Open access peer-reviewed chapter

Numerical Verification Method of Solutions for Elliptic Variational Inequalities

Written By

Cheon Seoung Ryoo

Reviewed: 22 October 2021 Published: 30 November 2021

DOI: 10.5772/intechopen.101357

From the Edited Volume

Simulation Modeling

Edited by Constantin Volosencu and Cheon Seoung Ryoo

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Abstract

In this chapter, we propose numerical techniques which enable us to verify the existence of solutions for the free boundary problems governed by two kinds of elliptic variational inequalities. Based upon the finite element approximations and explicit a priori error estimates for some elliptic variational inequalities, we present effective verification procedures that, through numerical computation, generat a set which includes exact solutions. We describe a survey of the previous works as well as show newly obtained results up to now.

Keywords

  • numerical verification method
  • variational inequalities
  • error estimates
  • fixed point formulation
  • newton-like method
  • finite element method

1. Introduction

Numerical verification methods of solutions for differential equations have been the subject of extensive study in recent years and much progress have been made both mathematically and computationally [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. However, for some problems governed by the elliptic variational inequalities, 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 [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

The authors have studied for several years the numerical verification method of solutions for elliptic variational inequalities using finite element method and the constructive error estimates combining with Schauder’s and Banach’s fixed point theorem. Several results in our research are already published in [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In this chapter, we briefly overview our resent research results including works not yet published.

The outline of this chapter is as follows. In Section 2, the two types of elliptic variational inequalities are considered. In Subsection 2.1, we describe the elliptic variational inequalities and give a fixed point formulation to prove the existence of solutions. In Subsections 2.2 and 2.3, the main tool of the verification method is explained at an abstract level. In Subsection 2.2, we present a simple iteration method for numerical verification of solutions for the elliptic variational inequalities. We construct the concepts of rounding and rounding error for functions and present a computer algorithm to construct the set satisfying the verification conditions. However, it is difficult to apply the method in Subsection 2.2 to a problem in which an associated operator is not retractive in a neighborhood of the solution, because it is based upon a simple iteration method. In Subsection 2.3, we propose another approach to overcome such a difficulty. This method can be applied to general elliptic variational inequalities without any retraction property of the associated operator. We introduce a Newton-like operator and reformulate the problem using it. Particularly, special emphasis is placed on the way to devise the Newton-like operator for a kind of non-differentiable map which defines the original problem. We introduce a computational verification condition. In order to show a concrete usage of the tool, in Section 3, we present an application to some problems governed by the elliptic variational inequalities. Many difficulties remain to be overcome in the construction of general techniques applicable to a broader range of problems. However, the authors have 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.

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2. Elliptic variational inequalities

The theory of elliptic variational inequalities has become a rich source of inspiration in both mathematical and engineering sciences. Elliptic variational inequalities are an effective tool for studying the existence of solutions of constrained problems arising in mechanics, optimization and control, operation research, engineering science, etc. [47, 48, 49, 50, 51, 52]. It is the aim of this chapter to introduce a numerical technique to verify the solutions for elliptic variational inequalities. The basic approach of this technique consists of the fixed point formulation of elliptic variational inequalities and construction of the function set, on computer, satisfying the validation condition of a certain infinite dimensional fixed point theorem. For fixed point formulation, we consider a candidate set which possibly contains a solution. In order to get such a candidate set, we divide the verification procedure into two phases: one is the computation of a projection into a closed convex subset of some finite dimensional subspace (rounding); the other is the estimation of the error for the projection (rounding error). Combining these methods with some iterative technique, the exact solution can be enclosed by sum of rounding parts, which is a subset of finite dimensional space, and the rounding error, which is indicated by a nonnegative real number. These two procedures enable us to treat infinite dimensional problems as finite procedures, thta is, by computer.

Notations

  • V: real Hilbert space with scalar product and associated norm ,

  • V: the dual space of V,

  • a:V×VR is a bilinear, continuous and V-elliptic from on V×V.

A bilinear form a is said to be V-elliptic if there exists a positive constant α such that avvαv2,vV..

In general we do not assume a to be symmetric, since in some applications nonsymmetric bilinear forms may occur naturally.

  • L:VR continuous, linear functional,

  • Kis a closed convex nonempty subset of V,

  • j:VR is a convex lower semicontinuous (l.s.c) and proper functional (j is proper if jv>,vV and j+).

The two types of elliptic variational inequalities.

We consider two classes of elliptic variational inequalities.

  • Elliptic variational inequalities of the first kind: FinduVsuch thatuis a solution of the problem

auvuLvu,vK,uK.

  • Elliptic variational inequalities of the second kind: FinduVsuch thatuis a solution of the problem

auu+jvjuLvu,vV,uV.

2.1 The problem and the fixed point formulation

Let us first set a few notations [1, 47, 49, 50, 53, 54, 55, 56, 57, 58, 59, 60, 61]. In what follows we shall make use of the Sobolev spaces Wk,pΩ of functions which possess generalized derivatives integrable with the pth power up to and including the kth order. For p=2, we shall write Wk,pΩ=HkΩ,H0Ω=L2Ω. Further, we introduce the scalar product in L2Ω by

fg=Ωfxgxdx.

The norm in HkΩ will be denoted by HkΩ. The symbol HkΩ will stand for the seminorm,

uHkΩ=α=kDαuL2Ω212,uHkΩ=j=0kuHjΩ212.

