Hybrid Discontinuous Finite Element/Finite Difference Method for Maxwell's Equations

A fully explicit, discontinuous hybrid finite element/finite difference method is proposed for the numerical solution of Maxwell's equations in the time domain. We call the method hybrid since the different numerical methods, interior penalty discontinuous finite element method, developed in [1], and finite difference method [2], are used in different parts of the computational domain. Thus, the flexibility of finite elements is combined with the efficiency of finite differences. Our numerical experiment illustrates stability of the proposed new method

Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging

This paper combines the use of high order finite element methods with parallel preconditioners of domain decomposition type for solving electromagnetic problems arising from brain microwave imaging. The numerical algorithms involved in such complex imaging systems are computationally expensive since they require solving the direct problem of Maxwell's equations several times. Moreover, wave propagation problems in the high frequency regime are challenging because a sufficiently high number of unknowns is required to accurately represent the solution. In order to use these algorithms in practice for brain stroke diagnosis, running time should be reasonable. The method presented in this paper, coupling high order finite elements and parallel preconditioners, makes it possible to reduce the overall computational cost and simulation time while maintaining accuracy

Adaptive Hybrid Finite Element/Difference Method for Maxwell’s Equations

An explicit, adaptive, hybrid finite element/finite difference method is proposed for the numerical solution of Maxwell's equations in the time domain. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. Thus, we combine the flexibility of finite elements with the efficiency of finite differences. Furthermore, an a posteriori error estimate is derived for local adaptivity and error control inside the subregion, where finite elements are used. Numerical experiments illustrate the usefulness of computational adaptive error control of proposed new method

Adaptive Hybrid Finite Element/Difference Method for Maxwell’s Equations

An explicit, adaptive, hybrid finite element/finite difference method is proposed for the numerical solution of Maxwell's equations in the time domain. The method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different parts of the computational domain. Thus, we combine the flexibility of finite elements with the efficiency of finite differences. Furthermore, an a posteriori error estimate is derived for local adaptivity and error control inside the subregion, where finite elements are used. Numerical experiments illustrate the usefulness of computational adaptive error control of proposed new method

Finite element methods for time-harmonic wave equations

This thesis concerns the numerical simulation of time-harmonic wave equations using the finite element method. The main difficulties in solving wave equations are the large number of unknowns and the solution of the resulting linear system. The focus of the research is in preconditioned iterative methods for solving the linear system and in the validation of the result with a posteriori error estimation. Two different solution strategies for solving the Helmholtz equation, a domain decomposition method and a preconditioned GMRES method are studied. In addition, an a posterior error estimate for the Maxwell's equations is presented. The presented domain decomposition method is based on the hybridized mixed Helmholtz equation and using a high-order, tensorial eigenbasis. The efficiency of this method is demonstrated by numerical examples. As the first step towards the mathematical analysis of the domain decomposition method, preconditioners for mixed systems are studied. This leads to a new preconditioner for the mixed Poisson problem, which allows any preconditioned for the first order finite element discretization of the Poisson problem to be used with iterative methods for the Schur complement problem. Solving the linear systems arising from the first order finite element discretization of the Helmholtz equation using the GMRES method with a Laplace, an inexact Laplace, or a two-level preconditioner is discussed. The convergence properties of the preconditioned GMRES method are analyzed by using a convergence criterion based on the field of values. A functional type a posterior error estimate is derived for simplifications of the Maxwell's equations. This estimate gives computable, guaranteed upper bounds for the discretization error

Discontinuous Galerkin Methods for Parallel Simulation of Ground-Penetrating Radar in 3D

In this work we examine different numerical methods for the simulation of Maxwell's equations in 3D with the application to ground-penetrating radar. In particular we consider an edge-based finite element and a discontinuous Galerkin method, both in the time domain. We implement these methods using the finite element framework Dune and the discretization module Dune-PDELab and test the implementations using two example problems. Finally, we apply them to a ground-penetrating radar problem derived from the ASSESS-GPR test site and compare the results to actual measurements made on the site

Validation of numerical approaches for electromagnetic characterization of magnetic resonance radiofrequency coils

Numerical methods based on solutions of Maxwell's equations are usually adopted for the electromagnetic characterization of Magnetic Resonance (MR) Radiofrequency (RF) coils. In this context, many different numerical methods can be employed, including time domain methods, e.g., the Finite-Difference Time-Domain (FDTD), and frequency domain methods, e.g., the Finite Element Methods (FEM) and the Method of Moments (MoM). We provide a quantitative comparison of performances and a detailed evaluation of advantages and limitations of the aforementioned methods in the context of RF coil design for MR applications. Specifically, we analyzed three RF coils which are representative of current geometries for clinical applications: a 1.5 T proton surface coil; a 7T dual tuned surface coil; a 7T proton volume coil. The numerical simulation results have been compared with measurements, with excellent agreement in almost every case. However, the three methods differ in terms of required computing resources (memory and simulation time) as well as their ability to handle a realistic phantom model. For this reason, this work could provide "a guide to select the most suitable method for each specific research and clinical applications at low and high field"

Improving the near-field transmission efficiency of nano-optical transducers by tailoring the near-field sample

Despite research efforts to find a better nano-optical transducer for light localization and high transmission efficiency for existing and emerging plasmonic applications, there has not been much consideration on improving the near-field optical performance of the system by engineering the near-field sample. In this work, we demonstrate the impact of tailoring the near-field sample by studying an emerging plasmonic application, namely heat-assisted magnetic recording. Basic principles of Maxwell's and heat transfer equations are utilized to obtain a magnetic medium with superior optical and thermal performance compared to a conventional magnetic medium