Multigrid finite element methods for electromagnetic field modeling COVER CONTENTS LIST OF FIGURES 2.1 2D TV edge elements (left) and NV edge elements (right). 2.2 Visual interpretation of the gradient curl and divergence matrices. 2.3 The node representation for the pth order non hierarchical scalar basis (p = 3). 2.4 The four first order node type scalar basis functions that form the subspace W1s n are assigned to the four vertices of 2.5 Assignment of edge type scalar basis functions in Wps e (p = 3) to nodes along the edges of the tetrahedron. 2.6 Assignment of facet type scalar basis functions of Wps f for (p = 3) to nodes on the facets of the tetrahedron. 2.7 The node representations for volume type scalar basis functions in Wps v (p = 4). 2.8 Vector basis functions in the edge type TV subspace associated with edge (m n). 2.9 Vector basis functions in the facet type TV subspace associated with facet (m n k). 2.10 Vector basis functions in the volume type TV subspace. 2.11 The six edge elements. 2.12 The six cross products of (2.85) depicted for two different orientations of the tetrahedron. 2.13 Facet type (left) and volume type (right) three dimensional normally continuous vector (NV) basis functions. 2.14 The four facet elements for the two different orientations of the tetrahedron. 2.15 Construction of the curl of an edge element from the linear combination of the two facet elements associated with the two facets that share the specific edge. 2.16 Visual aid for the interpretation of the way the gradient curl and divergence matrices are constructed. 3.1 Domain for the statement of the electrostatic BVP. 3.2 A two dimensional domain consisting of two elements. 3.3 a: Pictorial description of an unbounded domain. b: Definition of buffer and absorption layers for the application of coordinate stretching based domain truncation. 3.4 Domain for the statement of the magnetostatic BVP. 3.5 A two dimensional domain consisting of only two elements. 3.6 Geometry for the visualization of the numerical approximation of a solenoidal volume current density. 3.7 Domain for the statement of the magneto quasi static BVP. 3.8 A set of conductors in free space excited by current sources. 3.9 a: A row of incident matrix D. b: A column of loop matrix C. 3.10 Triangular mesh for a metal strip conductor. 3.11 Its corresponding circuit. 3.12 A conducting strip loop. 3.13 Its corresponding circuit. 4.1 Left column (from top to bottom): Plots of the smooth eigenvectors (modes) of A for a uniform grid oh. with 9 interior n 4.2 Left column (from top to bottom): Plots of the oscillatory eigenvectors of A for a uniform grid oh with 9 interior node 4.3 V cycle: N = 4 a = 1. 4.4 W cycle: N = 4 a = 2. 5.1 Coarse and fine grids used for a two level nested multigrid method. 5.2 Convergence of the three level V cycle MCGC for scattering by a circular PEC cylinder. 6.1 Transition between basis functions in coarse and fine grids; edge element (1 2). 6.2 Transition between basis functions in coarse and fine grids; edge element (1 3). 6.3 Transition between basis functions in coarse and fine grids; edge element (2 3). 6.4 Iterative finite element solution convergence for TEz plane wave scattering by a circular PEC cylinder at 1.5 GHz. (After 6.5 Convergence of NMGAV process for TEz plane wave scattering by a PEC circular cylinder at various frequencies. (After Zhu 6.6