Let V be a real Hilbert space with a scalar product V and an associated norm V,V its dual space. K denotes a nonempty closed convex subset of Va:V×VR is a bilinear, symmetric, continuous and elliptic form of V, a:V×VR is a bilinear, symmetric, continuous and elliptic form of V×V; that is, there exist constants α>0, and β>0 such thatauvαuVvV,u,vV and avvβvV2,vV. The pairing between V and V is denoted by <,>. Let Λ be a canonical isomorphism from V onto V defined, for gV, by <g,v>=ΛgvV,vV. We can easily see that ΛV=Λ1V=1. Now, let us consider the following variational inequality:

FinduKsuchthatauvu<fu,vu>,vK,E1

where f is a nonlinear operator such that fuV..

In order to obtain a fixed point formulation of variational inequality (1) we need the following standard result.

Lemma 1.LetKbe a closed convex subset ofV. Thenu=PKω, the projection ofωonK, if and only if

uK:uωvuV0,vK.E2

For some constant ρ>0, let us define a mapping G:VV by

Gu=PKΛΦu,E3

where uV, ΦuV is defined by

<Φu,v>=uvVρauv+ρ<fu,v>,vV.E4

For some constant ρ>0, problem (1) can be written as

uvuVuvuVρauvu+ρ<fuvu>0,vK.

Using (4) in the above inequality, problem (1) is equivalent to that of finding uK such that

uΛΦuvuV0,vK.E5

By (2) and (5), we now have the following fixed point problem for the operator G:

u=PKΛΦu=Gu.E6

Under appropriate conditions on the space V and the operator G:VV(e.g., continuity, compactness), which usually have to be verified by theoretical means, fixed point theorem yields the existence of a solution u of the problem (1) in some suitable set UV, provided that

GUU.E7

In order to compute an explicit inclusion, we must therefore construct U explicitly. For the numerical verification of condition (7), we have to use interval analysis on many levels between basic interval arithmetic and functional analysis. For the appropriate and suitable choice of the operator f, the form a, and the convex set K; one encounters problems governed by the elliptic variational inequality as special cases from the problem (1) [48, 49, 50, 51, 52]. Inbrief, it is clear that the problem (1) is the most common. Up to now, devising a verification technique for the problem (1) is still an open problem. It is an important and interesting area of future research to find the numerical inclusion methods for the problem (1) by using (6). In this paper, we suppose that VL2Ω and the nonlinear map f:VL2Ω satisfies the following assumptions.

A1.f is a continuous map from V to L2Ω.

A2. For each bounded subset WV,fW is also bounded in L2Ω.

If we restrict the nonlinear map f as above, then it can be shown that the problem (1) can be characterized by a class of variational inequality of the type,

finduKsuchthatauvufuvu,vK.E8

The problem (8) has the restricted condition; even so (8) is an important and very useful class of nonlinear problems arising in mathematical physics, mechanics, engineering sciences, etc. In Section 3, we briefly consider a particular example of interest in applications. Another example is given in [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In the special case in which KV, (8) yields the variational theory of the boundary value problems for partial differential equations. We will discuss existence and inclusion methods for problem (8). These are methods providing the existence of a solution of the problem (8) within explicitly computable bounds. As we have seen before, the transformation of problem (8) into some fixed point formulation (6) can be carried out in the same way. In a conclusion problem (8) is equivalent to the fixed point problem of finding uK such that

u=Su,E9

where S denotes a specific operator, not necessarily the same as in (6). In particular for a given problem, we reduced the problem (8) to the fixed point formulation (9) and the continuity and compactness of S is discussed. For this reason, we shall say nothing about this problem for which we refer to [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In order to simplify argument we assume that S is a continuous and compact operator. Since S is continuous and compact, as a result of Schauder’s fixed point theorem, if there exists a nonempty, bounded, convex, and closed subset U such that SUU, then there exists a solution of u=Su in U. In Sections 2.2 and 2.3, we describe how to construct U explicitly.

2.2 Verification by a simple iteration method

In this subsection, we describe a simple iteration method for numerical verification of solutions for elliptic variational inequalities. In order to treat functions and variational inequalities in the infinite dimensional space V by computer, we introduce two concepts, rounding and rounding error. Now, let Vh be a finite dimensional subspace of V dependent on h0<h<1 and let Kh be a nonempty closed convex subset of Vh. Usually, Vh is taken to be a finite element subspace with mesh size h. For the sake of simplicity, we shall define Kh, an approximate subset of K, by Kh=VhK.Kh is a closed convex subset of Vh. In practical applications, the construction of Kh is one of the difficulties presented by variational inequalities. For a given problem, several approximations are available. For a general study of the approximation of convex sets, we refer the reader to the work of Mosco [51]. We define the projection PKh from V into Kh [49, 50]. That is, vh=PKhu, the projection of u into Kh, is defined as follows:

u=Su,vhKh:vhζvhVuζvhV,ζKh.E10

To verify the existence of a solution of (9), we determine a set W for a bounded, convex, and closed subset UV as

W=vV:v=SuuU.

