Nadaniela Egidi

Special metrics on compact complex manifolds

Abstract We study the existence of special metrics on compact complex manifolds. We show that every considered metric can be caracterized using conditions on the space of positive currents. We investigate what happens under holomorphic submersions or modifications. We show which metrics exist on some classical examples.

Block decomposition techniques in the generation of adaptive grids

We consider the problem of the generation of adaptive grids, where, for a given domain @W, we have to compute a partition that depends on a given map f providing the desired local mesh size. We propose a numerical method for the solution of this problem in the case of planar quadrilateral grids. In particular, a block decomposition method is proposed, by which @W is first decomposed in several blocks depending on the shape of the domain and on the size map f, and then the whole grid is obtained as the union of the quadrilateral grids on each block. The algorithm concludes with a smoothing step that depends on the size map f. We present some simple numerical examples to test the proposed method.

Artificial boundary conditions for the Burgers equation on the plane

The numerical solution of the initial value problem for the two-dimensional Burgers equation on the whole plane is considered. Usual techniques, like finite difference methods and finite element methods cannot be directly applied for the solution of this problem, because the corresponding domain is unbounded. We propose a new method to overcome this difficulty. The efficiency of the proposed method is tested by several numerical examples.

The use of Sherman-Morrison formula in the solution of Fredholm integral equation of second kind

We consider a constructive method for the solution of Fredholm integral equations of second kind. This method is based on a simple generalization of the well-known Sherman-Morrison formula to the infinite dimensional case. In particular, this method constructs a sequence of functions, that converges to the exact solution of the integral equation under consideration. A formal proof of this convergence result is provided for the case of Fredholm integral equations with L^2 integral kernel. Finally, a boundary value problem for the Laplace equation is considered as an example of the application of the proposed method.

An efficient method for the solution of the inverse scattering problem for penetrable obstacles

Abstract: We consider a two-dimensional scattering problem for inhomogeneous media. This problem arises from the study of time-harmonic electromagnetic scattering problems for Transverse Magnetic (TM) waves. In the direct scattering problem we have to compute the scattered wave from the knowledge of the incident wave and of the inhomogeneity. In the inverse scattering problem we have to reconstruct the refractive index of the inhomogeneity from some knowledge of the scattered waves generated by the inhomogeneity itself with known incident waves. The complete formulation of this inverse problem is given by a system of two integral equations, where the unknown functions are the refractive index of the inhomogeneity, and the total field inside the inhomogeneity. Note that, the total field is defined as the sum of the incident field plus the scattered field. The numerical solution of this system has a high computational cost even for low or medium size discretization schemes. The main contribution of the present paper is given by an efficient method for the numerical solution of this problem, where, roughly speaking, the computation of the total field inside the inhomogeneity is avoided. Some numerical experiments are used to show the performance of the proposed method. In these experiments, we have used exact scattering data, and synthetic scattering data.

The efficient solution of direct medium problems by using translation techniques

We consider a Fredholm integral equation arising from a time-harmonic electromagnetic scattering problem for inhomogeneous media. The discretization of this equation usually produces a large dense linear system that must be solved by iterative methods. To speed up these methods we propose an efficient computation of the action of the corresponding coefficient matrix on a generic vector. This computation is mainly based on the well known addition formula for the Hankel functions and a simple translation argument. We present some numerical examples to show the efficiency of the proposed method.

A numerical solution of Richards equation: a simple method adaptable in parallel computing

ABSTRACT Numerical simulation models of water flow in variably saturated soils are important tools in water resource management, assessment of water-related disasters and agriculture. Richards equation is one of the most used models for the fluid flow simulation into porous media. It is a partial differential equation whereby analytical solutions are only possible after applying a number of restrictive assumptions. Therefore, the derivation of efficient numerical schemes for its approximated solution has to be computed by discretization methods. We propose a numerical procedure considering a simplified linearization scheme that makes it adaptable to parallel computing. A comparison in computational performances with three other numerical procedures is detailed for a large computation, including the assessment of the landslide hazard in real areas. We demonstrate the efficiency of the proposed numerical procedure by comparing the results we obtained with a parallel code.

An integral equation method for the numerical solution of the Burgers equation

Abstract We consider an initial–boundary value problem for the two-dimensional Burgers equation on the plane. This problem is reformulated by an equivalent integral equation on the Fourier transform space. For the solution of this integral equation, two numerical methods are proposed. One of these two methods is based on the properties of the Gaussian function, whereas the other one is based on the FFT algorithm. Finally, the Galerkin method with Gaussian basis functions is applied to the original initial–boundary value problem, in order to compare the performances of the proposed methods with a standard numerical procedure. Some numerical examples are given to evaluate the efficiency of the proposed methods. The performances obtained from these numerical experiments promise that these methods can be applied effectively to more complex problems, such as the Navier–Stokes equation and the turbulence flows.

Solution strategies for finite elements and finite volumes methods applied to flow and heat transfer problem in U-shaped geothermal exchangers

Matter of this paper is the study of the flow and the corresponding heat transfer in a U-shaped heat exchanger. We propose a mathematical model that is formulated as a forced convection problem for incompressible and Newtonian fluids and results in the unsteady Navier-Stokes problem. In order to get a solution, we discretise the equations with both the Finite Elements Method and the Finite Volumes Method. These procedures give rise to a non-symmetric indefinite quadratic system of equations. Thus, three regularisation techniques are proposed to make approximations effective and ideas to compare their results are provided.

Original article: A three-dimensional model for the study of the cooling system of submersible electric pumps

We study the cooling system for submersible electric pumps. This study aims to provide some guidelines to improve the existing cooling system of these electric pumps when they work partially or totally not immersed in the service fluid. Note that inefficient cooling systems cannot prevent the rise in temperature of the alternating current (AC) motor of the pump, and the consequent reduction of the service factor. We propose a three-dimensional model to describe the fluid flow in the cooling circuit and the heat transfer among the engine of the electric pump, the cooling system and the external environment. This model is discretized by using a finite element method, and the resulting algorithm has been implemented in a FORTRAN code. We show some preliminary results obtained by this code using two different geometries for the cooling circuit, and we compare these results with experimental data and with other numerical results obtained in a previous work, by using a two-dimensional model.