A new error estimate on uniform norm of a parabolic variational inequality with nonlinear source terms via the subsolution concepts

This paper deals with the numerical analysis of parabolic variational inequalities with nonlinear source terms, where the existence and uniqueness of the solution is provided by using Banach’s fixed point theorem. In addition, an optimally L∞-asymptotic behavior is proved using Euler time scheme combined with the finite element spatial approximation. The approach is based on Bensoussan–Lions algorithm for evolutionary free boundary problems using the concepts of subsolutions.

In this paper, we give an L ∞ -error estimate for the numerical approximation of the solution of problem (1.1). From [2] (see also [8]), we know that (1.1) can be approximated by the following parabolic variational inequality with nonlinear source terms (PVI): Find u(x, t) such that u ∈ L 2 (0, T; H 1 0 (Ω)), ∂u ∂t ∈ L 2 (0, T; L 2 (Ω)), and u(x, 0) = u 0 (x) in Ω, where a(·, ·) is a bilinear form continuous on H 1 (Ω) × H 1 (Ω) corresponding to elliptic operator L of second order defined as follows: with a jk (·), b j (·), a 0 (·), smooth coefficients satisfying the following conditions: and for each ξ ∈ R d and for almost every x ∈ Ω, d j,k=1 a jk (x)ξ j ξ k ≥ α 0 |ξ | 2 with constant α 0 > 0. (1.6) According to Theorem 2.3 in [8], there exists γ > 0 such that The function f (·) is a nondecreasing and Lipschitz continuous nonlinearity with Lipschitz constant α > 0, satisfying the following assumption: where β is the constant defined in (1.3). The symbol (·, ·) is the inner product in L 2 (Ω). Error estimates for piecewise linear finite element approximations of parabolic variational inequalities with linear source terms have been established in various papers: in [20] and [9] an L 2 -error estimate is given by using a backward differencing in time. Also an L 2 -error estimate is given in [23] by using a general finite difference discretization in time. Reference [4] gives an L 2 -error estimate using the discretized truncation method. In [1] and [22] a posteriori error estimates have been proved. An L ∞ -error estimate has been proved in [15] and [17]. Recently an L ∞ -asymptotic behavior has been considered in [2] by using a semi-implicit time scheme combined with the finite element spatial approximation.
In this paper, we introduce a new approach to derive optimal L ∞ -asymptotic behavior for parabolic variational inequality with nonlinear source terms. This approach is based on Bensoussan-Lions algorithm for evolutionary free boundary problems using the concepts of subsolutions.
The paper is organized as follows. In Sect. 2, we state the continuous problem and study some qualitative properties. In Sect. 3, we consider the discrete problem and set up analogous discrete qualitative properties. In Sect. 4, we derive an L ∞ -error estimate of the approximation and we give the main result of the paper.

Time discretization
In order to obtain a full discretization of (1.3), we consider a uniform mesh for the time variable t and define t n = n t, n = 0, 1, . . . , N , (2.1) t > 0 being the time-step, and N = [ T t ], the integral part of T t . Next, we replace the time derivative by means of suitable difference quotients, thus constructing a sequence u n (x) ∈ H 1 0 (Ω) that approaches u(t n , x). For simplicity, we confine ourselves to the so-called semi-implicit scheme, which consists in replacing (1.3) by the following scheme: Find u n ∈ H 1 0 (Ω) such that By adding ( u n-1 t , vu n ) to both parties of inequalities (2.2), we get ⎧ ⎨ As the bilinear form a(·, ·) is noncoercive in (2.5) Then the bilinear form b(u, v) is an elliptic, and therefore (2.4) can be written as the following coercive elliptic variational inequalities: Find u n ∈ H 1 0 (Ω) such that Remark 1 Equation (2.6) is called the coercive continuous problem of elliptic variational inequalities (VI).

Existence and uniqueness
Next, using the preceding assumptions, we prove the existence of a unique solution for problem (2.6) by means of Banach's fixed point theorem.

A fixed point mapping associated with continuous problem (2.6)
We consider the following mapping: is the solution to the following variational inequalities: Problem (2.10) being a coercive VI, thanks to [3] and [10], has one and only one solution.
Theorem 1 Under the preceding hypotheses and notation, the mapping T is a contraction in L ∞ (Ω) with a contraction constant ( α t+1 β t+1 ). Therefore, T admits a unique fixed point which coincides with the solution of problem (2.6).
Proof In [13], by taking λ = 1 t , we can easily get The mapping T clearly generates the following continuous algorithm.

A continuous iterative scheme
A continuous iterative scheme for the solution of problem (2.6) is given as follows.
Starting from u 0 = u 0 the solution of the following equation: Now, we give the following algorithm: where u n is the solution to (2.6).
Making use of the propriety of mapping T, we have the following geometric convergence result.
where u ∞ is the asymptotic solution of the problem of variational inequalities: Proof We adapt [2].
In what follows, we give some qualitative properties of the solution of problem (2.6).

Some qualitative properties of the solution of (2.6)
The solution u n of (2.6) possesses the following properties.

A continuous L ∞ -stability property
Proposition 2 Under conditions of Lemma 1, we have Then, from (1.5), it is easy to see that So, due to Lemma 1, it follows that Interchanging the role of F(u n ) andF n , we also get Then, from (2.8), it is easy to see that which completes the proof.