From Schauder’s fixed point theorem, if WU holds, then there exists a solution of (8) in the set U. Our goal is to find a set U which includes W. For any subset WV, we define RWKh by the projection of V to Kh, which is called the rounding of W. Additionally, we define RE(W), the rounding error of W, as a subset of V so that WRW+REW holds. Using RW+REW instead of W, the verification condition becomes

RW+REWU.E11

Let us describe the procedure more concretely. First, we consider the auxiliary problem: given gL2Ω,

finduKsuchthatauvugvu,vK.E12

We note that, by well known result [49], there is a unique element u which satisfies (12).

Secondly, we define the approximate problem corresponding to (12) as

auhvhuhgvhuh,vhKh,uhKhE13

and (13) admit one and only one solution [49]. Error estimates for the variational inequalities can be found in [48, 49, 52], etc. Now, using (10), (12), (13) and error estimates, we make the following assumption.

A3. For each uV, there exists a positive constant C, independent of u and h, such that

uPKhuVChgL2Ω.E14

In order to verify the solutions numerically, it is necessary to determine the constant C that appears in a priori error estimations; this constant will be discussed later.

In order to construct the set U satisfying the verification condition (11) in a computer, we use an iterative procedure, that is, the sequential iteration. We propose a computer algorithm to obtain the set U which satisfies the condition (11).

(1) First, we obtain an approximate solution vh0Kh to (8) by an appropriate method. Set Uh0=vh0 and α0=0..

(2) Next we will define RWi and REWi for i0, where Wi is the set defined as follows:

Wi=viV:vi=SuiuiUi.

RWi is defined by the subset of Kh which consists of all the elements vhiKh such that

avhiψvhifuiψvhi,ψKh,E15

holds for some uiUi. Note that RWi can be enclosed by RWij=1MAjϕj, where Aj=Aj¯Aj¯ are intervals, ϕjj=1M is a basis of Vh, and M=dimVh. For details of the interval calculation, we refer the reader to Nakao [6, 7, 12]. Next REWi is defined as

REWi=vV:vVChsupuiUifuiL2Ω.E16

Here, C is the same constant as in (14). Hence, WiRWi+REWi holds.

(3) Check the verification condition:

RWi+REWiUi.E17

If the condition is satisfied, then Ui is the desired set, and a solution to (8) exists in Wi, and hence in Ui.

(4) If the condition is not satisfied, we continue the simple iteration by using δinflation; that is, let δ be a certain positive constant given beforehand, and take

αi+1=ChsupuiUifuiL2Ω+δ,αi+1=vV:vVαi+1,Uhi+1=j=1MAj¯δAj¯+δϕj,Ui+1=Uhi+1+αi+1,

and then go back to the second step. The reader may refer to [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] for the details. If the condition (17) is satisfied, in our inclusion method of solutions for (9), the solution u is enclosed in the set Ui, which we call ‘a candidate set’ of the form Ui=Uhi+αi.

2.3 Verification by a Newton-like method

The significance of a Newton-like operator was already pointed out in [29, 43]. Hence we will not discuss it in detail here. In Subsection 2.1, numerical verification of solutions for elliptic variational inequalities using a finite element method have been discussed only for simple iteration method. The method proposed in Subsection 2.2 is such that Uhiαi always converges to the limit value Uhα from an arbitrary initial value Uh0α0 if S in (9) is retractive operator (we refer to Zeidler [59, 60, 61] for the definition of retraction), while no convergence can generally be expected if S is not retractive operator. Briefly, for not retractive operator in the neighborhood of the solution, it is difficult to use the previous scheme proposed in Subsection 2.2. To overcome such a difficulty, in this section, we newly formulate a verification method using the Newton-like method. This approach enables us to remove the restriction in Subsection 2.2 to the retraction property of the operator in the neighborhood of the solution. Namely, this technique can be applied to general variational inequalities without any retraction property of the associated operator S. We refer to [29, 43] for a detailed study of the properties of the Newton-like Method.

In this subsection, we use the notation of Section 2.2. We assume that Kh=VhK is a closed convex cone with vertex at 0 and Kh its dual. We note that Kh is also a closed convex cone with vertex at 0, which is the only point common to Kh and Kh. From (10) it follows that Kh is the set of points whose projections into Kh is 0. We need some additional lemma.

Lemma 2.AnyuVcan be uniquely decomposed into the sum of two orthogonal elements. That is,

u=PKhuIPKhu=PKhuPKhu.

Here,denotes the sum of two orthogonal elements in the sense ofV..

Note that (9) can be rewritten as the following decomposed form in Kh and Kh:

PKhu=PKhSu,IPKhu=IPKhSu.E18

In order to formulate a Newton-like verification condition for (18), we need a Fréchet derivative of the operator S. For most of the variational inequalities, the S in (9) is not Fréchet differentiable at all. Therefore, in order to use a Newton-like type method, a major difficulty in numerically solving the fixed point formulation u=Su is the treatment of the non-differentiable operator S. We need a suitable modification of the Fréchet derivative of S. Using some techniques, we can devise the approximate Fréchet derivative of S. Hence we shall assume that D˜Su is the approximate Fréchet derivative of the Su at u as the linear operator. Let D˜Su be designated as the Fréchet-like derivative of S at u.

To consider the Newton-like operator for (18), we define the nonlinear operator Nh:VVh as

NhuPKhuID˜Suhh1PKhPKhSu.

Here I is the identity operator and ID˜Suhh1 denotes the inverse on Vh of the restriction operator ID˜SuhVh. Note that we will verify the existence of the inverse operator ID˜Suhh1 from the nonsingularity of the matrix corresponding to ID˜SuhVh in actual calculations.

Next we define the operator T:VV as follows:

TuNhu+IPKhSu.E19

Then T is considered as the Newton-like operator for the former part of (18), but as the simple iterative operator for the latter part. T becomes a compact and continuous map on V by properties of S. Using some techniques, for a given problem we can not only define the Newton-like operator, but also devise a Newton-like Method. Furthermore, we obtain the following proposition and theorem.

Proposition 3.Given the assumption thatNhuKh,

u=Suu=Tu.E20

Theorem 4.If there exists a nonempty, bounded, convex, and closed subsetUKsuch thatTU=TuuUU, then by the Schauder fixed point theorem, there exists a solutionuUofu=Su.

When we decompose the set U as U=UhU in Theorem 8.1, where UhKh and UKh, the verification condition can be written by

NhUUh,IPKhSUU.E21

Here, Uh is represented as the linear combination of the base functions of Vh with interval coefficients, whereas U is the intersection of Kh with a ball in V. That is,

Uh=φhKh:φh=j=1MAjϕjwithajAj¯Aj¯,U=φKh:φVα,

respectively.

Note that NhU can be directly computed from Uh and U with additional information on the a priori error estimates. On the other hand, IPKhSU is evaluated using (14), by the following constructive error estimates for the finite approximate solution of variational inequality (8):

IPKhSUVChsupuUfuL2Ω.

Therefore, the former condition in (21) is validated as the inclusion relations of corresponding coefficient intervals; the latter part can be checked by comparing two nonnegative real numbers.

Next we show a computer algorithm to construct the set U which satisfies the verification condition (21). In order to realize it, we use the iteration method described in Subsection 2.2. Similarly to that in Subsection 2.2, we now generate the following iteration sequence Uhnαn for n=0,1,2,. For n1, the δ-inflation of Uhn1αn1 is denoted by U˜hn1α˜n1. Next, for the set U˜n1=U˜hn1α˜n1, define Uhnαn by

UhnNhU˜n1,αn=ChsupuU˜n1fuL2Ω.E22

Finally, the verification condition in a computer is given by the following theorem. The proof of Theorem 4 will be given here for the sake of completeness; it is based on Proposition 3 and Schauder’s fixed point theorem.

Theorem 5.For an integerN, if two relationships

UhNU˜hN1andαN<α˜N1E23

hold, then there exists a solution u of (8) in UhNαN. Here, the first term of (21) means the strict inclusion in the sense of each coefficient interval of UhN and U˜hN1.

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3. Applications

The study for the numerical verification method for elliptic variational inequalities has been still made less progress than for the differential equation case. The author’s method in the present chapter can be also applied, in principal, to the verification of solutions of the practical problems. Namely, in Section 3.1, we first give, a slightly detailed descroption of the basic principle and formulation of our numerical verification method for the solution of obstacle problems with a homogeneous condition. This should be an appropriate introduction to another applications of our idea. The basic approach of the method consisits of the fixed point formulation of the problems and construction of the function set, in a computer, satisying the validation condition of a certain infinite dimensional fixed point theorem. We also mention that it is possible to extend the method to more general problems with non-homogenerous obstacles. Moreover, in order to apply our method to the problem whose associated operator is not retractive in a neighborhood of the solution, a Newton-like method is introduced. Next, in Section 3.2, we apply our method to another type of free boundary problem with appears in the elasto-plastic deformation theory. This problem causes some properties of non-smoothness in tha associated finite dimensional equations. But, we can also overcome such a difficulty by appling the solution method for non-smooth problems developed by [29, 32, 33]. In the Section 3.3, we briefly remark that our enclosure method can also be applied to the so-called simplified Signorini problem which is a simplified version of a problem accurring in the elasticity theory [43]. Finally, in Section 3.4, we show the way to apply our approach to elliptic variational inequalities of the second kind appearing in the flow problems of a viscos-plastic fulied in a pipe.

3.1 Obstacle problems

We introduce the verification method for solutions of the obstacle problem which is known as a free boundary problem to cahracterize the contacted zone by an obstacle ψ in an elastic membrane region.

3.1.1 Homogeneous case

Here, ‘homogeneous’ stands for the case that obstacle ψ0 in the whole domain.

3.1.1.1 Basic formulation of verification

Though the basic idea of verification is given in other places [26, 27, 28], in order to keep the paper as self-contained as possible, we describe rather detailed formulation and verification procedure for the present case.

Let Ω be a bounded convex domain in Rn, 1n2, with piecewise smooth boundary ∂Ω. We set VH01Ω={vH1Ω:v∂Ω=0} and

auv=uv

which is adopted as the inner product on V, where stands for the inner product on L2Ω. We define KvV:v0a.e.onΩ..

First, we note that, by well-known result [49], for any gL2Ω, the problem:

auvugvu,vK,uK,E24

has a unique solution uVH2Ω, and the estimate

uH2ΩgL2ΩE25

holds [49], where wH2 implies the semi-norm of w in H2Ω defined by

wH2Ω2i,j=1n2wxixjL2Ω2.

