On the equivalence of the norms of the discrete diffrential forms in discrete exterior calculus

• DOI: 10.1007/s13160-018-0334-8
• Publication Date: 2018-11-13T10:45:21Z
• Authors: T. Satoh; T. Yaguchi
• Keywords: 0101 mathematics; 01 natural sciences

For analysis of electromagnetic fields, differential forms are useful mathematical tools. Discrete exterior calculus (DEC) is a finite-difference-type framework of discretization of the discrete differential forms. Although several numerical estimations of errors of DEC have been reported, a tangible theoretical error bound is not established. One reason of this is that there are two norms that have been commonly used for the discrete differential forms. In this paper, we show that on any quasi-uniform mesh, the two norms of the discrete differential forms in DEC are equivalent, and moreover, the order of accuracy of the discrete differential forms are independent of the choice of the norm. As an application, it is shown that the accuracy of the discrete Hodge operator of DEC is of the second-order if any quasi-uniform Delaunay mesh is adopted.