FEMSTER: An object oriented class library of discrete differential forms

摘要

The equations of electromagnetics can be simply and elegantly cast in the language of differential geometry, more precisely in terms of differential forms or p-forms [1], [2], [3]. In this geometrical setting, the fundamental conservation laws are not obscured by the details of coordinate system dependent notation; and, the governing equations can be reformulated in a more compact and clear way using well known differential operators of the exterior algebra such as the exterior derivative, the wedge product, and the Hodge star operator. In this context, a natural framework for the modeling of physical quantities is also provided. For example, the electric potentials can be represented by 0-forms; electric and magnetic fields by 1-forms; electric and magnetic fluxes by 2-forms; and. scalar charge density by 3-forms. Our primary motivation for the development of FEMSTER was the need for a common fi-nite element framework for electrostatics, magnetostatics, eddy current problems, Helmholtz equation, time-dependent Maxwell equations, etc. Recently, Hiptmair [4], motivated by the theory of exterior algebra of differential forms, presented a unified mathematical frame-work for the construction of conforming finite element spaces. Remarkably, both H(curl) and H(div) conforming finite element spaces and the definition of their degrees of free-dom and interpolation operators can be derived within this framework. Given a physical law expressed in the language of differential forms, it is quite straightforward to discretize the problem using our class library. Our second motivation was the need for high-order discretization which can reduce the mesh size, memory usage, and CPU time required to achieve a prescribed error tolerance. This is particularly true for electrically large problems due to numerical dispersion. The FEMSTER library contains implementations of finite element basis functions of arbitrary order. These implementations include both uniform and non-uniform interpolatory bases, the latter providing significantly improved numerical stability as the order is increased.