Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

We devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three-dimensional magnetostatic problems. In particular, we construct a finite element basis of the first de Rham cohomology group of the computational domain. The proposed method works for general topological configurations and does not need the determination of “cutting” surfaces.