Pär Ingelström

A new set of H(curl)-conforming hierarchical basis functions for tetrahedral meshes

A new set of H(curl)-conforming hierarchical basis functions for tetrahedral meshes is presented. Contrary to previous bases, this one is designed such that higher order basis functions vanish when they are projected onto a lower order finite-element space using the interpolation operator defined by Nedelec. Consequently, to increase the polynomial order and improve the accuracy of the interpolated field, only additional degrees of freedom (DOFs) of higher order are added, whereas the original DOFs (the coefficients for the basis functions) remain unchanged. This makes this basis very well suited for use with efficient multilevel solvers and goal-oriented hierarchical error estimators, which is demonstrated through numerical examples

Goal-oriented error-estimation for S-parameter computations

Accurate and efficient goal-oriented error estimates for adaptive finite-element computations of scattering parameters are presented. The estimates are derived using dual problems and computed using special hierarchical basis functions, tailored to agree with the Ne/spl acute/de/spl acute/lec interpolant. The method is tested on a two-dimensional waveguide cavity resonator with several reentrant corners. When uniform mesh refinement is used, the convergence rate is limited to h/sup 4/3/, independently of the element order due to singularities. However, with adaptive h-refinement guided by the presented error estimates, the convergence rate is increased to h/sup 2p/ for elements of (complete or incomplete) order p/spl les/3, even in the presence of singularities. Further, the accuracy of the scattering parameters can be increased significantly by adding the estimated errors to the computed results.

The Finite Element Method

The finite element method (FEM) is a standard tool for solving differential equations in many disciplines, e.g., electromagnetics, solid and structural mechanics, fluid dynamics, acoustics, and thermal conduction. Jin [40, 41] and Peterson [54] give good accounts of the FEM for electromagnetics. More mathematical treatments of the same topic are given in [12, 48]. This chapter gives an introduction to FEM in general and FEM for Maxwell’s equations in particular. Practical issues, such as how to handle unstructured grids and how to write FEM programs, will be discussed in some detail.

The Method of Moments

The previous chapters of this book are devoted to the solution of Maxwell’s equations on differential form, where the focus is on finite-difference schemes and the finite element method. In this chapter, Maxwell’s equations are reformulated as integral equations, where the field solution is expressed in terms of superpostion integrals that involve the sources and a so-called Green’s function. In this setting, we would typically have unknown sources that we wish to compute given that we have sufficient information that describes the known field. Typically, this type of formulation is useful for problems where the sources can be described by a relatively few degrees of freedom, when compared to the number of degrees of freedom that would be required for a corresponding description in terms of the fields.

Summary and Overview

The goal of any analysis or optimization is to achieve sufficient accuracy with minimum effort, where effort usually is interpreted as computational cost in terms of computational time and memory requirements. However, there may also be a considerable effort associated with other issues such as the programming of the numerical algorithm or the construction of geometrical descriptions suitable for the the computations at hand.

The Finite-Difference Time-Domain Method

The finite-difference time-domain (FDTD) scheme is one of the most popular computational methods for microwave problems; it is simple to program, highly efficient, and easily adapted to deal with a variety of problems. The FDTD scheme is typically formulated on a structured Cartesian grid and it discretizes Maxwell’s equations formulated in the time domain. The derivatives with respect to space and time are approximated by finite-differences, where the field components of the electric and magnetic field are staggered in space with respect to each other in a particular manner that is tailored for Maxwell’s equations.