An approach to the numerical verification of solutions for variational inequalities using Schauder fixed point theory
© Ryoo; licensee Springer. 2014
Received: 3 September 2014
Accepted: 20 October 2014
Published: 7 November 2014
In this paper, we describe a numerical method to verify the existence of solutions for a unilateral boundary value problems for second order equation governed by the variational inequalities. It is based on Nakao’s method by using finite element approximation and its explicit error estimates for the problem. Using the Riesz representation theory in Hilbert space, we first transform the iterative procedure of variational inequalities into a fixed point form. Then, using Schauder fixed point theory, we construct a high efficiency numerical verification method that through numerical computation generates a bounded, closed, convex set which includes the approximate solution. Finally, a numerical example is illustrated.
MSC: 65G20, 65G30, 65N15, 65N30.
A numerical verification method to verify the existence of solutions for mathematical problems is a new approach in the field of existence theory of solutions for mathematical problems that appear in mathematical analysis. Numerical verification methods of solutions for differential equations have been the subject of extensive study in recent years and much progress has been made both mathematically and computationally (see –etc.). These methods are known as new numerical approaches for the problems where it is difficult to prove analytically the existence of solutions for differential equations. However, for some problems governed by the variational inequality, there are very few approaches. As far as we know, it is hard to find any applicable methods except for those of Nakao and Ryoo. The theory of variational inequalities has become a rich source of inspiration in both mathematical and engineering sciences. So, a high efficiency numerical method for variational inequalities is often beneficial to the relevant subject. It is the aim of this paper to attempt a numerical technique to verify the solutions for elliptic equations of the second order with boundary conditions in the form of inequalities, that is, we construct a computing algorithm which automatically encloses the solution with guaranteed error bounds. In the following section, we describe the elliptic equations of the second order with boundary conditions in the form of inequalities considered and the fixed point formulation to prove the existence of solutions. In Section 3, in order to treat the infinite dimensional operator by computer, we introduce two concepts, rounding and rounding error, and a computational verification condition. In Section 4, we construct a concrete computing algorithm for the verification by computer, which is an efficient computing algorithm from the viewpoint of interval arithmetic. In order to verify solutions numerically, it is necessary to calculate the explicit a priori error estimates for approximate problems. These constants play an important role in the numerical verification method. In Section 5, we determine these constants. Finally, a numerical example is presented. Many difficulties remain to be overcome in the construction of general techniques applicable to a broader range of problems. However, the author has no doubt that investigation along this line will lead to a new approach employing numerical methods in the field of existence theory of solutions for various variational inequalities that appear in mathematical analysis. We hope to make progress in this direction in the future.
2 Problem and fixed point formulation
Here, we suppose the following conditions for the map f.
A1. f is the continuous map from V to .
A2. For each bounded subset , is also a bounded set in .
In order to obtain a fixed point formulation of variational inequality (2.5) we need the following standard result.
and the map is a compact operator (see ).
3 Rounding and verification conditions
It is easily seen that is a closed, convex, and nonempty subset of .
Here, has to be numerically determined.
With the above, we have the following as a result of the Schauder fixed point theorem.
If there exists a nonempty, bounded, convex, and closed subsetsuch that, then there exists a solution ofin U.
4 Computing procedures for verification
In this section, we propose a computer algorithm to obtain a set U which satisfies the condition of Theorem 3.1.
Here, , with and is the coefficient vector for corresponding to the function in (4.1). Further, is an m dimensional vector.
Let be an approximate solution of (4.4). Then note that or for each .
Problem (4.5) can also be reformulated by nonsmooth equations using other methods, e.g., . However, (4.5) is continuous and differentiable. Hence, to enclose solutions for (4.5), we use the following theorem proposed by .
then there exists anwith.
Let be an enclosure of a solution of the nonlinear system (4.5) by using Theorem 4.1, where and . Then we set or for each provided that or , respectively. If, for all , and hold, then it implies that the problem (4.7) has an optimal solution (cf.). As one can see, for the case that and are both close to zero, this algorithm would not work. Fortunately, we have never encountered such a difficulty up to now. But, in order to establish more general applications of our method, it should be necessary to consider the methods for nonsmooth problems such as in .
We now consider the fully automatic computer generation of the set U satisfying Theorem 3.1. First, we generate a sequence of sets , , which consists of subsets of V in the following manner.
Now we have the following verification condition on a computer.
For a convergence analysis of the iterative method for generating a sequence of set , we will prove that the concerned sequence converges for the case that the nonlinear operator in (2.9) is retractive around the solution u, and provided that the mesh size h is sufficiently small. We will leave such a general case as a further research topic.
5 Computation of the constants
In this section, we only deal with the one dimensional case. We give a bound of the constant of (3.3).
We now consider the estimates of optimal order (that is, ) of via a generalization of the Aubin-Nitsche method. The following result is given by arguments similar to those in , except for obvious modifications. Since the basic notations and results are also the same as that of Natterer , we do not discuss it further. The reader may refer to  for the details.
Regarding the approximation error , we then have the following.
Hence, we may takein (3.3).
Also, for we have , and similarly we obtain for .
6 Example of numerical verification
In this section, we provide some numerical examples of verification in the one dimensional case according to the procedure described in the previous section.
where is the space of polynomials of degree ≤1 on . We now choose the basis of as the usual hat functions.
The execution conditions are as follows:
Extension parameters: ;
Initial values: ; .
The results are as follows:
Iteration numbers for verification: ;
-error bound: 0.00002;
Maximum width of coefficient intervals in ;
Maximum width of the coefficient intervals
In the above calculations, we carried out all numerical computations using the usual double precision computer arithmetic instead of strict interval computations (e.g., ACRITH-XSC, PASCAL-XSC, FORTRAN-XSC, C-XSC, PROFIL, etc.). Therefore, we neglected the round-off error. The reason is that the main purpose of our numerical experiments is the estimation of the truncation errors which usually, roughly speaking, are over 10−10 times larger than the round-off errors. That is, there will be in general some rounding errors at each step.
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