An H(curl; Ω)-conforming FEM: Nédélec's elements of first type

The aim of this report is to give an introduction to Nédélec's H(curl; Ω)-conforming finite element method of first type. As the name suggests, this mentioned has been introduced in 1980 by J. C. Nédélec in 8. In the first section, we present the model problem and introduce the framework for its variational formulation. In the second section, we present Nédélec's elements of first type for H(curl; Ω). We start by considering the case of affine grids in two and three space dimensions. We introduce the Piola transformation for vector fields and discuss the choice of function spaces and degrees of freedom. These results are then extended to bilinear and trilinear grids. We explain the practical construction of global shape functions and conclude this section with some remarks on approximation results. Numerical results, which serve to illustrate the convergence of the method, are presented in the third section. In Appendix A, we demonstrate how solutions of the two-dimensional model problem can be constructed from solutions of the scalar Laplace equation. In Appendix B, we motivate the model problem studied in the report by considering the time-harmonic Maxwell's equations in the low-frequency case.
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