Now consider the following elliptic variational inequalities with nonlinear right-hand side;

FindwKsuchthatawvwfwvw,vK.E26

We take an appropriate finite dimensional subspace Vh of V for 0<h<1. Usually, Vh is taken to be a finite element subspace with mesh size h. We then define Kh, an approximation of K, by

Kh=VhK=vhvhVhvh0onΩ¯.

We also define the projection PK from V onto K. That is, v=PKw, the projection of wV into K, is defined as the unique solution of the following problem:

vK:avζvawζv,ζK.E27

And define the projection PKh from V onto Kh. That is, vh=PKhw, the projection of w into Kh, is defined as follows:

vhKh:avhζvhawζvh,ζKh.E28

Now, as one of the approximation properties of Kh, assume that.

For each wKH2Ω, there exists a positive constant C1, independent of h, such that

wPKhwVC1hwH2Ω.E29

Here, C1 has to be numerically determined. For example, it is known that we may take C1=5π for the linear element in one dimensional case [27]. Furthermore, it will be readily seen that the same constant can be taken for the two dimensional bilinear element from the consideration on the proof of Theorem 5.1 in [27]. To verify the existence of a solution of (26) in a computer, we use the fixed point formulation.

First, note that, for each wV, there exists a unique FwV such that

Fwv=fwv,vV,E30

which also implies that

ΔFw=fwinΩ,Fw=0on∂Ω.E31

Then the map F:VV is compact. By (30), the problem (26) is equivalent to finding wV such that

awvwaFwvw,vK.E32

Using the definition (27) and (32), we now have the following fixed point problem for the compact operator PKF.

FindwVsuchthatw=PKFw.E33

3.1.1.2 Verification condition

We introduce two concepts, rounding and rounding error, which enable us to deal with the infinite dimensional problem by finite procedures, that is, in a computer.

Now we define the dual cone of Kh by

Kh=wV:awv0vKh,

and note that Kh is also closed convex cone in V with vertex at 0 which is the only point common to Kh and Kh. From (28) it follows that Kh is the set of points whose projections into Kh is 0.

Lemma 6.AnywVcan be uniquely decomposed into the sum of two orthogonal elements. That is,

w=PKhwIPKhw=PKhwPKhw.

Here,denotes the sum of two orthogonal elements in the sense ofV.

For any wV, we now define the rounding RPKFwKh by the solution of the following problem:

aRPKFwvhRPKFwfwvhRPKFw,vhKh.

Next, for any subset WV, we define the rounding RPKFWKh by

RPKFW=whKh:wh=RPKFwwW.

Usually, RPKFW is enclosed and represented as a linear conbination of the base functions in Vh with interval coefficients.

Moreover, for WV, we define REPKFW, the rounding error of PKFW, as a subset of Kh, that is,

REPKFW=vKh:vVC0hfWL2,E34

where

fWL2supwWfwL2.

Here, C0C1C2, where C1 is the same positive constant as in (29), and C2 is determined by the following regularity estimate for the solution to (24) of the form

uH2C2gL2.E35

Thus we may take as C2=1 for the present case from (25). Then, we have

PKFwRPKFwREPKFw,wW.

Therefore, the following verification condition is obtained by Schauder’s fixed point theorem.

Lemma 7.If there exists a nonempty, bounded, convex, and closed subsetWKsuch that

RPKFWREPKFWW,E36

then there exists a solution of w=PKFw in W.

We sometimes refer the above set W as a candidate set, which we generate in computer so that it satisfies the condition (36).

3.1.1.3 Verification procedures

We describe the method to find a set W satisfying (36) in th ebelow.

Consider the following approximate solution whKh of (24):

awhvhwhgvhwh,vhKh,whKh.E37

Since the bilinear form a is symmetric, (37) is reduced to the quadratic programming problem:

minvKh12avv(gv).E38

Let ϕjj=1M be a basis of Vh with usual linear functions such that ϕjx0,xΩ and satisfying

ϕjxi=1,i=j,0,ij,

where xi is an interior node of the finite element mesh. Then (38) reduces to the following vector form:

minw012wDwPw,E39

where w0 means the componentwise relation. Here, Ddij1i,jM with dij=ϕiϕj, and w is the coefficient vector with ϕj of the function v in (38). Also, Pgϕj1jM.

Furthermore, we define for any αR+, nonnegative real number, we set

αϕKhϕVα.

Then, for a given candidate set W=Whα with WhKh, the computation of the rounding RPKFW reduces to enclose an interval vector Z=Zj and Y=Yj satisfying the following nonlinear system of equations [27]:

YDZ=fWϕj,1jM,YjZj=0,1jM.E40

Here, fWϕj is evaluated as an interval Bj such that fwϕjwWBj. In order to solve (40) with guaranteed accuracy, we use some interval approaches for nonlinear system of equations [19, 20]. Thus, using the solution of (40), we can enclose the set RPKFW in (36). Combining this with (34), we can successfully compute the left-hand side of (36) for any candidate set W=Whα.

Thus we can present a computational verification condition. In the actual computation, we use an iterative procedure with δ-inflation technique to find the set W satisfying (36). Several numerical examples for verification are presented in [27] for one dimensional problem using linear finite element.

