FINITE ELEMENT ESTIMATES FOR A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES

It is well known that a wide class of obstacle and unilateral problems arising in pure and applied sciences can be studied in a general and unifield framework of variational inequalities. In this paper, we derive the error estimates for the finite element approximate solution for a class of highly nonlinear variational inequalities encountered in the field of elasticity and glaciology in terms of wl’l(t2) and Ll0(t2)-norms. As a special case, we obtain the well-known error estimates for the corresponding linear obstacle problem and nonlinear problems.


INTRODUCTION.
Variational inequality theory is an interesting branch of applicable mathematics, which not only provides us with a uniform framework for studying a large number of problems occurring in different branches of pure and applied sciences, but also gives us powerful and new numerical methods of solving them. In this paper, we conider a broad class of highly nonlinear elliptic boundary value problems having some extra constrained conditions. A much used approach with any elliptic problem is to reformulate it in a weak or variational form aad then to approximate these. In the presence of a constraint, this approach leads to a variational inequality, which is the weak formulation.
In recent years, the finite element techniques are being applied tb compute the approximate solutions of various classes of variational inequalities. Relative to the linear variational inequalities, little is known about the accuracy and convergance properties of finite element approximation of nonlinear variational inequalities associated with nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. An error analysis of finite element method for the boundary value problem having nonlinear operator 7 (I x7 ulP-27) was derived by Glowinski and Marroco [1], which was an improvement of the results of Oden [2]. For piecewise linear finite element approximations, they obtained error estimates in the wl'l-norm of order h I[p-1, which were extended by Noor [3] for strongly nonlinear problems. Babuska [4] also obtained the same type of estimate for the finite element approximation of second order quasilinear elliptic problems.
Error estimates for various types of variational inequalities involving second order linear and nonlinear elliptic operators have been derived by many workers including Falk [5], Mosco and Strang [6], Janovsky and White,nan [7] and Noor ([8], [9]), under sufficient regular solutions. Oden and Reddy [10] obtained some general results for a class of highly nonlinear variational inequalities involving certain psuedo-monotone operators under the assumption that all the solutions (exact and approximate ones) of these variational inequalities are in the interior of a closed convex set in wI'p(). This assumption converts the variational inequalities into variational equations, which makes the error analysis a standard one as in the uncontrained case.
The most important and difficult part of the problem is when the solutions are not in the interior of a closed convex set, a case not covered by their analysis. It is also known that in the presence of the constraints, the approximate solution is no longer a projection of the exact solution as in the unconstrained case. This represents a major difficulty in obtaining the error estimates for the finite elemen approximation of nonlinear variational inequalities.
In the present study, our analysis is based on the existence theory of nonlinear operator equations put forward by Glowinski and Marroco [1]. We extend their results for a class of nonlinear obstacle problems arising in elasticity and glaciology in Section 2. Section 3 is devoted to an analysis of error estimates in finite element approximation for our model problem. Here we derive error estimates in the wl'p(f) and Ll0-norms using the ideas and technique of Mosco and Strang [6]. Our results represent a substantial generalization and improvement of the error analysis of finite element approximation of strongly nonlinear monotone operators and variational inequalities contributed by Glowinski  The mathematical model discussed in this paper arises in the field of elasticity and Oceanography, see [11]. We consider the problem of finding the velocity of the glacier, which is required to satisfy the nonlinear obstacle problem of the type < on/)f where is the cross-section of the glacier and is the given function, known as the obstacle. The presence of/' and 72u may be interpreted as body heating terms, these arises from resistivity and are local Joule heating effects. Also, in elasticity, the problem of torsional stiffness of a prismatic bar with a simply connected convex cross-section f and subject 'to steady creep, which is characterized by a power law, can be described by (2.1) and p is the exponent of the creep law.
The case/ and 72u 0 is related to the problem of capillarity and minimal surfaces, see Finn [12].
The problem (2.1) is a generalization of the nonlinear problem of finding u such that for which the error estimates have been derived by using the finite element approximation by Glowinski and Marroco [1]. The presence of the obstacle needs a different approach for deriving the error estimates and this is the main motivation of this paper.
Let, ft c R n be a bounded open domain with smooth boundary 0f. We consider wlo'l(f) a reflexive Banach space with norm Ilt, =(ftf VvlP) lip and the dual space w-l,q(), _+ 1. The pairing between wlo'P(gt) and W-l,q(fl)is denoted by < .,. >. For more details and notation, see Kikuchi and Oden [13].
We here study the problem (2.1) in the framework of variational inequalities. To do so, we consider that set K defined by K {v W'P(n): ,, , on n}, which is a closed convex set in W'P(9).
The energy (potential) functional l[v] sociated with the obstacle problem (2.1) is given by (2.10) We also remark that if the operator T satisfies the relations (2.7)-(2.10) and the bilinear form b(u,v) is positive continuous, then, using the techniques of Noor [14] and Kikuchi and Oden [13], we can prove the existence of a unique solution of (2.5). Furthermore, concerning the regularity of the solution u e K satisfying (2.5), we assume the following hypothesis: (A) {For pe Wlo'P(fl)Nw2'P(fl),u K satisfying (2.5) also lies in W2'P(fl)}.

