GALERKIN APPROXIMATION FOR A SEMI LINEAR PARABOLIC PROBLEM WITH NONLOCAL BOUNDARY CONDITIONS

We analyze a θ-method for the numerical solution of a semi linear parabolic problem with boundary conditions containing integrals over the interior of the interval. Existence and convergence are proved for θ ≥ 1/2, numerical application is given.


Introduction
This paper is concerned with the numerical solution of the semi linear parabolic problem ⎧ ⎪ ⎨ ⎪ ⎩ u t − (a(x)u x ) x − b(x)u x = F (u, x, t), α < x < β, 0 < t ≤ T, (1.1) subject to the nonlocal boundary conditions and the initial condition u(x, 0) = u 0 (x), α ≤ x ≤ β. (1.3)We assume that the function F is Lipschitz continuous on compact sets of R, the functions a and b x are supposed to be continuous and that there exist positive constants a 1 , a 2 , b 1 , b 2 such that This kind of initial-boundary-value problem arises in the quasi-static theory of thermoelasticity [2,3].The existence, uniqueness and some analytic properties of the solution of (1.1)- (1.3), have been studied by Day [2,3], Friedman [6], Kawohl [7], when b(x) = 0, α = 0 and β = 1 under the hypothesis that We assume that the functions u, u 0 , g 0 , g 1 are sufficiently smooth and the functions K 0 and K 1 satisfy Fairweather and López-Marcos [5] considered the Crank-Nicholson method and the extrapolated Crank-Nicholson method for the problem (1.1)- (1.3) when b(x) = 0, α = 0, β = 1 using an energy argument, they proved the convergence of both methods under the condition (1.6), Fairweather, López-Marcos and A.Boutayeb [11] analyzed an orthogonal spline collocation method for a quasi linear parabolic problem, recently the present authors [1] considered several methods of extrapolations for a linear parabolic problem with non local boundary conditions.The main purpose of this paper is to show the convergence of a θ-finite element Galerkin method (θ-method) for the solution of (1.1-1.3), and for θ ∈ [ 1  2 , 1], using similar energy argument to the one in [5].This paper is outlined as follows: Section 2 is devoted to the θ-method while the consistency, the stability, the convergence and the existence of the θ−method are treated in section 3, 4 and 5.In the last section we give some numerical results.

The θ-method
According to [5], let be (π h ) = (x j ) J j=0 a partition of [α, β] such that α = x 0 < x 1 < ... < x J = β, where h = max j (h j ), h j = x j − x j−1 and I j = [x j−1 , x j ], 1 ≤ j ≤ J, we consider a family of partitions (π h ) h∈H where H is a set of positif numbers with inf (H) = 0, we define the space S h in the following way

and S •
h , the subspace of S h , given by Where P I r (E) denotes the set of polynomials of degree at most r, r ≥ 2 for a closed interval E and n < r.We denote by k the time step, t m = mk, m = 0, ..., M = [T/k], t m+θ = (m + θ)k, and (., .) the usual inner product on L 2 ([α, β]).

With the nonlocal boundary conditions
For the analysis of this method we require to examine the consistency (sect.3)and the stability (sect.4) of the method, in order to prove the existence and convergence, according to the framework developed in [8,10].In the sequel C denotes a positive constant whose value is not the same on each occurrence and which is independent of h and k, we define the norms k.k and k.k ∞ , by where Wheeler [9]).
We define on S 0 h the linear form and the operators L 0 and L 1 We denote where (S 0 h ) * is the dual space of S 0 h .It is clear that We also define the operator by the relations It is easy to show that an element (U 0 , ..., U M ) ∈ (S h ) M +1 is a solution of the θ-method if and only if we also need in the analysis the norm It is well known that if K 0 and K 1 are zero then the θ−method is convergent for θ ≥ 1 2 and the order of convergence is 2 , our aim is to show that a similar result holds in the case of nonlocal boundary conditions.

Consistency
We consider the problem Whose solutions are the positive functions where and the functions φ 0 , φ 1 ∈ S h , defined by Clearly φ 0 and φ 1 are the elliptic projections of the functions v 0 and v 1 , respectively, and from (2.4) we have Furthermore, for ν > 0 there exists h 0 > 0 such that for each h ≤ h 0 , We have then the following theorem Theorem 3.1.Suppose that the solution of the problem (1. Proof.Integrating by parts in (1.1), we have Furthermore, using (2.3), (3.1), and (3.7)-(3.8), it follows that from the hypotheses of the theorem, we obtain On the other hand The following theorem ensures the existence of a suitable initial condition that satisfies (2.7), which is required for the θ-method.Theorem 3.2.Let V ∈ S h , for h sufficiently small, there exists a unique element V * ∈ S h such that (1.6) implies that By (3.6), and for h sufficiently small, we obtain (3.20) corollary 3.1.Let u 0 be the initial condition given in (1.3) and ũ0 h = ũh (t 0 ), for h sufficiently small, if the element ũ0 h, * is given by Proof.We use similar techniques as those in [[5], corollary 3.1].We define in S h the following norm Under the hypotheses of theorem 3.1, if ũh (x, t) is the function defined by (3.8) and the initial condition is chosen such that U 0 = ũh, * , where ũh, * ∈ S h is defined in corollary (3.1), and θ ≥ 1/2, then

Existence and convergence
We first, show a stability result of the θ-method.
Proof.Noting that for V ∈ S h , V ∈ S • h is given by taking into account (3.1) we get ´, and It follows by replacing all these inequalities in (4.4) that for h sufficiently small, the last three terms of (4.6) are bounded by combining (1.6), the Cauchy-Schwartz inequality, and the fact that (a + b) 2 ≤ (1 + δ) a 2 + ³ 1 + 1 δ ´b2 , for any real numbers a, b and for any δ > 0 we get If we substitute (4.7)-(4.8)into (4.6),we obtain 0 ≤ m ≤ M − 1.If ν and δ are chosen such that (1 + ν)µ 2 (1 + δ 4 ) < 1 and using the fact that the function F is Lipschitz continuous (4.8) takes the form ´− ³ On the other hand, using Sobolev's inequality we get ´, (4.12) and to conclude, we use Gronwal's inequality (discrete form), which gives, for k sufficiently small, Analogous to theorem 5.1 of [5], the following result holds: Theorem 4.2.Let be (ũ 0 h , ..., ũM h ) defined as in (3.9) and Φ hk the operator defined as in (2.5) with U 0 = ũ0 h, * if R is a fixed positif real, for h and k sufficiently small, (2.8) possesses a unique solution (U 0 , ..., U M ) ∈ B((ũ 0 h , ..., ũM h ), R) that satisfies The following corollary is a consequence of the above theorem.corollary 4.1.For θ ≥ 1/2, under the hypotheses of theorem 4.2, if (U 0 , ..., U M ) is the solution of (2.8), and consider where u is the solution of (1.1-1.3), then
Below are tables, representing sample calculations using three values for the parameter θ of the described methods.The first and second column in the table represents the mesh parameter M and the time steps N .The third column shows the error measured in maximum norm on the M × N mesh, covering [0, 1] × [0, T ] and the last column shows the CPU-time used, in seconds, for the computation.Comparing the above three tables, we find that the Crank-Nicholson method is, unsurprisingly, more efficient than the others.