3.1.2 Non-homogeneous case

In this subsection, we consider the two-dimensional case. In order to verify solutions numerically, it is necessary to determine some constants that appear in the a priori error estimates. For the non-homogeneous case, we define KvV:vψa.e.onΩ, where ψ is a given H2Ω function such that ψ0 on ∂Ω and is not identically equal to 0. Let Ω be a square with side 1 and let Th be the uniform triangulation of Ω. We introduce Σh=ppΩ¯pisavertexofTTh and define the approximate Vh of H01Ω by Vh={vhvhH01ΩC0Ω¯vhTP1,TTh}. Here, vhT denotes the restriction of vh to T and P1 representing the space of polynomials in two variables of degree 1. It is then quite natural to approximate K by

Kh=vhVhvhpψppΣh.

Note that, in general, KhVhK. Then, PK and PKh are similarly defined as before, and we also have the constructive error estimates of the form, vhKh and vK,

uhuH01ΩCgψh,E41

where,

CgψhsupgL2Ω0.4942h2uH22+2gL2+AuL20.4942h2uH2+6h2ψH2.

We provide a numerical example of verification in the two-dimensional case according to the procedures described in the previous section. Let Ω=01×01. We consider the case fu=Ku+sinπxsin2πy and ψ=sinπxsinπy. For simplicity, we only consider the uniform mesh here. First, we divide the domain into small triangles with a uniform mesh size h and choose the basis of Vh as the pyramid functions.

The execution conditions are as follows (Figures 13):

Figure 1.

Obstacle function ψ.

Figure 2.

Approximate solution uh0.

Figure 3.

Illustration of the contact zone.

K=0.1,dimVh=10Obstaclefunctionψ=sinπ xsinπ ytheoutlineofψisshowninFigure1.Initialvalue:uh0=Galerkinapproximation,α0=0theoutlineofuh0isshowninFigure2.IllustrationofcontactzonebetweenobstacleandapproximatesolutionisshowninFigure3.Extensionparameters:δ=105.

Results are as follows:

Iterationnumbersforverification:2H01Ωerrorbound:0.15437MaximumwidthofcoefficientintervalsinAjN=0.00001.

Detailed arguments and with numerical examples are presented in [42].

3.1.3 A Newton-type verification method

The idea of the enclosure method for solutions of obstacle problems is based upon simply sequential iterations for the original fixed point operator PKF. Therefore, it is difficult to apply the method to the problem of which associated operator is not retractive in a neighborhood of the solution. In order to overcome such a difficulty, we introduce an another formulation using a Newton-like operator. The essential point is the way to devise the Newton-like operator for a kind of non-differentiable map which defines the original problem.

To formulate a Newton-type verification condition, we need a Fréchet derivative of the operator PKF. However, PKF is not Fréchet differentiable at all. Therefore, we define the approximate Fréchet-like derivative D˜KFuh on Vh for some uhKh instead of the Fréchet derivative. Assume that ϕjj=1M is a basis of Vh, where M=dimVh, such that ϕjx0 on Ω and satisfying

ϕjxi=1,i=j,0,ij,

where xi is an interior node of the finite element mesh.

And, for vhVh, we represent it such as

vh=j=1Mvhjϕj.

Here, vhjj=1,,M is called as the coefficient vector of vh. Now we take a fixed subset N012M, define Vh,N0, the closed subspace of Vh, by

Vh,N0=vhvhVhvhj=0forjN0.

And let Ph,N0 be a H01-projection from V onto Vh,N0 defined by

auPh,N0uv=0,vVh,N0,Ph,N0uVh,N0.

In order to define D˜KFuh:VhVh,N0, we differentiate the first equation of (40) in W at W=uh to get, for arbitrary δVh,

YDZ=f'uhδϕj1jM.E42

Here, Y=Y˜j1jM and Z=Z˜j1jM, where Y˜j=0 for jN0 and Z˜j=0 for jN0, respectively.

Then we define the approximate Fréchet-like derivative of PKFu at u=uh, as the linear map D˜KFuh:VhVh,N0 such that, for each δVh,

D˜KFuhδj=1MZj˜ϕj.

We now assume that.

A4. The restriction to Vh,N0 of the operator Ph,N0ID˜KFuh:VhVh,N0 has the inverse operator

Ph,N0D˜KFuhh1:Vh,N0Vh,N0.

Here, I means the identity map on Vh.

By using the above approximate Fréchet-like derivative, we define the Newton-like operator Nh:VVh by

NhwPKhwPh,N0D˜KFuhh1Ph,N0PKhPKhPKFw).

Next we define the operator T:VV as follows:

TwNhw+IPKhPKFw.

Then T becomes a compact map on V and it follows the fixed point problem w=PKFw is equivalent to w=Tw. Detailed arguments and with numerical examples are presented in [35].

3.2 Elasto-plastic torsion problems

In this subsection, we consider an enclosure metnod of solutions for elasto-plastic torsion problems governed by an elliptic variational inequalities [25, 32, 33]. The nonlinear elasto-plastic torsion problem is defined as the same type elliptic variational inequalities as (26) with

KvH01Ωv1a.e.onΩ.E43

As is well known [56, 58], two sub-domains Ωp and Ωe defined by

Ωp=xxΩu=1,

and

Ωe=Ω\Ωp=xxΩu<1

correspond to the plastic and elastic regions, respectively. The elastic region Ωe and the plastic region Ωp are not known beforehand and should be determined, therefore ΩeΩp is actually the free boundary of the problem (26). The problem (26) has been formulated as the problem of finding u satisfying

Δu=fuinΩe,u=1inΩp,u=0on∂Ω.E44

The finite dimensional convex subset Kh is also defined similarly as before:

KhVhK=vhvhVhvh1a.e.onΩ.E45

In order to formulate the verification procedure, we need a verified computational method for solving the finite dimensional part (rounding) and a constructive estimates for infinite dimensional part (rounding error) as in the previous subsection.

