On the numerical solution of axisymmetric domain optimization problems by dual finite element method

Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.     

A class of random variational inequalities and simple random unilateral boundary value problems - existence,discretization, finite element approximation

The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation     

Remark on a trace theorem for transmission problems

In this paper we discuss elliptic transmission problems of second order in plane non-smooth domains with non-homogeneous boundary data. It is known that if the boundary data are homogeneous, then the variational solution of the transmission problem admits a representation as a sum of a regular function and certain singular functions, analogous to that encountered in boundary value problems on corner domains. We determine for which domains arbitrary non-homogeneous boundary data can be reduced to the homogeneous ones preserving availability of the representation formula. In the remaining cases we find the compatibility conditions for the data which also yield such a reduction.     

Characterization of the coincidence set for mixed signorini problems

For a mixed Signorini problem, reduction to a boundary variational inequality is derived. It is shown that its solution is a function constituted, on one portion of the boundary, of the upper-skirting function of solutions family of some associated linear mixed boundary value problems and, on the other portion, of the lower-skirting function of the same family. Qualitative behavior of the solution on different portions when perturbing the unknown boundary is analyzed. This shows in particular the usefulness of reduction to the boundary in some linear and nonlinear elliptic problems, even when usual variational methods cannot be applied for such purpose.     

Approximation of time-dependent free boundaries

Summary Two problems are considered in the paper: the first of them is connected with elliptic variational inequalities and consists in developing a moving obstacle algorithm for approximating the unknown free boundary; the other problem is linked with numerical solution of the Stefan problem, which is formulated in the similar way as in the elliptic case. Some computational aspects are also discussed in the paper.     

THE ELLIPTIC VARIATIONAL INEQUALITIES WITH DOUBLE DEGENERATE AND GENERAL GROWTH CONDITIONS

A class of quasilinear elliptic variatlonal inequalities with double degenerate is discussed in this paper. We extend the Keldys-Fichera boundary value problem and the first boundary problem of degenerate elliptic equation to the variatlonal inequalities. We establish the existence and uniquenees of the weak solution of conespondlng problem under monstandard growth conditions.     

Mixed variational formulations in locally convex spaces

The main purpose of this paper is to extend to the setting of locally convex spaces the study of the mixed variational formulation of some elliptic boundary value problems, the so-called Babuška–Brezzi theory. This study consists of characterizing the existence of a solution and giving conditions that guarantee the stability of the corresponding Galerkin method.     

Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.     

Nontrivial solutions for a class of quasilinear problems with jumping nonlinearities via a cohomological local splitting

We establish the existence of a nontrivial solution for a class of quasilinear elliptic boundary value problems with jumping nonlinearities by means of variational arguments and a cohomological local splitting.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

Projector preconditioning for partially bound‐constrained quadratic optimization

Preconditioning by a conjugate projector is combined with the recently proposed modified proportioning with reduced gradient projection (MPRGP) algorithm for the solution of bound‐constrained quadratic programming problems. If applied to the partially bound‐constrained problems, such as those arising from the application of FETI‐based domain decomposition methods to the discretized elliptic boundary variational inequalities, the resulting algorithm is shown to have better bound on the rate of convergence than the original MPRGP algorithm. The performance of the algorithm is illustrated on the solution of a model boundary variational inequality. Copyright © 2007 John Wiley & Sons, Ltd.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

Functional a posteriori estimates for elliptic variational inequalities

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlét) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 147–164.     

Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the nonlinearities.     

Estimates of Deviations from Exact Solutions of Elliptic Variational Inequalities

In this paper, we present a method of deriving majorants of the difference between exact solutions of elliptic type variational inequalities and functions lying in the admissible functional class of the problem under consideration. We analyze three classical problems associated with stationary variational inequalities: the problem with two obstacles, the elastoplastic torsion problem and the problem with friction type boundary conditions. The majorants are obtained by a modification of the duality technique earlier used by the author for variational problems with uniformly convex functionals. These majorants naturally reflects properties of exact solutions and possess necessary continuity conditions. Bibliography: 15 titles.     

Regularization and existence of solutions of three-dimensional elastoplastic problems

We prove the existence of solutions to the three-dimensional elastoplastic problem with Hencky's law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two dimensional crack. It is shown, in particular, that the variational solution satisfies all boundary conditions. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Multiple and sign‐changing solutions for the multivalued p ‐Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the first eigenvalue as well as the Fu?ik spectrum of the p ‐Laplacian we prove the existence of a positive, a negative and a sign‐changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant‐sign solutions plays a key role in the proof of sign‐changing solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

We consider a nonconforming hp -finite element approximation of a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H¹ as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved.