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# An approach to the numerical verification of solutions for variational inequalities using Schauder fixed point theory

- Cheon Seoung Ryoo
^{1}Email author

**2014**:235

https://doi.org/10.1186/s13661-014-0235-y

© Ryoo; licensee Springer. 2014

**Received:**3 September 2014**Accepted:**20 October 2014**Published:**7 November 2014

## Abstract

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.

## Keywords

- numerical verification
- error estimates
- variational inequalities
- unilateral boundary value problems for second order equations
- finite element method
- Schauder fixed point theory

## 1 Introduction

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 [1]–[11]*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

*p*th power up to and including the

*k*th order. For $p=2$, we shall write ${W}^{k,p}(\mathrm{\Omega})={H}^{k}(\mathrm{\Omega})$, ${H}^{0}(\mathrm{\Omega})={L}^{2}(\mathrm{\Omega})$. Further, we introduce the scalar product in ${L}^{2}(\mathrm{\Omega})$ by

*V*, respectively, as follows:

*J*is continuous, strictly convex, and coercive in the space

*V*. From these properties of

*J*and results of optimization theory [12], it follows that the minimization problem is finding

*u*such that

*u*such that

*J*at

*u*. Since $({J}^{\prime}(u),v)=a(u,v)-(g,v)$ and $a(u,v)$ is symmetric, problem (2.2) is equivalent to that of finding $u\in K$ such that

Here, we suppose the following conditions for the map *f*.

A1. *f* is the continuous map from *V* to ${L}^{2}(\mathrm{\Omega})$.

A2. For each bounded subset $U\in V$, $f(U)$ is also a bounded set in ${L}^{2}(\mathrm{\Omega})$.

In order to obtain a fixed point formulation of variational inequality (2.5) we need the following standard result.

### Lemma 2.1

[13]

*Let*

*K*

*be a closed convex subset of*

*V*.

*Then*$u={P}_{K}\omega $,

*the projection of*

*ω*

*on*

*K*,

*if and only if*

and the map $F:V\to V$ is a compact operator (see [6]).

Then, using the fixed point problem (2.9), we can construct the numerical procedure to verify the existence of a solution for the variational inequality (2.5).

## 3 Rounding and verification conditions

*i*associated with the internal nodes ${x}_{i}$ of the domain Ω and we shall denote by ${I}_{\mathrm{\Gamma}}=\{{m}_{0}+1,{m}_{0}+2,\dots ,m\}$ the set of all node indices

*i*associated with the boundary nodes ${x}_{i}$ of the domain Ω and we let $I={I}_{\mathrm{\Omega}}\cup {I}_{\mathrm{\Gamma}}$. Here, for the sake of simplicity, let us assume that ${I}_{{\mathrm{\Gamma}}_{0}}=\{{m}_{0}+1,{m}_{0}+2,\dots ,{m}_{0}+j\}$, ${I}_{{\mathrm{\Gamma}}_{+}}=\{{m}_{0}+j+1,\dots ,m\}$, and ${I}_{\mathrm{\Gamma}}={I}_{{\mathrm{\Gamma}}_{0}}\cup {I}_{{\mathrm{\Gamma}}_{+}}$. In what follows, we shall consider only a regular system of triangulations. In other words, when refining the partition of $\overline{\mathrm{\Omega}}$, the triangles of the given triangulation do not reduce to segments. Let $\{{\mathcal{T}}_{h}\}$ be a regular system of triangulations of $\overline{\mathrm{\Omega}}$. The nodes of a triangulation lying on ${I}_{\mathrm{\Gamma}}$ will be denoted by ${p}_{{m}_{0}+1},{p}_{{m}_{0}+2},\dots ,{p}_{m}$. We then approximate

*V*by

*k*on the definition domain

*T*. We then define ${K}_{h}$, an approximate subset of

*K*, by

It is easily seen that ${K}_{h}$ is a closed, convex, and nonempty subset of ${V}_{h}$.

Let *u* be the solution of (2.4) and ${u}_{h}\in {K}_{h}$ be the approximate solution of (3.1). Now, as one of the approximation properties of ${K}_{h}$, assume the following.

Here, $C(h)$ has to be numerically determined.

With the above, we have the following as a result of the Schauder fixed point theorem.

### Theorem 3.1

*If there exists a nonempty*, *bounded*, *convex*, *and closed subset*$U\subset K$*such that*$\mathcal{R}({P}_{K}FU)+\mathcal{RE}({P}_{K}FU)\subset U$, *then there exists a solution of*$u={P}_{K}F(u)$*in* *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, ${D}_{II}\equiv {({a}_{ij})}_{i,j\in I}$, with ${a}_{ij}=(\mathrm{\nabla}{\varphi}_{i},\mathrm{\nabla}{\varphi}_{j})$ and ${z}_{I}$ is the coefficient vector for $\{{\varphi}_{i}\}$ corresponding to the function ${u}_{h}$ in (4.1). Further, ${P}_{I}\equiv {((g,{\varphi}_{i}))}_{i\in I}$ is an *m* dimensional vector.

