Thesis Summary:

The main objective of this doctoral thesis is the mathematical and numerical analysis of two important problems in electromagnetism; The first is related to various applications of magnetohydrodynamics, while the second has to do with the study of induced currents. As regards the first problem, the study focuses on the numerical approximation of the eigenvalues ​​of the rotational operator, whose solutions are called Beltrami fields and arise in various areas of physics. To carry out this study, the spectral problem is first mathematically analyzed, for which a mixed variational formulation is proposed by which a complete characterization of the solutions of the eigenvalue problem is obtained. In addition, it is considered a primal formulation which turns out to be "equivalent", under certain hypotheses, to that problem. For the numerical approximation of the eigenvalue problem, finite element schemes associated to each of these formulations are considered. In both cases spectral approximations are obtained with optimal convergence order, which are corroborated by numerical examples. In the second part of the thesis the mathematical and numerical analysis of diverse problems of currents induced in transitory regime, assuming axisymmetric domains, is approached. The materials considered are non-linear and may or may not have magnetic hysteresis. For this, and motivated by the physical applications, two data types are considered: the first one corresponds to a non-homogeneous Dirichlet condition at the domain boundary (usually the current intensity), while the second consists of assuming known The magnetic flux that crosses a southern section of the domain. In both cases, a formulation is proposed in terms of the magnetic field, and it is considered that the relationship between this field and the magnetic induction is given either by a non-linear function or by a hysteresis operator. Initially the nonlinear problem of induced currents is studied considering the magnetic flux as data. The existence and uniqueness of solution of the corresponding variational formulation is demonstrated by an abstract result. For the numerical approximation a spatial discretization is proposed by finite elements for which solution existence and an error estimate are proved. The above scheme is combined with an implicit Euler scheme for temporal discretization and optimal error estimates are shown. Next, the problem of induced currents with non-homogeneous Dirichlet condition is analyzed. In this case, the existence and uniqueness of solution are based on techniques of temporal discretization, a priori estimates and step to the limit by compactness. The numerical approximation of this problem is studied considering an implicit Euler scheme for the temporal discretization, which is later combined with a finite element method in space. As in the previous problem, optimal estimates of error are demonstrated in the appropriate standards, for both the semi-discretisation and the completely discrete problem. For both problems we show convergence tests that confirm the theoretical results obtained. Finally, we study the axisymmetric problem of induced currents in the case where the relationship between the magnetic field and the magnetic induction is given by a hysteresis operator. It is demonstrated the existence of solution of the problem considering a general hysteresis operator and the different boundary conditions. As in the problem without hysteresis, the study of the existence of solution is based on an implicit discretization of time; This approach procedure is frequently used in the analysis of equations that include memory operators. For the numerical approximation, a completely discrete scheme is considered by finite elements and implicit Euler, with a particular choice of hysteresis operator given by the classical Preisach operator. Contrary to problems without hysteresis, the convergence analysis of the scheme used is only performed by numerical examples.

ISI Publications from the Thesis

Rodolfo RODRíGUEZ, Pablo VENEGAS: Numerical approximation of the spectrum of the curl operator. Mathematics of Computation, vol. 83, 286, pp. 553-577, (2014).

Alfredo BERMúDEZ, Dolores GóMEZ, Rodolfo RODRíGUEZ, Pilar SALGADO, Pablo VENEGAS: Numerical solution of a transient non-linear axisymmetric eddy current model with non-local boundary conditions. Mathematical Models and Methods in Applied Sciences, vol. 23, 13, pp. 2495-2521, (2013).