On variational inequalities with vanishing zero term
© Boulbrachene; licensee Springer. 2013
Received: 19 April 2013
Accepted: 8 August 2013
Published: 16 September 2013
In this paper, we are concerned with variational inequalities (VIs), where the ‘discount factor’ (i.e., the zero-order term) is set to zero. Especially, we introduce a new method for studying the finite element approximation, based on an algorithm of Bensoussan-Lions type and the concept of subsolutions.
as α tends to 0+.
Such problems play a fundamental role in the solution of problems of stochastic control with ergodic control type payoffs (see  and the references therein).
For , there exists a unique solution of (1.1) which belongs to , (see ).
where C is a constant independent of both α and h and denotes the -norm.
The finite element approximation of variational inequalities with vanishing zero-order term was first studied in  and error estimates were derived by means of the concept of subsolutions. In , a quasi-optimal convergence order was derived by adapting an algorithmic approach due to  for quasi-variational of impulse control problems. This method combines the approximation of both the continuous and discrete solutions of VIs (1.4) and (1.7) by monotone geometrically convergent iterative schemes of Bensoussan-Lions type and an estimation of the error in the maximum norm between the n th iterate of the iterative scheme and its finite element counterpart.
In this paper, this algorithmic approach is improved and optimal convergence order is derived. Besides the optimal convergence order (1.9) and (1.10), the other important novelty in this paper is the optimal error estimate that is established between the n th iterate of the iterative scheme and its finite element counterpart, which we achieve by employing the concept of subsolutions.
It is worth mentioning that the approach introduced in this paper is new and different from the one employed in . Moreover, it also has the merit of providing a basic computational scheme for the solution of (1.8). It may also be extended to the quasi-variational inequality of ergodic impulse control problems studied by Lions and Perthame .
The paper is organized as follows. In Section 2, we construct a monotone iterative scheme and establish its geometrical convergence to the solution of VI (1.4). In Section 3, the same study is reproduced for the discrete problems. Section 4 is devoted to the finite element error analysis and proofs of the main results of this paper.
2 The continuous problem
Notice that the bilinear form (2.2) is independent of α, as its zero-order term is equal to 1.
2.1 Construction of monotone sequences for VI (1.4)
Thanks to , problem (2.6) has a unique solution which belongs to , . Moreover, is independent of α, as the bilinear form (2.2) is itself independent of α.
Lemma 1 Let . Then the mapping T is increasing, concave, and satisfies , .
Proof It is an easy adaptation of . □
As a result of Lemma 1, it is clear that both sequences and are well defined in ℂ. Moreover, they are monotone decreasing and increasing, respectively.
Next, we shall establish the geometrical convergence of these sequences.
2.2 Geometrical convergence
Lemma 2 The solution of VI (1.4) or (2.1) satisfies .
because and .
which completes the proof. □
Remark 1 The constant μ defined in (2.11) is independent of α as ψ, f, and are themselves independent of α.
Thus, (2.15) follows. □
Next, we shall give the existence and uniqueness for VI (1.5).
Theorem 2 The solution of VI (1.4) converges uniformly in and strongly in , , to , the unique solution of VI (1.5).
Proof For uniqueness, see . Let us give the existence.
proving the strong convergence in . □
3 The discrete problem
We assume that Ω is polyhedral. The extension to the general case can be set up by the usual techniques (see ). Let be a regular and quasi-uniform triangulation of Ω consisting of triangles of diameter less than h. Let also , , be the basis functions of , and be the stiffness matrix associated with the bilinear form .
As in the continuous case, we shall construct two discrete sequences and prove their geometrical convergence to the solution of VI (1.7).
3.1 Construction of monotone sequences for VI (1.7)
Lemma 4 Let . Then, under the DMP, the mapping is increasing, concave, and satisfies , .
Thanks to Lemma 4, the sequences and are well defined in . Moreover, they are monotone decreasing and increasing, respectively.
3.1.1 Geometrical convergence
As in the continuous case, in order to establish the geometrical convergence of sequences (3.6) and (3.7), we shall need the following lemmas. Their proofs will be omitted as they are very similar to those of their respective continuous counterparts.
Lemma 5 Let the DMP hold. Then the solution of VI (1.7) or (3.1) satisfies .
Lemma 6 Let . Then, under the DMP, the mapping is increasing, concave, and satisfies , .
Remark 2 Thanks to Lemma 6, sequences (3.6) and (3.7) are well defined in . Moreover, they are monotone decreasing and increasing, respectively.
Remark 3 The constant μ is independent of α as ψ, f, and are themselves independent of α.
4 -Error estimates
This section is devoted to proving the main results of this paper. For that, let us recall some useful properties enjoyed by elliptic variational inequalities of obstacle type.
4.1 Elliptic variational inequality
Theorem 4 
Let X denote the set of continuous subsolutions. Then the solution ω of VI (4.1) is the least upper bound of the set X.
Theorem 5 
Theorem 6 Let denote the set of discrete subsolutions. Then, under the DMP, the solution of VI (4.3) is the least upper bound of the set .
Let us now introduce two auxiliary variational inequalities.
4.2 Two auxiliary sequences of variational inequalities
where is defined in (2.7).
Proof Since (independent of α, h, and n) and (independent of α, h, and n), making use of , we get both (4.6) and (4.7). □
4.3 Optimal -error estimates
Next, we shall estimate the error in the maximum norm between the n th iterates and defined in (2.7) and (3.6), respectively.
In order to prove Theorem 8, we need the following lemma.
Theorem 10 The solution of VI (1.7) converges, as , uniformly in to , the solution of discrete VI (1.8).
The rest of the proof is similar to that of Theorem 2. □
Next, combining Theorems 2, 9, and 10, we are in a position to derive the main result of this paper.
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