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.

A Note on Iterations-based Derivations of High-order Homogenization Correctors for Multiscale Semi-linear Elliptic Equations

This Note aims at presenting a simple and efficient procedure to derive the structure of high-order corrector estimates for the homogenization limit applied to a semi-linear elliptic equation posed in perforated domains. Our working technique relies on monotone iterations combined with formal two-scale homogenization asymptotics. It can be adapted to handle more complex scenarios including for instance nonlinearities posed at the boundary of perforations and the vectorial case, when the model equations are coupled only through the nonlinear production terms.

Aplicação da metodologia numérica “fast bounds crack” para uma estimativa eficiente da evolução do tamanho de trinca

A significant part of the life of a mechanical component occurs, the crack propagation stage in fatigue. Currently, it is had several mathematical models to describe the crack growth behavior. These models are classified into two categories in terms of stress range amplitude: constant and variable. In general, these propagation models are formulated as an initial value problem, and from this, the evolution curve of the crack is obtained by applying a numerical method. This dissertation presented the application of the methodology "Fast Bounds Crack" for the establishment of upper and lower bounds functions for model evolution of crack size. The performance of this methodology was evaluated by the relative deviation and computational times, in relation to approximate numerical solutions obtained by the Runge-Kutta method of 4th explicit order (RK4). Has been reached a maximum relative deviation of 5.92% and the computational time was, for examples solved, 130,000 times more higher than achieved by the method RK4. Was performed yet an Engineering application in order to obtain an approximate numerical solution, from the arithmetic mean of the upper and lower bounds obtained in the methodology applied in this work, when you don’t know the law of evolution. The maximum relative error found in this application was 2.08% which proves the efficiency of the methodology "Fast Bounds Crack".

An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.

Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions

We develop the energy norm a-posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the errors measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

Pricing recreation use of national parks for an efficient nature conservation : an application to the portuguese case

Current institutions, research, and legislation have not yet been sufficient to achieve the conservation level of Nature as required by the society. One of the reasons that explains this relative failure is the lack of incentives to motivate local individual and Nature users in general, to adopt behaviour compliant with Nature sustainable uses. Economists believe that, from the welfare point of view, pricing is the more efficient way to make economic actors to take more environmental friendly decisions. In this paper we will discuss how efficient can be the act of pricing the recreation use of a specific natural area, in terms of maximising welfare. The main conservation issues for pricing recreation use, as well as the conditions under which pricing will be an efficient and fair instrument for the natural area will be outlined. We will conclude two things. Firstly that, from the rational utilitarian economic behaviour point of view, economic efficiency can only be achieved if the natural area has positive and known recreation marginal costs under the relevant range of the marshallian demand recreation curve and if price system management is not costly. Secondly, in order to guarantee equity for the different type of visitors when charging the fee, it is necessary to discuss differential price systems. We shall see that even if marginal recreation costs exist but are unknown, pricing recreation is still an equity instrument and a useful one from the conservation perspective, as we shall demonstrate through an empirical application to the Portuguese National Park. An individual Travel Cost Method Approach will be used to estimate the recreation price that will be set equal to the visitor’s marginal willingness to pay for a day of visit in the national park. Although not efficient, under certain conditions this can be considered a fair pricing practice, because some of the negative recreation externalities will be internalised. We shall discuss the conditions that guarantee equity on charging for the Portuguese case.

Efficient microwave hydrothermal preparation of nanocrystalline anatase TiO2 colloids

Titanium dioxide nanocrystals are an important commercial product used primarily in white pigments and abrasives, however, more recently the anatase form of TiO2 has become a major component in electrochemical and photoelectrochemical devices. An important property of titanium dioxide nanocrystals for electrical applications is the degree of crystallinity. Numerous preparation methods exist for the production of highly crystalline TiO2 particles. The majority of these processes require long reaction times, high pressures and temperatures (450–1400 °C). Recently, hydrothermal treatment of colloidal TiO2 suspensions has been shown to produce quality crystalline products at low temperatures (<250 °C). In this paper we extend this idea utilising a direct microwave heating source. A comparison between convection and microwave hydrothermal treatment of colloidal TiO2 is presented. The resulting highly crystalline TiO2 colloids were characterised using Raman spectroscopy, XRD, TEM, and electron diffraction. The results show that the microwave treatment of colloidal TiO2 gives comparable increases in crystallinity with respect to normal hydrothermal treatments while requiring significantly less time and energy than the hydrothermal convection treatment.