Real life electromagnetic modeling problems often combine complex physics, complicated geometry, and high accuracy requirements. Under such conditions many solution methods fail or perform poorly.; The main goal of this work was to develop efficient and reliable practical methodology for solving several classes of 3D engineering electromagnetic problems. This methodology is based on the adaptive Finite Element - multigrid approach and features a unique combination of state-of-the-art techniques, such as multigrid preconditioners, adaptive mesh refinement, local error estimates. The major advantages of the adaptive multigrid method are its high speed and flexibility in handling complex physical and geometric characteristics of the model.; The dissertation presents the mathematical fundamentals of the method and its contributing techniques. The main emphasis, however, is made on the applications. Several engineering problems are chosen to demonstrate how the method is applied and customized to suit a particular situation.; The first example is from the area of geophysical measurements---namely, computation of fields and currents produced by a galvanic measuring device in ground formations. The problem features a large number of unknowns, complex geometry, nonuniform materials, and an unbounded domain. Several test runs confirmed the high efficiency and accuracy of the adaptive multigrid method supplemented with a spatial mapping technique.; The second example is related to micromagnetic simulations. Minimization of the free energy functional is required in order to determine the domain structure of magnetic materials. The adaptive multigrid technique is applied to the computation of the demagnetizing field, which is a crucial part of each minimization step. The method provides a near optimal solution speed without imposing restrictions on the geometry of the problem.; Another solution method considered and implemented in the dissertation was the Generalized Finite Element Method by Partition of Unity (GFEM-PU). This method permits the use of any reasonable approximating functions (not necessarily piecewise polynomial as in the standard FEM) thus providing a high level of flexibility and customization. In the dissertation, the method is applied to the modeling of magnetically driven deposition of nanoparticles. The behavior of the field near the particles is qualitatively known a priori and is incorporated into the set of approximating functions. This approximation leads to accurate numerical results even for coarse meshes that do not resolve the geometry of the particles.
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