k ≤

k2

X

k=k1+1

τkD^kbuk ≤



(k2−k1)τ

k2

X

k=k1+1

τkD^kbuk²





1/2

,

which provides (3.22), using (3.20).

We may now prove the following convergence result.

Theorem 3.6 Under Hypothesis 2.6, let T > 0, f ∈ L²(0, T;H) and u⁰ ∈ B be given. Let (Hb_n)n∈N^? be a sequence of closed subspaces of H, and let Bb_n=Hb_n∩B. We assume that

n→∞lim inf

w∈Bbn

kw−vk+ (ψ(w)−ψ(v))⁺

= 0, ∀v∈B. (3.27)

For all n∈N^?, let bu⁰_n satisfy

ub⁰_n∈Bbn, kbu⁰_n−u⁰k →0 as n→ ∞, G:= sup

n

ψ(ub⁰_n)<∞, (3.28) which is possible according to (3.27). Let τ =T /n, let ub_n andD_tub_n be defined by (3.15)-(3.18) where Hb has to be replaced by Hbn and ub⁰ = bu⁰_n. Then ubn(t) weakly converges in H to u(t) as n → ∞, uniformly with respect to t ∈ [0, T], and u is solution to Problem (3.3). Moreover, u satisfies

ψ(u(t))≤ 1

2C2, ∀t∈[0, T], (3.29)

k∂_tuk²_L2(0,T;H) ≤C2, (3.30) ku(t₂)−u(t₁)k ≤(C₂)^1/2|t₂−t₁|^1/2, ∀t₁, t₂∈[0, T], (3.31) where C2 = 2G+kfk²_L2(0,T;H).

Remark 3.7 Letting Hbⁿ = H and ub⁰_n = u⁰ for all n ∈ N^? (semi-discretization in time), Theorem 3.6 provides the existence of a solution to Problem (3.3), and allows for taking G = ψ(u⁰) and C₂= 2ψ(u⁰) +kfk²_L2(0,T;H) in (3.29)-(3.31), since the solution is unique.

Remark 3.8 Theorem 3.11 shows indeed stronger convergence properties.

Proof of Theorem 3.6. Applying Lemma 3.5, we get that the hypotheses of the Ascoli- type Lemma 5.3 (provided in appendix) are fulfilled, from which we deduce that there exists u ∈ C⁰([0, T];H) and of a subsequence of (ub_n)n∈N^?, again denoted (ub_n)n∈N^?, such that bu_n(t) converges tou(t) weakly inH, uniformly for t∈[0, T]. Note that, according to (3.28), we have u(0) = u⁰. Using (3.19) and the lower semi-continuity of ψ, we get that (3.29) holds, and in particular thatu(t)∈B andu∈B_T. Next, (3.31) comes directly from (5.10). Then, according to (3.20), we have that (Dtubn)n∈N is bounded in the Hilbert space L²(0, T;H). Hence extract- ing a subsequence, Dtbun weakly converges in L²(0, T;H) to some w ∈ L²(0, T;H) satisfying kwk²_L2(0,T;H)≤C₂. We then have, for a given ϕ∈C_c¹(]0, T[),

− Z T

0 ub_n(t)ϕ⁰(t)dt=−

n

X

k=1

ub^k Z kτ

(k−1)τ

ϕ⁰(t)dt=

n

X

k=1

τ D^kbu ϕ((k−1)τ).