In this paper we proposed a modi ed Lagrangian functional for obstacle problem, investigated its properties. Then we construct Uzawa method for nding a saddle point, proved the convergence theorems. Some numerical examples are provided. We consider a modi ed duality method for solving the obstacle problem. The obstacle problem is a typical example of the elliptic variational inequality. Many important problems ranging from contact problems in continuum mechanics to option pricing in computational nance can be formulated as the obstacle problem. See for instance the book [1] where many of these applications are described, as well as the classical literature on this problem. Finally, apart from their practical relevance, obstacle problems are fascinating mathematical objects of their own value. The basic properties of the solution, including existence and uniqueness, were established by Lions and Stampacchia [2]. Many approaches for the numerical solution of obstacle problems have been suggested and pursued [3{5]. The main existing numerical methods for the solution of obstacle problems in particular, are mathematical programming approach, and schemes based on penalty formulations and Lagrangian multiplier formulations [6{8]. The modi ed Lagrangian functional for the rst time were developed and investigated for solving the problem of nite-dimensional optimization. Their emergence was related to the fact that classical Lagrangian functionals that are linear functions of the dual variables are not suitable for solving the singular optimization problems. The construction of modi ed Lagrangian function (functional) actually comprises regularization of dual variables. In last time the Lagrangian multiplier method is successfully applied to the solution of in nite-dimensional variational inequalities in mechanics [9{ 11]. In this paper the duality scheme based on the modi ed Lagrangian functional is examined for the obstacle problem. The paper is structures as follows. In Sect. 2, we introduce the obstacle problem. In Sect. 3, we present the sensitivity functional for the obstacle problem and we investigate In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Copyright ⃝c by the paper's authors. Copying permitted for private and academic purposes. its properties. In Sect. 4, we propose the modi ed Lagrangian functional and construct the Uzawa method for nding a saddle point, prove the convergence theorems. Finally, in Sect. 5 we present the numerical examples. Sect. 6 contains some concluding remarks. 2
The Obstacle Problem Let us consider a simple model for a problem with obstacle. There are horizontal circular wire and a membrane hanging on this wire (see g. 1(a)). We assume that this membrane is horizontal and above a plate. This plate is a obstacle for plate's de ection. When we load the membrane with a force f in the vertical direction, it undergoes de ection ( g. 1(b) | the case without the plate). If there is the place we get a contact area between the membrane and the obstacle which is the plate. This contact area is called the coincidence set ( g 1(c)).
The obstacle problem can be described as follows: nd the equilibrium position u(x), x 2 Ω R2 of an elastic membrane constrained to lie above a given obstacle (x) under an external force f (x). Then u(x) is the formal solution of the boundary problem ∆u(x) = f (x) a.e. in N = fx 2 Ω : u(x) > (x)g; u(x) = (x) in = fx 2 Ω : u(x) =
(x)g; u(x) = 0; x 2 : Here Ω R2 be a bounded and open domain with smooth boundary ; where f is an element L2(Ω); is an element of H01(Ω) with 0 on .
Therefore the obstacle problem can be posed as a variation problem. Set and
1 ∫ 2
{ Find v 2 K such that
J (v) min :
(
The functional J (v) is strongly coercive in H01(Ω), it means J (v) ! +1 for
∥v∥H1(Ω) ! 1. Hence the problem (
L(v; l) = J (v) + l (
v) dΩ; v 2 H01(Ω); l 2 L2(Ω) is consider, and it shows that a point (v ; l ) = (u; ∆u of the Lagrangian functional L(v; l), f ) is a unique saddle point L(u; l)
L(u; ∆u f )
L(v; ∆u f ); 8 v 2 H01(Ω); l 2 (L2(Ω))+ where (L2(Ω))+ = fw 2 L2(Ω) : w 0 a.e. in Ωg:
It is known that duality methods based on classical Lagrangian functionals don't guarantee the convergence of the available methods for nding saddle points in variational inequalities in mechanics. The convergence of duality methods can be established according to direct variable only. In addition the step according to dual variable must be sufficiently small. 3
The Sensitivity Functional
Km = fv 2 H01(Ω) : v
m a.e. on Ωg (m) = { inf J (v); if Km ̸= ∅; v2Km +1; otherwise:
It is easy to see that if a function m 2 L2(Ω) is lower bounded on Ω, the corresponding set Km is not empty and inf J (v) > 1 [14]. The set Km can be empty v2Km if m 2 L2(Ω) n H1(Ω) and not lower bounded on Ω. Then (m) is a proper convex functional on L2(Ω), but it's effective domain dom = fm 2 L2(Ω) : (m) < +1g does not coincide with L2(Ω). Notice that dom is a convex but not closed set. In this case, dom = L2(Ω).
Theorem 1. The sensitivity functional (m) is weakly lower semicontinuous on L2(Ω).
Since (m) is convex functional, it is suffices to show that (m) is lower semicontinuous on L2(Ω) (in norm space L2(Ω)).
L2(Ω) such that m = lim mi. We can show i!1 that the conditions 1) lim (mi) = +1 if m 2= dom ;
i!1 2) lim (mi) (m) if m 2 dom
i!1 are satis ed. Desirable property of lower semicontinuous for (m) follows from these conditions. The proof is complete.
Fl(m) = (m) + Ω m2 dΩ; where r > 0 is a constant. The functional Fl(m) is very important for constructing the duality methods based on modi ed Lagrangian functionals [11].