FINITE ELEMENT APPROXIMATIONS.
In this section, we derive the error estimates for the finite element approximation of variational inequalities of type (2.5). To do so, we consider a finite dimensional subspace Sh C wol'P(fi) of continuous piecewise linear functions on the triangulation of the polygonal domain t2 vanishing on its boundary gf. Let *h be the interpolant of ,p such that Ch agrees at all the vertices of the triangulation. For our purpose, it is enough to choose the finite dimensional convex subset K h Shf3{v h > Ph only at the vertices of the triangulation}, as in Berger and Falk [15]. For other choices of convex subsets, see ([5], [7], [8], [13] We also note that in certain cases, the equality holds instead of inequality in (2.6). This happens when v, together 2u-v, also lies in K. In this case, we get <Tu, v-u> +b(u, v-u)= < f, v-u>.

(3.2)
Furthermore, if W is the interpolant of u, which agrees at every vertex of fl, then " lies in K h. It is well known from approximation theory, see Ciarlet [16] that u-W _< ch II u 2" Finally, let M and M h be the cones composed of non-negative functions on wlo'P(f) and its subspace Sh" Thus, it is clear that From these relations, it follows that u--u h U -Uh + l,b h.
(3) For p 4, we have II '-'h z, 0(hi/3), which is proved by Oden and Reddy [0] Wo 4(fl) in finite elasticity under the assumption that the solution lies in the interior of th convex set K. Thus our results represents an improvement of the previous results For < p < 2, there is no counterpart in the linear theory and our results appear to bc new ones.
Similarly, we can show that, 0(hZ-t), for < p<2. (1) For v 2, the results obtained in this paper are exactly those of Falk [5] and Mosco and Strang [6]. (2) In the absence of the constraints, our results reduce to the well known result., of Glowinski and Marroco [i] and Babuska [4].
(3) For p 4, we have u u h , 0(hi Wo in finite elasticity under the assumption that the solution lies in the interior ol ,he convex set K. Thus our results represents an improvement of the previous res Its.
For < p < 2, there is no counterpart in the linear theory and our results appear t,. be new ones.
Using the one-sided approximation result of Mosco and Strang [6] and Aubin-Nitsche trick [16], and the techniques of Noor [17] and Mosco [18], we can derive the fo, llowing error estimate for the finite element approximation of variational inequality (2.6) in the Lp-norm.  [10] under the assumption that all the solutions lie in the interior of the closed convex set K in wl'P-space. In this way, our results represent an improvement of their result. For < p _< 2, our results appear to be new ones and there is no counterpart in the linear theory. 4. CONCLUSION.
In this paper, we have obtained the error estimates of the finite element approximations of the solutions of a class of highly nonlinear variational inequalities in the w l'p and tv-norms, which appear to be new ones. These estimates are distinctly nonlinear in character. In particular, for p 2, corresponding to the linear elliptic theory, we obtain an error of order h, which agrees with the recent results.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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