Convergence of double step scheme for a class of parabolic Clarke subdifferential inclusions

In this paper we deal with a first order evolution inclusion involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a double step time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem.


Introduction
We consider a class of evolutionary inclusions of parabolic type.A multivalued term, which appears in our problem, has a form of Clarke subdifferential of a locally Lipschitz function.Such problems are closely related to so called hemivariational inequalities (HVIs) in the sense that the solution of HVI can be found by solving a corresponding inclusion, as is presented in Section 6. HVIs are generalization of variational inequalities and they play very important role in modelling of various problems arising in mechanics, physics, and the engineering sciences.
The theory of HVIs has been introduced by Pangiotopulos in 1980s [19] in order to describe several important mechanical and engineering problems with nonmonotone phenomena in solid mechanics.The concept of HVIs is based on the notion of Clarke subdifferential of a locally Lipschitz functional that may be nonconvex -see [6].The theory of HVIs has been rapidly developed in the last decades as is illustrated in [7,10,16] for example.For more recent results on theory and applications of HVIs see [15,21].In addition to the theory, numerical aspects of discussed problems began to be developed especially in the last few years.They involve various kinds of discrete schemes based on temporal and spatial discretization.In our paper we deal with the first kind of discretization, known as Rothe method.We are motivated by [12], in which Rothe method was used for the first time to solve an evolutionary inclusion involving Clarke subdifferential.In the quoted paper the time discretization technique based on the implicit (backward) Euler scheme was used.In this approach the time derivative of unknown function u is approximated by a finite difference according to the role u ′ (t n ) ≃ 1 τ (u n − u n−1 ), where τ represents the time step.The result presented there has been developed in [13] by applying more general θ-scheme.Both results deal with evolutionary inclusions of parabolic type.In [4,5], Rothe method has been applied for more general parabolic problem, namely variational-hemivariational inequalities.The techniques introduced in [12,13] has been successfully adapted to a second order evolutionary inclusion in Chapter 5 of [10] and in [2,3].We also refer to [11,14,17,18,20] for more result concerning Rothe method in analysis of various kinds of evolutionary hemivariational inequalities.In this paper we deal with the parabolic Clarke inclusion which has been already studied in [12].But this time we use two steps backward differential formula (BDF) based on the idea that the time derivative u ′ (t n ) is approximated by the derivative of the unique second order polynomial that interpolates the approximate solution in three points, namely Hence, it seems to be more accurate then the standard implicit Euler scheme used in [12], which involves only two points to approximate the time derivative.On the other hand, two steps BDF scheme is more complicated and its analysis requires more sophisticated technique than the one used in previous related papers.Our approach relies on techniques from [9], where BDF for non-Newtionian fluids was considered.Nevertheless we think that our efforts are fruitful and the presented result opens new perspectives in numerical analysis of nonlinear evolutionary inclusions.The rest of the paper is structured as follows.In Section 2 we introduce preliminary materials and recall basic results to be used letter.In Section 3 we formulate the problem that is an abstract evolution inclusion of parabolic type involving Clarke subdifferential.We also list all assumptions on the data of the problem.In Section 4 we introduce a time semidiscrete Rothe scheme corresponding to the original problem and provide the existence result for the approximate one.Moreover we derive a-priori estimates for the approximate solution.In Section 5 we provide the convergence result, which shows that the sequence of solutions of semidiscrete problem converges weakly to the solution of the original problem as the discretization parameter converges to zero.Finally, in Section 6, we provide a simple example of parabolic boundary problem for which our theoretical result can be applied.

