Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls

Summary. An optimal control problem for impressed cathodic systems in electrochemistry is studied. The control in this problem is the current density on the anode. A matching objective functional is considered. We first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition. We then prove the existence of an optimal solution. Next, we derive a necessary condition of optimality and establish an optimality system of equations. Finally, we define a finite element algorithm and derive optimal error estimates. Received March 10, 1993 / Revised version received July 4, 1994     

A posteriori error estimates for nonconforming finite element methods

Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz decomposition. Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001     

Semiconvexity of invariant functions of rectangular matrices

The paper deals with the semiconvexity properties (i.e., the rank 1 convexity, quasiconvexity, polyconvexity, and convexity) of rotationally invariant functions f of matrices. For the invariance with respect to the proper orthogonal group and the invariance with respect to the full orthogonal group coincide. With each invariant f one can associate a fully invariant function of a square matrix of type where It is shown that f has the semi convexity of a given type if and only if has the semiconvexity of that type. Consequently the semiconvex hulls of f can be determined by evaluating the corresponding hulls of and then extending them to matrices by rotational invariance. Received: 10 October 2001 / Accepted: 23 January 2002 // Published online: 6 August 2002 RID="*" ID="*" This research was supported by Grant 201/00/1516 of the Czech Republic.     

Numerical analysis of the primal problem of elastoplasticity with hardening

Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening. Received May 7, 1997 / Revised version received August 31, 1998     

Locking and robustness in the finite element method for circular arch problem

Summary. In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order , and is uniformly robust with order for which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. Received June 5, 1992 / Revised version received May 17, 1994     

An isoperimetric estimate and W^{1,p}-quasiconvexity in nonlinear elasticity

A class of stored energy densities that includes functions of the form with , g and h convex and smooth, and is considered. The main result shows that for each such W in this class there is a such that, if a 3 by 3 matrix satisfies , then W is -quasiconvex at on the restricted set of deformations that satisfy condition (INV) and a.e. (and hence that are one-to-one a.e.). Condition (INV) is (essentially) the requirement that be monotone in the sense of Lebesgue and that holes created in one part of the material not be filled by material from other parts. The key ingredient in the proof is an isoperimetric estimate that bounds the integral of the difference of the Jacobians of and by the -norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids. Received January 5, 1998 / Accepted March 13, 1998     

New properties of a nonlinear conjugate gradient method

Summary. This paper provides several new properties of the nonlinear conjugate gradient method in [5]. Firstly, the method is proved to have a certain self-adjusting property that is independent of the line search and the function convexity. Secondly, under mild assumptions on the objective function, the method is shown to be globally convergent with a variety of line searches. Thirdly, we find that instead of the negative gradient direction, the search direction defined by the nonlinear conjugate gradient method in [5] can be used to restart any optimization method while guaranteeing the global convergence of the method. Some numerical results are also presented. Received March 12, 1999 / Revised version received April 25, 2000 / Published online February 5, 2001     

On constrained Newton linearization and multigrid for variational inequalities

Summary. We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments. Received March 22, 1999 / Revised version received February 24, 2001 / Published online October 17, 2001     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994     

Rate of convergence estimates for the spectral approximation of a generalized eigenvalue problem

Summary. The aim of this work is to derive rate of convergence estimates for the spectral approximation of a mathematical model which describes the vibrations of a solid-fluid type structure. First, we summarize the main theoretical results and the discretization of this variational eigenvalue problem. Then, we state some well known abstract theorems on spectral approximation and apply them to our specific problem, which allow us to obtain the desired spectral convergence. By using classical regularity results, we are able to establish estimates for the rate of convergence of the approximated eigenvalues and for the gap between generalized eigenspaces. Received February 6, 1996 / Revised version received November 28, 1996     

Young measure approximation in micromagnetics

Summary. Modeling of micromagnetic phenomena typically faces the minimization of a non-convex problem, which gives rise to highly oscillatory magnetization structures. Mathematically, this necessitates to extend the notion of Lebesgue-type solutions to Young-measure valued solutions. The present work proposes and analyzes a conforming finite element method that is based on an active set strategy to compute efficiently discrete solutions of the generalized minimization problem. Computational experiments are given to show the efficiency of the scheme. Received January 20, 2000 / Published online May 30, 2001     

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Summary. A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly. Received September 28, 1995 / Revised version received May 6, 1996     

Numerical approach of temperature distribution in a free boundary model for polythermal ice sheets

Summary. A numerical method for solving the thermal subproblem appearing in the modelization of polythermal ice sheets is described. This thermal problem mainly involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two phase Stefan formulation of the polythermal regime. The stationary temperature is obtained as the limit of an evolutive problem which is discretized in time with an upwind characteristics scheme and in space with finite elements. The nonlinearities are solved either by Newton-Raphson method or by duality techniques applied to maximal monotone operators. The application of the algorithms provides the dimensionless temperature distribution approximation and allows to identify the cold and temperate ice regions. Received February 1, 1998 / Published online July 28, 1999     

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Summary. Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a ``crystalline' approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension. Received January 28, 1993     

A posteriori error estimates for mixed FEM in elasticity

A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Received July 17, 1997     

Approximation of conformal mappings of annular regions

Summary. In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping of an annular region onto a circular annulus. In particular, we consider the case where has an analytic extension in compl() and, for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence. We also estimate the rate of uniform convergence in the case where the annular region has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations to the conformal module of . Received February 2, 1996     

Upper and lower error bounds for plate-bending finite elements

Summary. We compare the robustness of three different low-order mixed methods that have been proposed for plate-bending problems: the so-called MITC, Arnold-Falk and Arnold-Brezzi elements. We show that for free plates, the asymptotic rate of convergence in the presence of quasiuniform meshes approaches the optimal O(h) for MITC elements as the thickness approaches 0, but only approaches for the latter two. We accomplish this by establishing lower bounds for the error in the rotation. The deterioration occurs due to a consistency error associated with the boundary layer – we show how a modification of the elements at the boundary can fix the problem. Finally, we show that the Arnold-Brezzi element requires extra regularity for the convergence of the limiting (discrete Kirchhoff) case, and show that it fails to converge in the presence of point loads. Received June 9, 1998 / Published online December 6, 1999     

Design principles and error analysis for reduced-shear plate-bending finite elements

Summary. In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic analysis, that convergence of the finite element approximations will occur uniformly as , even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze both straight and curved elements. We also analyze some other elements. Received June 19, 1995     

Line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions

Summary. Many successful quasi-Newton methods for optimization are based on positive definite local quadratic approximations to the objective function that interpolate the values of the gradient at the current and new iterates. Line search termination criteria used in such quasi-Newton methods usually possess two important properties. First, they guarantee the existence of such a local quadratic approximation. Second, under suitable conditions, they allow one to prove that the limit of the component of the gradient in the normalized search direction is zero. This is usually an intermediate result in proving convergence. Collinear scaling algorithms proposed initially by Davidon in 1980 are natural extensions of quasi-Newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations, and that they interpolate values of both the gradient and the function at the current and new iterates. Line search termination criteria that guarantee the existence of such a normal conic local approximation, which also allow one to prove that the component of the gradient in the normalized search direction tends to zero, are not known. In this paper, we propose such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions. Received February 1, 1997 / Revised version received September 8, 1997     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994