ABSTRACT

This paper, we develop a numerical method for solving a unilateral obstacle problem by using the cubic spline collocation method and the generalized Newton method. This method converges quadratically if a relation-ship between the penalty parameter and the discretization parameter h is satisfied. An error estimate between the penalty solution and the discret penalty solution is provided. To validate the theoretical results, some numerical tests on one dimensional obstacle problem are presented.

1. Introduction

Let be a bounded open domain in with smooth boundary, and let be an element of with on. Set

We consider the following variational inequality problem:  

(1)

where f is an element of. This problem is called a unilateral obstacle problem. It is well known that problem (1) admits a unique solution u, and if, then u is an element of (see [1,2]). There are several alternative solution methods of the obstacle problem; see, e.g., [1,3-5]. Numerical solution by penalty methods have been considered, e.g. by [4,6]. In this paper we develop a numerical method for solving a one dimentional obstacle problem by using the cubic spline collocation method and the generalized Newton method. First, problem (1) is approximated by a sequence of nonlinear equation problems by using the penalty method given in [2,7]. Then we apply the spline collocation method to approximate the solution of a boundary value problem of second order. The discret problem is formulated as to find the cubic spline coefficients of a nonsmooth system, where. In order to solve the nonsmooth equation we apply the generalized Newton method (see [8-10], for instance). We prove that the cubic spline collocation method converges quadratically provided that a property coupling the penalty parameter and the discretization parameter h is satisfied.

Numerical methods to approximate the solution of boundary value problems have been considered by several authors. We only mention the papers [11,12] and references therein, which use the spline collocation method for solving the boundary value problems.

The present paper is organized as follows. In Section 2, we present the penalty method to approximate the obstacle problem by a sequence of second order boundary value problems. In Section 3 we construct a cubic spline to approximate the solution of the boundary problem. Section 4 is devoted to the presentation of the generalized Newton method. In Section 5 we show the convergence of the cubic spline to the solution of the boundary problem and provide an error estimate. Finally, some numerical results are given in Section 6 to validate our methodology.

2. Penalty Problem

Let be an element of with on. Assume that is an element of, then the solution u of problem (1) is an element of and can be characterized as (see [1], for instance):

(2)

The penalty problem is given by the following boundary value problem (see [10], p. 107, [12]):

(3)

where is a sequence of Lipschitz functions which tend to the function defined by

(4)

almost everywhere on, as goes to zero. Assume that the function, , is uniformly Lipschitz, non increasing and satisfy. Then problem (3) admits a unique solution (see [2] p. 107). We now specify the function

(5)

We have the interesting properties.

Theorem 1 ([2,7]) Let u denote the solution of the variational inequality problem (1) and, , denotes the solution of the penalty problem (3) with defined by relation (5). Then is a nondecreasing sequence and

3. Cubic Spline Collocation Method

In this section we construct a cubic spline which approximates the solution of problem (3), with is the interval and is the function given by (5).

Cubic Spline Solution

Let

be a subdivision of the interval I. Without loss of generality, we put, where and.

Denote by the space of piecewise polynomials of degree 3 over the subdivision and of class everywhere on. Let, , be the B-splines of degree 3 associated with. These Bsplines are positives and form a basis of the space. If we put

(6)

then problem (3) becomes

(7)

It is easy to see that is a nonlinear continuous function on; and for any two functions and, satisfies the following Lipschitz condition:

(8)

where

Now, we define the following interpolation cubic spline of the solution of the nonlinear second order boundary value problem (7).

Proposition 2 Let be the solution of problem (7). Then, there exists a unique cubic spline interpolant of which satisfies:

where, , , and.

Proof Using the Schoenberg-Whitney theorem (see [13]), it is easy to see that there exits a unique cubic spline which interpolates at the points, .  

If we put, then by using the boundary conditions of problem (7) we obtain

, and

Hence

Furthermore, since the interpolation with splines of degree d gives uniform norm errors of order for the interpolant, and of order for the derivative of the interpolant (see [13], for instance), then for any we have

(9)

The cubic spline collocation method, that we present in this paper, constructs numerically a cubic spline which satisfies the Equation (7) at the points,. It is easy to see that

and the coefficients, , satisfy the following nonlinear system with n + 1 equations:

(10)

Relations (9) and (10) can be written in the matrix form, respectively, as follows

(11)

where

and is a vector where each component is of order. It is well known that, where A is a matrix independent of h given as follows:

Then, relation (11) becomes

(12)

with is a vector where each one of its components is of order.

The results of this work are basically based on the invertibility of the matrix A. Then, in order to prove that A is invertible we give the flowing lemma.

Lemma 3 (de Boor [13]) Let such that on where. If S admits r zeros in then.

Proposition 4 The matrix A is invertible.