Notation and preliminaries
In this section we introduce notation and recall several known results that will be used in the rest of the paper.
Let X be a real normed space.Everywhere in the paper we will use the symbols • X , X * and •, • X * ×X to denote the norm in X, its dual space and the duality pairing of X and X * , respectively.Moreover, if Y is a normed space and f ∈ L(X, Y ), we will briefly write f instead of f L(X,Y ) and we will use notation f * : Y * → X * for the adjoint operator to f .We start with the definition of the Clarke generalized directional derivative and the Clarke subdifferential.Definition 2.1 Let ϕ : X → R be a locally Lipschitz function.The Clarke generalized directional derivative of ϕ at the point x ∈ X in the direction v ∈ X, is defined by The Clarke subdifferential of ϕ at x is a subset of X * given by In what follows, we recall the definition of pseudomonotone operator in both single-valued and multivalued cases.
1) A has values which are nonempty, bounded, closed and convex.
2) A is upper semicontinuous (usc, in short) from every finite dimensional subspace of X into X * endowed with the weak topology.
3) For any sequence The following two propositions provide an important class of pseudomonotone operators that will appear in the next section.They correspond to Proposition 5.6 in [10] and Proposition 1.3.68 in [8], respectively.Proposition 2.4 Let X and U be two reflexive Banach spaces and ι : X → U a linear, continuous and compact operator.Let J : U → R be a locally Lipschitz functional and assume that its Clarke subdifferential satisfies with c > 0. Then the multivalued operator M : X → 2 X * defined by Proposition 2.5 Assume that X is a reflexive Banach space and A 1 , A 2 : X → 2 X * are pseudomonotone operators.Then the operator In what follows we introduce the notion of coercivity.
Definition 2.6 Let X be a real Banach space and A : X → 2 X * be an operator.We say that A is coercive if either D(A) is bounded or D(A) is unbounded and where, recall, The following is the main surjectivity result for multivalued pseudomonotone and coercive operators.
Proposition 2.7 Let X be a real, reflexive Banach space and A : X → 2 X * be pseudomonotone and coercive.Then A is surjective, i.e., for all b ∈ X * there exists v ∈ X such that Av ∋ b.
Let X be a Banach space and T > 0. We introduce the space BV (0, T ; X) of functions of bounded total variation on [0, T ].Let π denotes any finite partition of [0, T ] by a family of disjoint subintervals {σ i = (a i , b i )} such that [0, T ] = ∪ n i=1 σi .Let F denote the family of all such partitions.Then, for a function x : [0, T ] → X and for 1 ≤ q < ∞, we define a seminorm and the space BV q (0, T ; X) = {x : [0, T ] → X| x BV q (0,T ;X) < ∞}.
Then M p,q (0, T ; X, Z) is also a Banach space with the norm given by • L p (0,T ;X) + • BV q (0,T ;Z) .
The following proposition will play the crucial role for the convergence of the Rothe functions which will be constructed later.For its proof, we refer to [12].
The following version of Aubin-Celina convergence theorem (see [1]) will be used in what follows.Let x n : (0, T ) → X, y n : (0, T ) → Y , n ∈ N, be measurable functions such that x n converges almost everywhere on (0, T ) to a function x : (0, T ) → X and y n converges weakly in ) for all n ∈ N and almost all t ∈ (0, T ), then y(t) ∈ F (x(t)) for a.e.t ∈ (0, T ).
We conclude this section with a well known Young's inequality

Problem formulation
In this section we formulate an abstract evolutionary inclusion of parabolic type involving Clarke subdifferential.We also impose assumptions on the data of the problem.Let V be a real, reflexive, separable Banach space and H be a real, separable Hilbert space equipped with the inner product (•, •) H and the corresponding norm given by v H = (v, v) H for all v ∈ H.For simplicity of notation we will write Identifying H with its dual, we assume, that the spaces V, H and V * form an evolution triple, i.e., V ⊂ H ⊂ V * with all embeddings being dense and continuous.Moreover, we assume, that the embedding V ⊂ H is compact.Let i : V → H be an embedding operator (for v ∈ V we still denote iv ∈ H by v).For all u ∈ H and v ∈ V , we have u, v = (u, v).We also introduce a reflexive Banach space U and the operator ι ∈ L(V, U ).For T > 0, we denote by [0, T ] a time interval and introduce the following spaces of time dependent functions: Hereafter, v ′ denotes the time derivative of v in the sense of distribution.
We consider the operator A : V → V * and the functions f : [0, T ] → V * , J : U → R. Using the above notation, we formulate the following problem.
Problem P. Find u ∈ V such that u(0) = u 0 , and Now we impose assumptions on the data of Problem P.