Following [49, 56], we define the Lagrangian functional L associated with (1) by

Lvμ=12Ωv2dxgv+12Ωμv21dx.

It follows, from [49, 56], that if L has a saddle point uλH01Ω×L+Ω, then u is a solution of (1), where L+Ω=qLΩq0a.e.inΩ. We use the Uzawa algorithm to solve (1). Thus we can claculate the rounding RPKFW, for a candidate set W, by solving the following problem with guaranteed error bounds:

FinduhλhKh×Λhsuchthatλh=maxλh+ρuh210withρ>0.Ω1+λhuhvhdx=fWvh,vhVh,uhVh,E46

The problem (46) can be formulated as a system of nonlinear and nonsmooth (nondifferentiable) equations. A verification method for nonsmooth equations by a generalized Krawczyk operator is studied in [1, 55]. We briefly describe the method presented by [55] in the below.

We consider the following equivalent system of nonlinear(and nondifferentiable) equation to (46) for a fixed wW

Hx=0.E47

Here, we assume that H:RnRn is locally Lipschitz continuous. The equivalence means that x solves (46) if and only if x solves (47). The method is based on the mean value theorem for local Lipschitz functions of the form

HxHycoHxxy,forallx,yx,

where [x] stands for an interval vector, “co” denotes the convex hull, and H the generalized Jacobian in Clarke’s sense [57], which is also considered as a slope function, and

coHxcoVHxxx.

Let Lx be an interval matrix such that coHxLx. Then for any x,yxRn it holds that HxHyLxxy.

Then an interval operator for nonsmooth equations is defined by

GxAxxA1Hx+IA1Lxxx.E48

The mapping GxAx is called a generalized Krawczyk operator. Therefore, the verification condition of solutions for (46) in x is given by

GxAxxD.

Thus, we can compute the solution of (46) with guaranteed accuracy. That is, we can enclose the rounding RPKFU. On the other hand, in order for the calculation of the rounding error REPKFU, the similar arguements can also be applied for one dimensional problem. Actually, we can prove that the same constant C0=5π is also valid for the present problem in one dimensinal case, which implies that we can give a verification procedure besed on the same principle as before [25, 32, 33]. In [33], we extended the approach to the numerical proof of existence of solutions for elasto-plastic torsion problems as well as gave a numerical example for one dimensional case. The verification method in [33] is based on the generalized Krawczyk operator for solving a system of nonsmooth (nondifferentiable) equations. In order to use the generalized Krawczyk operator, we need to calculate the Jacobian. In that case, we need some complicated techniques. However, in many cases, calculating the generalized Jacobian is very difficult. To overcome such difficulties, we proposed a numerical verification method without using the generalized Krawczyk operator. This method is attractive, since calculating the generalized Jacobian is not required in the computational performance. Furthermore, up to know, our verification methods are mainly based on the enclosure of solutions in the sense of L2 or H1 norms. We considered a numerical verification method with guaranteed L error bounds for the solution of elasto-plastic torsion problem.

3.3 Simplified Signorini problems

A simplified Signorini problem is also given by the elliptic variational inequalities of the form (26) with

KvH01Ωv0on∂ΩE49

and

auv=Ωuvdx+Ωuvdx.E50

where

uv=ux1vx1+ux2vx2.

As well known, the solution u of this elliptic variational inequalities can be characterized as a solution of the following free boundary problem finding u and two subsets Γ0 and Γ+ such that Γ0Γ+=∂Ω and Γ0Γ+=Ø

Δu+u=fuinΩ,u=0onΓ0,un0onΓ0,u>0onΓ+,un=0onΓ+,E51

where n the outer normal derivative on ∂Ω. In the present case, the approximation subspace Kh is taken as

KhVhK=vhvhVhvh0on∂Ω.E52

For a candidate set W, the computation of rounding RPKFW is also reduced to the quadratic programming problem as in the Section 3.1 [56].

Since the constant C2 in (25) is easily estimated as C2=1, the standard approximation property of the interpolation by Kh gives a constructive error estimates to compute the rounding error REPKFW. For a simplified Signorini problem [43], we constructed a computing algorithm which automatically encloses the solution within guaranteed error bounds. In particular, the method proposed in [43] enables us to verify the free boundary of a simplified Signorini problem, which has been impossible so far. Concerning the numerical verification of solutions for elliptic variational inequalities, we would like to mention that the inclusion method described in this article can be applied to the solution of the elliptic variational inequalities on large space domains.

3.4 Some other problems

In this subsection, we show that our idea of verification method can also be applied to the elliptic variational inequalities of the second kind.

Now, we define the functional jv=Ωvdx. We consider the following problem of the flow of a viscous plastic fluid in a pipe:

FinduH01Ωsuchthatauvu+jvjufuvu,vH01Ω.E53

As in the previous section, we consider the following auxiliary problem associated with (53) for a given gL2Ω:

auvu+jvjugvu,vH01Ω,uH01Ω.E54

By the well known result, we have the following lemma.