The concept of continuous subsolution property
(2.16) Theorem 2 (cf. [6]) Let X denote the set of such subsolutions, then the solution of (2.6) is the least upper bound of X.

The discrete problem
Let Ω be decomposed into triangles, and let τ h denote the set of all those elements; h > 0 is the mesh size. We assume that the family τ h is regular and quasi-uniform. We consider φ l , l = 1, 2, . . . , m(h), the usual basis of affine functions defined by φ l (M s ) = δ l,s , where M s is a vertex of the considered triangulation. Let us V h denote the standard piecewise linear finite element space such that The interpolation operator is applied to the function v continuous, defined by and B is the matrix with generic entries In the sequel of the paper, we use the discrete maximum assumption (d.m.p.). In other words, we assume that the matrix B is an M-matrix (cf. [14]).
Remark 2 Under the d.m.p., we achieve a similar study to that devoted to the continuous problem; therefore the qualitative properties and results stated in the continuous case are conserved in the discrete case.
As in the continuous situation, one can tackle the discrete problem by considering the equivalent formulation: Existence and uniqueness of a solution of problem (3.4) can be shown similarly to that of the continuous case provided the discrete maximum principle is satisfied.

A fixed point mapping associated with discrete problem (3.4)
We consider the following mapping: where ξ n h = σ h (f n (w), r h ψ) is a solution of the following discrete coercive VI: As in the continuous situation, one can define the following discrete iterative scheme.

A discrete iterative scheme
A discrete iterative scheme for the solution of problem (3.4) is given as follows.
Starting from u 0 h = u 0,h , the solution of the following equation: Now, we give the following algorithm: where u n h is a solution of problem (3.4). Using the above result, we are able to establish the following geometric convergence of sequence u n h .

is the asymptotic solution of problem of variational inequalities: Find u
(3.10) Proof It is very similar to that of the continuous case.
Under the d.m.p., the solution of discrete problem (3.4) possesses analogous properties to those of the continuous problem.

Some qualitative properties of the solution of (3.4)
As in the continuous situation, the solution u n h of system (3.4) possesses the following properties.

A monotonicity property
Proof It is very similar to that of the continuous case.

The concept of discrete subsolution
Definition 2 z n h ∈ V h is said to be a discrete subsolution for the system of quasi-variational inequalities (3.4) if

Finite element error analysis
This section is devoted to deriving an error estimate, in the maximum norm, between the nth iterates u n and their finite element counterpart u n h . For that we first introduce two auxiliary sequences.

A discrete sequence
We define the following discrete sequence {ū n h } n≥1 , whereū n h is a solution to the following discrete problem of variational inequalities (VI): where u n is the solution to (2.6).

Lemma 3 (cf. [13]) There exists a constant C independent of h, n, and t such that
Proposition 5 There exists a sequence of discrete subsolutions {α n h } n≥1 such that where the constant C is independent of h, t, and n.
Proof For n = 1, we consider the discrete problem of VI: Then asū 1 h is a solution to a discrete VI, it is also a subsolution, i.e., Using the Lipschitz continuity of f (·), we have On the other hand, due to [11] u 0u 0,h ∞ ≤ C h 2 | log h|.
So,ū 1 h is a discrete subsolution for the VI whose solution isŪ 1 h = ∂ h (f (u 0,h ) + C h 2 | log h|, r h ψ). Then u 1 h = ∂ h (f (u 0,h ), r h ψ), and making use of Proposition 4, we have Hence, making use of Theorem 4, we havē Putting Using Lemma 3, we get For n + 1, let us now assume that and we consider the discrete problem Using the Lipschitz continuity of f (·), we have Using (4.2), we have So,ū n+1 h is a discrete subsolution for the VI whose solution isŪ n+1 , r h ψ), making use of Proposition 4, we have Hence, applying Theorem 4, we get Using Lemma 3, we obtain which completes the proof.

A continuous sequence
We define the following continuous sequence {ū n (h) } n≥1 , whereū n (h) is a solution to the following continuous problem of variational inequalities (VI): where u n h is the solution of discrete problem (3.4).
Lemma 4 (cf. [13]) There exists a constant C independent of h, k, and n such that where the constant C is independent of h, n, and t. where the constant C is independent of h, t, and n.
Proof For n = 1, we consider the continuous problem of VI Then, asū 1 (h) is a solution to a continuous VI, it is also a subsolution, i.e., Using the Lipschitz continuity of f (·), we have On the other hand, due to [11] u 0u 0,h ∞ ≤ C h 2 | log h|.
Thus, we get u n hu n ≤ C h 2 | log h| 2 .
Therefore u nu n h ∞ ≤ C h 2 | log h| 2 , which completes the proof.

L ∞ -Asymptotic behavior
Now we estimate the order of the difference between u h (T, ·), the discrete solution calculated at the moment T = n t, and u ∞ , the solution of problem (2.13).
Theorem 6 (The main result) Under the conditions of Proposition 3 and Corollary 1, the following inequality holds: u h (T, ·)u ∞ (·) ∞ ≤ C h 2 | log h| 2 + α t + 1 β t + 1 N . (4.9) Proof We have u n h (t, ·) = u h (t, ·) for all t ∈ (n -1) t, n t , Indeed, applying the previous results of Proposition 3 and Corollary 1, we get Then the following result can be deduced: which completes the proof.