H(A)
The operator A :

H(J)
The functional J : U → R is such that (i) J is locally Lipschitz, (ii) ∂J satisfies the following growth condition for all u ∈ U, ξ ∈ ∂J(u) with d > 0.

H(ι)
The operator ι : V → U is linear, continuous and compact.Moreover, there exists a Banach space Z such that V ⊂ Z ⊂ H, where the embedding V ⊂ Z is compact, the embedding Z ⊂ H is continuous, and the operator ι can be decomposed as ι = ι 2 • ι 1 , where ι 1 : V → Z denotes the (compact) identity mapping and ι 2 ∈ L(Z, U ).
In the rest of the paper we always assume that assumptions H(A), H(J), H(ι), H(f ) and H(0) hold.
The direct consequence of Corollary 3.1 is following.Corollary 3.2 For every ε > 0 we have

The Rothe problem
In this section we consider a semidiscrete approximation of Problem P known as Rothe problem.
Our goal is to study a solvability of the Rothe problem, to obtain a-priori estimates for its solution and to study the convergence of semidiscrete solution to the solution of the original problem as the discretization parameter converges to zero.We start with a uniform division of the time interval.Let N ∈ N be fixed and τ = T /N be a time step.In the rest of the paper we denote by c a generic positive constant independent on discretization parameters, that can differ from line to line.Let u 0 τ ∈ V be given and assume that (4.1) We define the sequence {f n τ } N n=0 by the formula We now formulate the following Rothe problem.
and for n = 2, ..., N Now we formulate the existence result for Problem P τ .Theorem 4.1 There exists τ 0 > 0 such that for all 0 < τ < τ 0 , Problem P τ has a solution.
Proof.We can formulate (4.3) as: find where, recall, i : V → H is the identity mapping, and i * denotes its adjoint operator.We observe, that T 1 is pseudomonotone.In fact, the operator V ∋ v → i * iv ∈ V * is pseudomonotone, as it is linear and monotone (cf.[22]).Moreover, the operator τ A is pseudomonotone by assumption H(A)(i) and the operator τ ι * ∂J(ιv) is pseudomonotone by by assumption H(J)(ii) and by Proposition 2.4.Hence, the operator T 1 is pseudomonotone by Proposition 2.5.In order to check coercivity, let η ∈ T 1 v, which means, that η = i * iv + τ Av + τ ι * ξ, whith ξ ∈ ∂J(ιv).Then, using H(A)(ii) and (3.3), we have Taking ε and τ small enough, we find out that Then, it follows that T 1 is coercive.Hence, by Proposition 2.7 operator T 1 is surjective for τ < τ 0 , where τ 0 > 0 and problem (4.7) has a solution.
Next we proceed recursively.For n = 2, ..., N having u n−1 τ and u n−2 τ we formulate (4.5) as: find where 3 τ Av + 2 3 τ ι * ∂J(ιv) for all v ∈ V .Proceeding analogously as in the first part of the proof, we can easy show that T 2 satisfies assumptions of Proposition 2.7 for τ small enough, which guaranties the solvability of (4.8) and completes the proof.
Now we provide a result on a-priori estimates for the solution of Problem P τ .We denote by c a generic positive constant independent of τ that may differ from line to line.Moreover, if c depends on ε, we write c(ε).

