Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

The tangential variation of a localized flux-type eigenvalue problem

In this work the differentiability of the principal eigenvalue lambda = lambda(1)(Gamma) to the localized Steklov problem -Delta u + qu = 0 in Omega, partial derivative u/partial derivative nu = lambda chi(Gamma)(x)u on partial derivative Omega, where Gamma subset of partial derivative Omega is a smooth subdomain of partial derivative Omega and chi(Gamma) is its characteristic function relative to partial derivative Omega, is shown. As a key point, the flux subdomain Gamma is regarded here as the variable with respect to which such differentiation is performed. An explicit formula for the derivative of lambda(1) (Gamma) with respect to Gamma is obtained. The lack of regularity up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem. Therefore, the whole analysis must be done in the weak sense of H(1)(Omega). The study is of interest in mathematical models in morphogenesis. (C) 2011 Elsevier Inc. All rights reserved.

On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L2-setting. An effective way of discretizing these boundary integral equations based on the Nystr¨om method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary displacements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [16] applied to Cauchy problems in linear elasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed method.

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

An alternating potential based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut

We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

Type-2 TOPSIS:a group decision problem when ideal values are not extreme endpoints

In the traditional TOPSIS, the ideal solutions are assumed to be located at the endpoints of the data interval. However, not all performance attributes possess ideal values at the endpoints. We termed performance attributes that have ideal values at extreme points as Type-1 attributes. Type-2 attributes however possess ideal values somewhere within the data interval instead of being at the extreme end points. This provides a preference ranking problem when all attributes are computed and assumed to be of the Type-1 nature. To overcome this issue, we propose a new Fuzzy DEA method for computing the ideal values and distance function of Type-2 attributes in a TOPSIS methodology. Our method allows Type-1 and Type-2 attributes to be included in an evaluation system without compromising the ranking quality. The efficacy of the proposed model is illustrated with a vendor evaluation case for a high-tech investment decision making exercise. A comparison analysis with the traditional TOPSIS is also presented. © 2012 Springer Science+Business Media B.V.

An iterative procedure for solving a Cauchy problem for second order elliptic equations

An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.