Multilevel Projection Algorithm for Solving Obstacle Problems

K e y w o r d s O b s t a c l e problem, Multilevel projection, Convergence, Finite element, Error estimate. 1 . I N T R O D U C T I O N The obstacle problem we consider here can be described as follows: find the equilibrium position u = u(x), x E f~ C R 2 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 problem A u _> f , in ~, (1) u _> ~, in f~, (2) ( A u f ) (u g2) = 0, in f~, (3) u = 0, on F, (4) u = ~, on r*, (5) O~u = 0 ~ , on r*, (6) where A is the Laplace operator and F is the boundary of f~. The interface between the sets {x e f~: u(x) = ~(x)} and {z e 12: u(x) > ~Y(x)} is the free boundary F*, and 0~ is the normal derivative operator along F*. Notice tha t F* is unknown of the problem. Problem (1)-(6) can be posed as a problem in the calculus of variations. I t is solved by the unique solution of the minimization problem J(u) = min J(v), (7) v E K Supported by the MICS Program of the U.S. Department of Energy under Grant DE-FG02-90ER25084 and by Los Alamos National Laboratories under Contract Number C738100182X. 0898-1221/01/$ see front matter Q 2001 Elsevier Science Ltd. All rights reserved. Typeset by ~4j~.9-TEX PII: S0898-1221 (01)00115-8 1506 Y. ZHANG where J(v) = -~ 1~Tvl2 dxf v d x , (8) and K is a convex set of functions in H~(12) greater than or equal to ~, i.e., K = {v • H~(12) : v _> • in g~}. It is well known that this problem is equivalent to a variational inequality, to find u • K such that a(u, v u) > ( f , v u), for all v • K, where (., .) is the L 2 inner product and a(., .) is of the Dirichlet form a(v, w) = ~ Vv. Vw dx, • Obstacle problems are a type of free boundary problem. They are of interest both for their intrinsic beauty and for the wide range of applications they describe in subjects from physics to finance. Many important problems can be formulated by transformation to an obstacle problem, e.g., the filtration dam problem [1], the Stefan problem [1], the subsonic flow problem [2], American options pricing model [3], etc. The basic properties of the solution, including existence and uniqueness, were established by Lions and Stampacchia [4]. Since obstacle problems are highly nonlinear, the computation of approximate solutions can be a challenge. Multilevel (or multigrid) methods have been proven very robust in modern scientific computing since the proof of convergence of the multigrid method in linear equations by Bank and Dupont [5]. The purpose of this paper is to extend this method to solve the obstacle problem and to establish the convergence theorems of this method. The outline of this paper is as follows. In Section 2, the finite element discretization of the obstacle problem is defined. In Section 3, a multilevel projection method is introduced. A property of the projection gradient operator is studied. The convergence of the multilevel projection algorithm is presented in Section 4. The error estimates are obtained in both Euclidean and L 2 norms. In Section 5, we give an illustrative numerical example in two-dimensional space. Our numerical result agrees well with the theory of Section 4. The example also shows that the multilevel projection algorithm is very robust in practice. The final section is reserved for conclusions and discussion. 2. FINITE E L E M E N T DISCRETIZATION Suppose henceforth that the boundary of 12, F, is polygonal. For a triangulation T of ~, let h = h(T) be the max of the lengths of the edges. Then T satisfies the shape regularity and the maximum angle condition if (a) there is a positive constant p such that for any ~E T, there is a disk B of radius r with B C T and ph < r < h, and (b) maximum angle < ~r/2. We call a family of triangulations regular if each triangulation in it satisfies (a) and (b) with p uniform for the family. Given a triangulation Th, let Vh = Vh(Th) denote the collection of all H I ( ~ ) functions which are affine on each triangle in Th; Vh is the space of continuous piecewise linear functions over Th. Take V h = VhOHI(~) . For v E C°(~), let Irh(V) • Vh be the interpolant of V; V = "ffh(V) at each vertex in Th. Let ~h = Irh(~), and define K h = {Vh • Vh : Vh ~_ ~ h } . The discrete approximation of u is given by Uh E Kh such that J ( u h ) = min J ( v h ) . (9) Vt, E Kh It is convenient to be able to express the above problem on R s. To this end, let p~, i = 1 , . . . , s be the vertices of Th that are in ~; these are usually called the interior vertices. Take ¢i E Vh Obstacle Problems 1507 to be such tha t ¢i vanishes at all vertices of Th except p~ and ¢~(Pi) = 1. Then Vh = span {¢h}, and the ¢~s provide the usual nodal basis for Vh. Define the s x s matr ix A h = (ahj) and the s × 1 matr ix fh __ (fh) by h % = a (¢3, ¢~), fh ---(f, ¢i). With V h = R s, set q~h = ffyh(Pi ) and take K h = {v h • V h : v h > ~ h i = l , . , s} With this notation, V h and K h are just the coefficients with respect to the basis {¢i} of Vh and Kh, respectively. Take Vh ~ v h, i.e., Vh = ~ vh¢~. The energy can be writ ten in terms of