Lemma 4.3
The following convergence holds Proof.For the proof it is enough to show, that for any subsequence of the sequence {u 1 τ − u 0 τ } one can find a subsequence, which converges to 0 strongly in H. Suppose than, that {u 1 τ − u 0 τ } is any subsequence of the original sequence (denoted by the same symbol for simplicity).From (4.25) we know that {u 1 τ − u 0 τ } is bounded in H, hence there exists η ∈ H, such that for a subsequence (again denoted by the same symbol), there holds (4.33) From (4.12) we have Hence u 1 τ − u 0 τ → 0 strongly in V * , hence also weakly in V * .On the other hand, from (4.33), u 1 τ − u 0 τ → η weakly in H, hence also weakly in V * .From the uniqueness of weak limit in V * we have η = 0, hence (4.34) In what follows we will use the identity After some reformulation, we have from (4.3) From (4.1) and (4.34) we have Now we estimate the right-hand side of (4.36) in two steps.First we get By Jensen's inequality and by (4.9) τ u 1 τ 2 ≤ c.Hence, from (4.38) we have as τ → 0.Then, using H(A)(iii) and a slight modification of (3.3) we get From (4.36),(4.37),(4.39),(4.40)and the squeeze theorem we get Combining it with (4.35) we obtain the thesis.
Basing on the solution of the Rothe problem P τ we define the following functions ūτ , u τ : Lemma 4. 4 We have the following convergence result: Proof.We observe that Hence, we conclude Hence, the result follows from (4.12)-(4.14).
At the end of this section, we observe, that due to ( On the other hand, by the definition of function u τ , we have Hence, by Lemma 4.3 and by (4.1), we get u τ (0) → u 0 strongly in H, so also weakly in H. Comparing it with (5.9) we conclude, from uniqueness of the weak limit in H that u τ (0) → u 0 = u(0) strongly in H. (5.10)It follows from (5.1) that (5.11) We pass to the limit with (5.11) as τ → 0. From (5.7) we have (5.12) From (5.8) we have By standard arguments Then, we have (5.15)fτ → f strongly in V * .
From (4.47) and Proposition 2.9 we have for a subsequence where ῑ : V → L 2 (0, T ; Z) is the Nemytskii operator corresponding to ι 1 .We have First of all we will deal with two parts separately, namely To deal with the first term, we will use (5.6) and the convergence |u τ (0) − u(0)| → 0 (see (5.10)).Hence ) .
Keeping in mind the initial condition, we claim that u solves Problem Q V .Hence, in order to get a weak solution of Problem Q, it is enough to solve Problem QV .That's why we will concentrate on solvability of the last problem.First of all we notice that it corresponds to Problem P of Section 3, with ι = γ.Moreover, we will show, that Theorem 5.2 can be applied in our case.To this end we will examine all assumptions stated in Section 3. First we notice, that operator A is pseudomonotone, as it is linear and monotone.Hence H(A)(i) holds.Moreover, an elementary calculation shows that for all v ∈ V we have Hence, the assumption H(A)(ii) holds with the constants a = 0, b = 1.It is also clear that for all v ∈ V one has Av, v = v 2 − |v| 2 , so H(A)(iii) holds with α = 1 and β = 1.Now we study the properties of functional J.By Theorem 3.47 (iii) of [16], it is locally Lipschitz, hence H(J)(i) holds.Moreover, an elementary calculation shows that J satisfies H(J)(ii) with the constant d = √ 2d j max{1, |Γ C |}.Now let us show that the trace operator γ satisfies assumption corresponding to H(ι).Taking Z = H δ (Ω) with δ ∈ [ 1  2 , 1], we notice that the embedding V ⊂ Z is compact.Hence, letting ι 1 : V → Z be the identity mapping, we claim that it is compact.Moreover, we define an auxiliary trace mapping ι 2 : Z → U .Then, clearly γ = ι 2 • ι 1 and the assumption H(ι) holds fo ι = γ.It is also clear that assumption H(f ) is a consequence of H(f 0 ) and H(f N ).Now it is enough to impose u 0 ∈ H to fulfil the last assumption H(0).
In this way we have shown that Theorem 5.2 is applicable in our case, namely the weak solution of Problem Q can be approximated by means of a double step Rothe scheme described in Section 4.
At the end of this section we provide a simple example of a function j : R → R which satisfies assumptions H(j).Let it be defined by the formula j(s) = 0 for s < 0 −de −s + 1 2 ds 2 + d for s ≥ 0.
Then the Clarke subdifferential of j is given by for s = 0 de −s + s for s ≥ 0.
It is easy to check that the function j satisfies assumptions H(j)(i)-(ii).In particular, the constant d from the assumption H(j)(ii) and the constant d used in the formula of j coincide.

Proposition 2 .
10 Let X and Y be Banach spaces, and F : X → 2 Y be a multifunction such that (a) the values of F are nonempty, closed and convex subsets of Y , (b) F is upper semicontinuous from X into w − Y .