*i.e.*, an element of the power set ${2}^{{V}_{h}}$ in the following sense:

Here ${P}_{I}\equiv {((f(U),{\varphi}_{i}))}_{i\in I}$.

Let $\tilde{x}:=({\tilde{z}}_{1},{\tilde{z}}_{2},\dots ,{\tilde{z}}_{{m}_{0}},{\tilde{z}}_{{m}_{0}+1},\dots ,{\tilde{z}}_{m},{\tilde{w}}_{{m}_{0}+1},{\tilde{w}}_{{m}_{0}+2},\dots {\tilde{w}}_{m})$ be an approximate solution of (4.4). Then note that ${\tilde{z}}_{i}\approx 0$ or ${\tilde{w}}_{i}\approx 0$ for each ${m}_{0}+1\le i\le m$.

Problem (4.5) can also be reformulated by nonsmooth equations using other methods, *e.g.*, [15]. However, (4.5) is continuous and differentiable. Hence, to enclose solutions for (4.5), we use the following theorem proposed by [5].

### Theorem 4.1

*Let*$\mathcal{F}:{\mathbb{R}}^{2m-{m}_{0}}\to {\mathbb{R}}^{2m-{m}_{0}}$

*be a function with continuous first derivative and let*$M\in {\mathbb{R}}^{(2m-{m}_{0})\times (2m-{m}_{0})}$ (

*real*$(2m-{m}_{0})\times (2m-{m}_{0})$

*matrix*), $\tilde{x}\in {\mathbb{R}}^{2m-{m}_{0}}$.

*Denote the Jacobian matrix of*ℱ

*by*${\mathcal{F}}^{\prime}\in {\mathbb{R}}^{(2m-{m}_{0})\times (2m-{m}_{0})}$

*and for*$X\in I{\mathbb{R}}^{2m-{m}_{0}}$ (

*real interval vectors with*$2m-{m}_{0}$

*components*)

*define*${\mathcal{F}}^{\prime}(X):=\bigcap \{Y\in I{\mathbb{R}}^{m+{m}_{0}}:{\mathcal{F}}^{\prime}(x)\in Y\mathit{\text{for all}}x\in X\}$.

*If for some*$X\in I{\mathbb{R}}^{m+{m}_{0}}$

*with*$0\in X$

*then there exists an*$\stackrel{\u02c6}{x}\in \tilde{x}\phantom{\rule{0.2em}{0ex}}+\stackrel{\circ}{X}$*with*$\mathcal{F}(\stackrel{\u02c6}{x})=0$.

Let $X=(Z,W)$ be an enclosure of a solution of the nonlinear system (4.5) by using Theorem 4.1, where $Z:=({Z}_{1},{Z}_{2},\dots ,{Z}_{{m}_{0}},{Z}_{{m}_{0}+1},\dots ,{Z}_{m})\in I{\mathbb{R}}^{m}$ and $Y:=({W}_{{m}_{0}+1},\dots ,{W}_{m})\in I{\mathbb{R}}^{m-{m}_{0}}$. Then we set ${Z}_{i}:=0$ or ${W}_{i}:=0$ for each ${m}_{0}+1\le i\le m$ provided that ${\tilde{z}}_{i}\approx 0$ or ${\tilde{w}}_{i}\approx 0$, respectively. If, for all $i\in {I}_{\mathrm{\Gamma}}$, $\{{Z}_{i}=0\text{and}inf({W}_{i})0\}$ and $\{inf({Z}_{i})>0\text{and}{W}_{i}:=0\}$ hold, then it implies that the problem (4.7) has an optimal solution $x\in X$ (*cf.*[5]). As one can see, for the case that ${\tilde{z}}_{i}$ and ${\tilde{w}}_{i}$ 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 [15].

We now consider the fully automatic computer generation of the set *U* satisfying Theorem 3.1. First, we generate a sequence of sets $\{{U}^{(i)}\}$, $i=0,1,\dots $ , which consists of subsets of *V* in the following manner.

*cf.*[6], [7]). For $i=0$, we choose appropriate initial values ${u}_{h}^{(0)}\in {K}_{h}$ and ${\alpha}_{0}\in {\mathbb{R}}^{+}$, and define ${U}^{(0)}\subset V$ by

*δ*-inflation of $({u}_{h}^{(n-1)},{\alpha}_{n-1})$ by

Now we have the following verification condition on a computer.