Lemma 8. There exists a unique solution uH01ΩH2Ω of (54) for any gL2, such that

uH2ΩĈgL2Ω.

When we denote the solution u of (54) by u=Ag and define the composite map F on H01Ω by FuAfu, which is a little bit of different from the previously appeared symbol F in Section 2, we have.

Theorem 9.F is compact on H01Ω and the problem (53) is equivalent to the fixed point problem

u=Fu.

Proof. First, for a bounded subset UL2Ω, we show that AUH01Ω is relatively compact. Secondly, prove that A:L2ΩH01Ω is continuous. By Lemma 3, AUH2ΩH01Ω and AU is bounded in H2Ω. Since U is bounded in L2Ω, by the Sobolev imbedding theorem, we have AU is relatively compact in H01Ω. Next, for arbitrary f1,f2L2Ω, setting u1=Af1 and u2=Af2, by using (54), we obtain

au1u2u1+ju2ju1f1u2u1,au2u1u2+ju1ju2f2u1u2.

With the above inequalities, we obtain au2u1u2u1=au1u2u1+au2u2u1ju2Th Hence, by the Poincaré inequality, we have

u2u1H01Ω2f2f1L2Ωu2u1L2ΩC¯f2f1L2u2u1H01Ω.

Therefore, we obtain

u2u1H01ΩC¯f2f1L2Ω.

That is, A is Lipschitz continuous as a map L2ΩH01Ω. Hence A is compact. The latter half in the theorem is straightforward from the definition of F.

We now define the approximate problem corresponding to (54) as

auhvhuh+jvhjuhgvhuh,vhVh,uhVh.E55

In order to apply our verification method to enclose the solutions of (53), we need a guaranteed computation of the exact solution of the problem (55), a rounding procedure, as well as the constructive error estimates between the solution of (54) and (55), rounding error estimates.

A major difficulty in solving the problem (55) numerically is the processing of the nondifferentiable term ju=Ωudx. One approach is the method of Lagrange multiplier on that term, whose continuous version is as follows [56].

Let us define Λ=qqL2Ω×L2Ωqx1a.e.xΩ with qx=q1x2+q2x2. Then the solution u of (54) is equivalent to the existence of q satisfying

auv+Ωqv=gv,vH01Ω,uH01Ω,qu=ua.e.,qΛ.E56

Moreover, it is known that (56) is equivalent to the following problem:

auv+Ωqv=gv,vH01Ω,uH01Ω,q=q+ρusup1q+ρu.E57

Here ρ is a positive constant. Let Th be a triangulation of Ω, and let define Lh and Λh (approximation of LΩ×LΩ and Λ, respectively) by

Lh=qhqh=τThqτχτqτR2andΛh=ΛLh,respectively,

where χτ is the characteristic function of τ.

Then our first purpose, computing the rounding RFU, is to enclose the solution of the following approximation problem of (57):

auhvh+Ωqhvh=gvh,vhVh,uhVh,qh=qh+ρuhsup1qh+ρuh.E58

The Eq. (58) leads to a kind of finite dimensional, nonlinear but nondifferentiable problem. We use a slope function method proposed by Rump [18, 19, 20] to enclose the solutions of (58) with g=fW for a candidate set W. On the other hand, the rounding error REFU can be computed by using the following constructive error estimates:

Theorem 10. Let u and uh be solutions of (54) and (55), respectively. If gL2Ω, then there exists a constant Ch such that

uhuH01ΩChgL2Ω.

Here, we may take Ch=5πh for the linear element in one dimensional case, and C is also numerically estimated such that ChOh12 for the two dimensional linear element. A proof of this theorem is described in Ryoo and Nakao [34]. Thus we can also implement the verification algorithm for the solution of (53) as in the previous section. For details on this subsection, please refer to Ref. [47].

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4. Conclusions

We have surveyed numerical verification methods for differential equations, especially around partial differential equations, variational inequalities and the author’s works. But the period of this research is shorter than the history of the numerical methods for differential equations by computer and we can say it is still in the stage of case studies. Indeed, recently, this kind of studies have been referred little by little for practical applications in PDEs and variational inequalities but there are many open problems to be resolve. Therefore, we can make no safe prediction that these approaches will grow into really useful methods for various kinds of equations and variational inequalities in mathematical analysis. Also, since the program description of the verification algorithm is very complicated in general, there is another problem like software technology associated with assurance for the correctness of the verification program itself. Actually, some of the mathematician would not give credit the computer assisted proof in analysis as correct as they believe the theoretical proof, which might cause a kind of seriously emotional problem in the methodology of mathematical sciences. And there is another difficulty from the huge scale of numerical computations which often exceed the capacity of the concurrent computing facilities.

However, in the twenty-first century, the computing environment would make more and more rapid progress, which should be beyond conception in the present state. In any case, a realistic study for partial differential equations and variational inequalities should be the future subject of the numerical computations with guaranteed accuracy. The authors believe that numerical methods with guaranteed accuracy for differential equations and variational inequalities would highly improve the reliability in the numerical simulation of the complicated phenomena in both mathematical and engineering sciences.

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Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).

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Mathematics Subject Classification (2000)

65N15; 65G20

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Written By

Cheon Seoung Ryoo

Reviewed: 22 October 2021 Published: 30 November 2021