Geometric Interpretation of Discrete Approaches to Solving Magnetostatic Problems

The finite-element approach can be considered as a tool for constructing finite dimensional systems of equations that approximate to field problems on the discrete level. Although the finite-element technique is explained typically in terms of variational or weighted-residual approaches, another, less familiar way is available to introduce the same ideas geometrically. Using magnetostatics as an example, we will show how finite-element-type system matrices can be developed by exploiting the geometric properties of so-called Whitney elements. The simple interpretation we gain thereby enables us to view also the convergence properties of finite-element-type schemes in an intuitive way.