Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W ^{1, p} -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.     

A Proof of the Lewy-Stampacchina's Inequality by a Penalization Method

In this paper we prove the Lewy–Stampacchia's inequality for elliptic variational inequalities with obstacle involving fairly general Leray–Lions operators. The main novelty of the paper is the method of proof, which uses the natural penalization. One of the steps of the proof consists in proving, again thanks to the natural penalization, that the nonnegative cone of W ₀ ^1,p () is dense in the nonnegative cone of W^-1,p().     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

Three-step iterative methods for general variational inclusions in L P spaces

In this paper, we show that the general variational inclusions are equivalent to the fixed point problem. We use this equivalence to discuss the existence of the variational inclusions in L ^p spaces. Using the technique of the updating solution, we suggest some three-step iterative methods for solving the general variational inclusion. We also consider the convergence analysis of the proposed iterative methods under some mild conditions. Since the general variational inclusions include several classes of variational inequalities and optimization problems as special cases, results proved in this paper continue to hold for these problems.     

Optimal successive overrelaxation iterative methods for P-cyclic matrices

Summary We consider linear systems whose associated block Jacobi matricesJ _p are weakly cyclic of indexp. In a recent paper, Pierce, Hadjidimos and Plemmons [13] proved that the block two-cyclic successive overrelaxation (SOR) iterative method is numerically more effective than the blockq-cyclic SOR-method, 2<qp, if the eigenvalues ofJ _p ^p are either all non-negative or all non-positive. Based on the theory of stationaryp-step methods, we give an alternative proof of their theorem. We further determine the optimal relaxation parameter of thep-cyclic SOR method under the assumption that the eigenvalues ofJ _p ^p are contained in a real interval, thereby extending results due to Young [19] (for the casep=2) and Varga [15] (forp>2). Finally, as a counterpart to the result of Pierce, Hadjidimos and Plemmons, we show that, under this more general assumption, the two-cyclic SOR method is not always superior to theq-cyclic SOR method, 2<qp.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch supported in part by the Deutsche Forschungsgemeinschaft     

Finite element solution of nonlinear elliptic problems

Summary The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solutionuH ¹(), the convergence of the method is proved; under the additional assumptionuH ²(), the rate of convergenceO(h) is derived without the use of Green's theorem.     

On finite element approximation in theL ∞-norm of variational inequalities

Summary We are interested in the approximation in theL -norm of variational inequalities with non-linear operators and somewhat irregular obstacles. We show that the order of convergence will be the same as that of the equation associated with the non-linear operator if the discrete maximum principle is verified.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

On a computational method for the second fundamental tensor and its application to bifurcation problems

Summary An algorithm is presented for the computation of the second fundamental tensorV of a Riemannian submanifoldM ofR ⁿ . FromV the riemann curvature tensor ofM is easily obtained. Moreover,V has a close relation to the second derivative of certain functionals onM which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, thed-dimensional manifoldM is defined implicitly as the zero set of a submersionF onR ⁿ . In this case, the principal cost of the algorithm for computingV(p) at a given pointpM involves only the decomposition of the JacobianDF(p) ofF atp and the projection ofd(d+1) neighboring points ontoM by means of a local iterative process usingDF(p). Several numerical examples are given which show the efficiency and dependability of the method.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThis work was in part supported by the National Science Foundation (DCR-8309926) and the Office of Naval Research (N-00014-80-C09455). The second author began some of the work while visiting the University of Heidelberg/Germany as an Alexander von Humboldt Senior U.S. Scientist     

Optimal control of the obstacle in semilinear variational inequalities

We consider an optimal control problem where the state satisfies an obstacle type semilinear variational inequality and the control function is the obstacle. The state is chosen to be close to a desired profile while the obstacle is not too large in H ₀ ¹ (), and H ²-bounded. We prove that an optimal control exists and give necessary optimality conditions, using approximation techniques.     

Error estimates for the finite element solution of variational inequalities

Summary We analyze the convergence of finite element approximations of some variational inequalities namely the obstacle problem and the unilateral problem. OptimalO(h) andO(h^3/2–) error bounds for the obstacle problem (for linear and quadratic elements) and anO(h) error bound for the unilateral problem (with linear elements) are proved.Supported in part by the Institut de Recherche d'Informatique et d'Automatique and by National Science Foundation grant MCS 75-09457     

A continuation method for (strongly) monotone variational inequalities

We consider the variational inequality problem, denoted by VIP(X, F), whereF is a strongly monotone function and the convex setX is described by some inequality (and possibly equality) constraints. This problem is solved by a continuation (or interior-point) method, which solves a sequence of certain perturbed variational inequality problems. These perturbed problems depend on a parameter > 0. It is shown that the perturbed problems have a unique solution for all values of > 0, and that any sequence generated by the continuation method converges to the unique solution of VIP(X,F) under a well-known linear independence constraint qualification (LICQ). We also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Error estimates for the finite element solution of variational inequalities

Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called obstacle problem and unilateral problem. Optimal error bounds are obtained in both cases.Supported in part by National Science Foundation grant MCS 75-09457, and by Office of Naval Research grant N00014-76-C-0369     

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Summary In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ¹ discretized by using piecewise quadratic finite elements and in ² by using the five-point difference approximation are presented.     

Theh−p version of the finite element method for elliptic equations of order2m

Summary The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown in [3, 4, 5] that the solutions of the second order and fourth order equations belong to the spacesB ¹ where the weighted Sobolev norms of thek-th derivatives are bounded byCd ^k–l (k–l)!,kl, l2 whereC andd are constants independent ofk. In this case theh–p version of the finite element method leads to an exponential rate of convergence measured in the energy norm [6, 12, 13]. Theh–p version was implemented in the code PROBE¹ [18] and has been very successfully used in the industry.We will discuss in this paper the generalization of these results for problems of order2m. We will show also that the exponential rate can be achieved if the exact solution belongs to the spacesB ¹ where the weighted Sobolev norm of thek-th derivatives is bounded byCd ^k–l (k–l)!,kl=m+1, C andd are independent ofk. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spacesB ^m+1 .Problems of this type are related obviously to many engineering problems, such as problems of plates and shells, and are also important in connection with well-known locking problems.Dedicated to Professor Ivo Babuka on the occasion of his 60^th birthdaySupported by the Air Force Office of Science Research under grant No. AFOSR-80-0277 NOETIC TECHNOLOGIES, Inc., St. Louis, MO     

Convergence of a penalty-finite element approximation for an obstacle problem

Summary This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is where is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.     

An algorithm for best approximate solutions ofAx=b with a smooth strictly convex norm

Summary In this paper, overdetermined systems ofm linear equations inn unknowns are considered. With ^m equipped with a smooth strictly convex norm, ·, an iterative algorithm for finding the best approximate solution of the linear system which minimizes the ·-error is given. The convergence of the algorithm is established and numerical results are presented for the case when · is anl _p norm, 1<p<.Portions of this paper are taken from the author's Ph.D. thesis at Michigan State University     

A generalized f-projection algorithm for inverse mixed variational inequalities

In this paper, a new inverse mixed variational inequality is introduced and studied in Hilbert spaces, which provides a model for the study of traffic network equilibrium control problems. An iterative algorithm involving the generalized $f$ -projection operator for solving inverse mixed variational inequalities is constructed and the convergence of sequences generated by the algorithm is given under some suitable conditions.