For a xed l 2 L2(Ω), we examined the functional Fl(m) for m 2 L2(Ω). From theorem 1 follows that Fl(m) is a weakly semicontinuous functional on L2(Ω). Theorem 2. The functional Fl(m) is coercive in L2(Ω).
Since (m) is a lower semicontinuous functional, then the epigraph of sensitivity functional epi f(v; a) 2 L2(Ω)
R : (v) ag is a convex closed set in L2(Ω) R. According Mazur separation theorem [15, p. 164] there are 2 L2(Ω) and 2 R, such that m dΩ + (m) + 0 8 m 2 dom :
∫ is satis ed and Fl(m) ! +1 if ∥m∥L2(Ω) ! 1.
m(l) = arg m2mLi2n(Ω) Fl(m): (l) = (m(l)); 8l 2 L2(Ω): Theorem 3. The function
(l) is continuous in L2(Ω).
Since Fl(m) is a strongly convex functional, then the inequality (m(l)) + l m(l) dΩ + (m(l))2 dΩ + 2r ∥m(l) m∥2L2(Ω) (m) +
Ω ∫ Ω r ∫ 2
Ω r ∫ 2
Ω Ω (m′) + l′ m′ dΩ + (m′)2 dΩ + r 2 ∥m′ m′′∥2L2(Ω) (m′′) + l′ m′′ dΩ + (m′′)2 dΩ; (m′′) + l′′ m′′ dΩ + (m′′)2 dΩ + r 2 ∥m′′ m′∥2L2(Ω) (m′) + l′′ m′ dΩ + (m′)2 dΩ:
r∥m′ m′′∥2L2(Ω) (l′ l′′)(m′′
m′) dΩ; ∥m′ m′′∥L2(Ω)
l′′∥L2(Ω): Ω
Ω l′(m′′ m′) dΩ + ((m′′)2 (m′)2) dΩ: ∫ Ω
9 m2 dΩ= = ;
inf m2L2(Ω)
Fm(l):
If (v ; l ) is a saddle point of M (v; l), then v is a solution of problem (
M(l) max :
(
From theorems 2, 3 and inequality (
Theorem 4. The dual functional M(l) is Gateaux differentiable in L2(Ω) and its derivative ∇M(l) satis es the Lipschitz condition with the constant 1=r; that is, for all l′, l′′ 2 L2(Ω), it holds that
l′′∥L2(Ω): i=0,1,2,. . . ; l0 2 L2(Ω) is given.
We can show that the sequence fuk; lkg is bounded sequence in H1(Ω)
Theorem 6. The algorithm (
Indeed, the sequence flkg is bounded sequence in L2(Ω), and the functional (m) is weakly lower semicontinuous on L2(Ω), which yields
8 lim < (m(lk)) + k!1 : ∫ Ω
9 (m(lk))2 dΩ= = ; On the other hand, we have from the de nition M(l) =
Ω Ω
9 m2 dΩ= ; (0); k = 0; 1; 2; : : : : 8 lim < (m(lk)) + k!1 : ∫ Ω Ω 9 (m(lk))2 dΩ= ; (0):
8 lim < (m(lk)) + k!1 : ∫ Ω Now, theorem 5 implies that Ω
9 (m(lk))2 dΩ= = (0) = J (u ): ; lim J (uk) = lim k!1 k!1 (m(lk)) = (0) = J (u):
M(lk)
M(l )
∥m(lk)∥L2(Ω)∥lk
l ∥L2(Ω):
Hence, lim M(lk) = M(l ). It means that algorithm (
A Numerical Example
In this section we present some numerical experiments in solving an obstacle problem
by using the algorithm (
For the numerical realization (
We call a family of triangulations regular if each triangulation in it satis es (a) and (b) with uniform for the family. Given a triangulation Th, let Vh = Vh(Th) denote the collection of all H1(Ω) functions which are affine on each triangle in Th; Vh is the space of continuous piecewise linear functions over Th. Take Vh = Vh \ H01(Ω). For v 2 C0(Ω), let vh 2 Vh be the interpolant of v; v = vh at each vertex in Th. De ne Kh = fvh 2 Vh: vh hg.
The discrete approximation of u is given by uh 2 Kh, where uh is a solution the next problem { Find vh 2 Kh such that
J (vh) min :
Let k be an integer number denoting the iteration parameter. The algorithm presented in Sect. 4, to solve the obstacle problem, can be expressed as follows.
Step 1. Find a solution ukh+1 2 Vh of the problem 8 >< M(vh; lk) = J (vh) + >: vh 2 Vh:
In Fig. 3 a dashed line (blue) is a graph solution of the problem without obstacle, a thin solid line (green) is a graph of the function that de nes an obstacle, and a thick line (red) a graph solution of the obstacle problem.
The numerical experiments demonstrate the possibility to use Uzawa algorithm for modi ed Lagrange functional. In paper [18] it was found that for problems with constraints on the boundary the best convergence rate is achieved for large values r (r = 106, 108). In this work we we have shown that the optimum value of the parameter r is in the range 20 200. 6
In this paper, we have considered a modi ed Lagrangian functional for the obstacle problem. The modi ed Lagrangian functionals considered in the present paper are analogs of the corresponding modi ed functions constructed to solve the nitedimensional optimization problems. The duality methods based on the modi ed Lagrangian functionals offer a convenient and efficient tool to solve the in nite-dimensional variational inequalities of mechanics.
The author makes no attempt to compare the effectiveness of the proposed method with other methods of numerical optimization with constraints. The objective of this paper was demonstrate the application of the duality scheme based on the modi ed Lagrangian functional for the problem with constraints in the domain.