Proof Let be a vector of

such that. If we put, then we have and for any. Since then. If we assume that in, then using the above lemma and the fact that has zeros in, we conclude that, which is impossible. Therefore for each. This means that the function S is a piecewise linear polynomial in I. Since, then we obtain for any. Consequently and the matrix A is invertible.

Proposition 5 Assume that the penalty parameter and the discretization parameter h satisfy the following relation:

(13)

Then there exists a unique cubic spline which approximates the exact solution of problem (7).

Proof From relation (12), we have. Let be a function defined by

(14)

To prove the existence of cubic spline collocation it suffices to prove that admits a unique fixed point. Indeed, let and be two vectors of. Then we have

(15)

Using relation (8) and the fact that, we get

where. Then we obtain

From relation (15), we conclude that

Then we have

With

by relation (13). Hence the function admits a unique fixed point.

In order to calculate the coefficients of the cubic spline collocation given by the nonsmooth system

(16)

we propose the generalized Newton method defined by

(17)

where is the unit matrix of order and is the generalized Jacobian of the function, (see [8-10], for instance).

4. Generalized Newton Method

Let be a function. Consider the equation

The Newton method assumes that F is Fréchet differentiable, and is defined by

(18)

where is the inverse of the Jacobian of the function F. However, in nonsmooth case may not exists. The generalized Jacobian of the function F may play the role of in the relation (18). Rademacher's theorem states that a locally Lipschitz function is almost everywhere differentiable (see [14], for instance). Assume that F is a locally Lipschitz function and let be the set where F is differentiable. We denote

The generalized Jacobian of F at, , in the sense of Clarke [15] is the convex hull of:

(19)

For nonsmooth equations with a locally Lipschitz function F, the generalized Newton method is defined by

(20)

where is an element of. If the function F is semismooth and BD-regular at x, then the sequence in (20) superlinearly converges to a solution x (see [8,9, 16,17]). A Function F is said to be BD-regular at a point x  if all the elements of are nonsingular, and it is said to be semismooth at x if it is locally Lipshitz at x and the limit

exists for any. The class of semismooth functions includes, obviously smooth functions, convex functions, the piecewise-smooth functions, and others (see [10,18], for instance). Since the function defined by (6) is a Lipshitz and piecewise smooth function on, then the function given by (14) is also a Lipshitz and piecewise smooth function on. Hence we may apply the generalized Newton method to solve the problem (16).

5. Convergence of the Method

Theorem 6 If we assume that the penalty parameter and the discretization parameter h satisfy the following relation

(21)

then the cubic spline converges to the solution.

Moreover the error estimate is of order.

Proof From (12) and Lemma 4, we have

Since is of order, then there exists a constant such that. Hence we have

(22)

On the other hand we have

Since is the cubic spline interpolation of, then there exists a constant such that

(23)

Using the fact that

(24)

then, we obtain

By using relation (22) and assumption (21) it is easy to see that

(25)

We have

Then from relations (23), (24) and (25), we deduce that is of order. Hence the proof is complete.

Remark 7 Theorem 6 provides a relation coupling the penalty parameter and the discretization parameter h, which guarantees the quadratic convergence of the cubic spline collocation to the solution of the penalty problem.

6. Numerical Examples

In this section we give numerical experiments in order to validate the theoretical results presented in this paper. We report numerical results for solving a one dimensional obstacle problem by using the cubic spline method to approximate the solution of the penalty problem (7), and the generalized Newton method (20) to determine the coefficients of the cubic spline collocation. Consider the obstacle problem (1) with the following data:, and

The true solution of this problem is given by

As a stopping criteria for the generalized Newton’s iterations, we have considered that the absolute value of the difference between the input coefficients and the output coefficients is less than.

Tables 1-4 show, for different values of the discretization parameter h, the error between the cubic spline collocation and the true solution u. We note the convergence of the solution to the function u depends on the discretization parameter h and the penalty parameter. Theorem 6 implies that for a fixed h, this convergence is guaranteed only if there exists such that. Some experimental values of are given in Tables 1-4.

Theorems 1 and 6 imply that we have the error estimate between the exact solution and the discret penalty solution is given by. The obtained results show the convergence of the discret penalty solution to the solution of the original obstacle problem as

the parameters h and get smaller provided they satisfy the relation (21). Moreover, the numerical error estimates behave like which confirms what we were expecting.

7. Concluding Remarks

In this paper, we have consider an approximation of a unilateral obstacle problem by a sequence of penalty problems, which are nonsmooth equation problems, presented in [2,7]. Then we have developed a numerical method for solving each nonsmooth equation, based on a cubic collocation spline method and the generalized Newton method. We have shown the convergence of the method provided that the penalty and discret parameters satisfy the relation (21). Moreover we have provided an error estimate of order with respect to the norm. The obtained numerical results show the convergence of the approximate penalty solutions to the exact one and confirm the error estimates provided in this paper.

REFERENCES

NOTES

^*Corresponding author.