### Theorem 4.2

*If for an integer*

*N*,

*the two relationships*

*hold*, *then there exists a solution* *u* *of* (2.9) *in*${u}_{h}^{(N)}+[{\alpha}_{N}]$. *Here*, *the first term of* (4.11) *means the inclusion in the sense of each coefficient interval of*${u}_{h}^{(N)}$*and*${\tilde{u}}_{h}^{(N-1)}$.

For a convergence analysis of the iterative method for generating a sequence of set $\{{U}^{(i)}\}$, we will prove that the concerned sequence converges for the case that the nonlinear operator ${P}_{K}F$ 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 $C(h)$ of (3.3).

*M*be an integer >0 and let $h=\frac{1}{M}$. We consider ${x}_{i}=ih$ for $i=0,1,2,\dots ,M$ (that is, a uniform partition of Ω) and ${e}_{i}=({x}_{i-1},{x}_{i})$, $i=1,2,\dots ,M$. We then approximate ${H}^{1}(\mathrm{\Omega})$ by

*K*by

We now consider the ${L}^{2}(\mathrm{\Omega})$ estimates of optimal order (that is, $O({h}^{2})$) of ${u}_{h}-u$ via a generalization of the Aubin-Nitsche method. The following result is given by arguments similar to those in [13], except for obvious modifications. Since the basic notations and results are also the same as that of Natterer [16], we do not discuss it further. The reader may refer to [13] for the details.

Regarding the approximation error ${\parallel {u}_{h}-u\parallel}_{{L}^{2}(\mathrm{\Omega})}$, we then have the following.

### Theorem 5.1

*Let*

*u*

*and*${u}_{h}$

*be solutions of problems*(5.1)

*and*(5.3),

*respectively*.

*If*$g\in {L}^{2}(\mathrm{\Omega})$,

*then we have*

*Hence*, *we may take*$C(h)=\frac{{h}^{2}}{{\pi}^{2}}$*in* (3.3).

### Proof

*e*be such that $Ae=g$ and ${e}^{\prime}(a)={e}^{\prime}(b)=0$. Hence, for $w\in {H}^{1}(\mathrm{\Omega})$,

Also, for $z(b)>0$ we have ${z}^{\prime}(b)\le 0$, and similarly we obtain ${z}^{\prime}(a)\ge 0$ for $z(a)>0$.

*v*by ${r}_{h}u$ in [13], Theorem 1], we obtain

*y*by ${r}_{h}z$, $y-z=0$ on $\{a,b\}$ in [13], Theorem 2], (5.6), and (5.7), we obtain

□

## 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 ${P}_{1}({e}_{i})$ is the space of polynomials of degree ≤1 on ${e}_{i}$. We now choose the basis ${\{{\varphi}_{i}\}}_{i=1}^{M}$ of ${V}_{h}$ as the usual hat functions.

The execution conditions are as follows:

$\text{Numbers of elements}=300$;

$K=1/10$;

Extension parameters: $\u03f5={10}^{-3}$;

Initial values: ${u}_{h}^{(0)}=\text{Galerkin approximation (4.6)}$; ${\alpha}_{0}=0$.

The results are as follows:

Iteration numbers for verification: $N=3$;

${L}^{2}$-error bound: 0.00002;

Maximum width of coefficient intervals in $\{{A}_{j}^{(N)}\}=0.0000454239$;

**Coefficient intervals**

${\mathit{x}}_{\mathit{j}}$ | Coefficient intervals |
---|---|

0.0000 | [0.00000,0.00000] |

0.0033 | [−0.01562,−0.01562] |

0.0067 | [−0.03123,−0.03123] |

0.0100 | [−0.04683,−0.04683] |

0.0134 | [−0.06242,−0.06241] |

0.0167 | [−0.07798,−0.07798] |

0.0201 | [−0.09352,−0.09352] |

0.0234 | [−0.10904,−0.10903] |

0.0268 | [−0.12452,−0.12451] |

0.0301 | [−0.13997,−0.13996] |

*h*from $1/100$ to $1/300$. In Table 2, we show the values of

*α*and $max|{A}_{j}^{(n)}|$, which is the maximum width of the coefficient intervals on the nodes.

**Maximum width of the coefficient intervals**

h | $\mathbf{max}\mathbf{|}{\mathit{A}}_{\mathit{j}}^{\mathbf{(}\mathit{n}\mathbf{)}}\mathbf{|}$ | α |
---|---|---|

1/100 | 0.0000088369 | 0.00016 |

1/200 | 0.0000502130 | 0.00004 |

1/300 | 0.0000454239 | 0.00002 |

### Remark 6.1

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.

## Declarations

## Authors’ Affiliations

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