A two-level ILU preconditioner for electromagnetic applications

[EN] Computational electromagnetics based on the solution of the integral form of Maxwell s equations with boundary element methods require the solution of large and dense linear systems. For large-scale problems the solution is obtained by using iterative Krylov-type methods provided that a fast method for performing matrix vector products is available. In addition, for ill-conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance. For many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix. In this context, approximate inverse preconditioners based on Frobenius-norm minimization have emerged as a robust and highly parallel alternative. In this work we propose a two-level ILU preconditioner for the preconditioned GMRES method. The computation and application of the preconditioner is based on graph partitioning techniques. Numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ILU preconditioners that perform competitively compared with Frobenius-norm minimization preconditioners.This work was supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2014-58159-P and MTM2015-68805-REDT.Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2017). A two-level ILU preconditioner for electromagnetic applications. Journal of Computational and Applied Mathematics. 309:371-382. https://doi.org/10.1016/j.cam.2016.03.012S37138230

Iterative Solutions of Large-Scale Soil-Structure Problems

Ph.DDOCTOR OF PHILOSOPH

Geophysics for Mineral Exploration

This Special Issue contains ten papers which focus on emerging geophysical techniques for mineral exploration, novel modeling, and interpretation methods, including joint inversions of multi physics data, and challenging case studies. The papers cover a wide range of mineral deposits, including banded iron formations, epithermal gold–silver–copper–iron–molybdenum deposits, iron-oxide–copper–gold deposits, and prospecting forgroundwater resources

Development of provably stable A-phi formulation time domain integral equations

Applications involving quantum physics are becoming an increasingly important area for electromagnetic engineering. To address practical problems in these emerging areas, appropriate numerical techniques must be utilized. However, the unique needs of many of these applications require the development of new computational electromagnetic solvers. The A-Phi formulation is a novel approach that can address many of these needs. This formulation utilizes equations developed in terms of the magnetic vector potential (A) and electric scalar potential (Phi). The resulting equations overcome many of the limitations of traditional solvers and are ideal for coupling to quantum mechanical calculations. The main novelty of this thesis is the extension of the A-Phi formulation to two sets of time domain integral equations. These integral equations are provably stable and constitute robust numerical techniques that can be utilized in many applications. To validate the proposed time domain integral equations, numerical results are presented which demonstrate the stability and accuracy of the developed methods

Discontinuous Galerkin Method Applied to Navier-Stokes Equations

Discontinuous Galerkin (DG) finite element methods are becoming important techniques for the computational solution of many real-world problems describe by differential equations. They combine many attractive features of the finite element and the finite volume methods. These methods have been successfully applied to many important PDEs arising from a wide range of applications. DG methods are highly accurate numerical methods and have considerable advantages over the classical numerical methods available in the literature. DG methods can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders. Furthermore, DG methods provide accurate and efficient simulation of physical and engineering problems, especially in settings where the solutions exhibit poor regularity. For these reasons, they have attracted the attention of many researchers working in diverse areas, from computational fluid dynamics, solid mechanics and optimal control, to finance, biology and geology. In this talk, we give an overview of the main features of DG methods and their extensions. We first introduce the DG method for solving classical differential equations. Then, we extend the methods to other equations such as Navier-Stokes equations. The Navier-Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing

Least-squares methods for computational electromagnetics

The modeling of electromagnetic phenomena described by the Maxwell's equations is of critical importance in many practical applications. The numerical simulation of these equations is challenging and much more involved than initially believed. Consequently, many discretization techniques, most of them quite complicated, have been proposed. In this dissertation, we present and analyze a new methodology for approximation of the time-harmonic Maxwell's equations. It is an extension of the negative-norm least-squares finite element approach which has been applied successfully to a variety of other problems. The main advantages of our method are that it uses simple, piecewise polynomial, finite element spaces, while giving quasi-optimal approximation, even for solutions with low regularity (such as the ones found in practical applications). The numerical solution can be efficiently computed using standard and well-known tools, such as iterative methods and eigensolvers for symmetric and positive definite systems (e.g. PCG and LOBPCG) and reconditioners for second-order problems (e.g. Multigrid). Additionally, approximation of varying polynomial degrees is allowed and spurious eigenmodes are provably avoided. We consider the following problems related to the Maxwell's equations in the frequency domain: the magnetostatic problem, the electrostatic problem, the eigenvalue problem and the full time-harmonic system. For each of these problems, we present a natural (very) weak variational formulation assuming minimal regularity of the solution. In each case, we prove error estimates for the approximation with two different discrete least-squares methods. We also show how to deal with problems posed on domains that are multiply connected or have multiple boundary components. Besides the theoretical analysis of the methods, the dissertation provides various numerical results in two and three dimensions that illustrate and support the theory