Introduction

In this chapter, based on the general form of Maxwell's equations, the various static formulations (electrostatic, electrokinetic and magnetostatics) are presented. In addition to the homogenous boundary conditions, a special focus is dedicated to the boundary conditions which impose global variables of the flux or circulation type. Indeed, in many electromagnetic systems, we often have to solve a problem where a condition of the flux or potential difference type of a vector field acts as a source term.

Subsequently, the source fields and the concepts of scalar potential and vector potential are introduced. These "mathematical beings" implicitly verify one of the equations of the system to be solved. Their use notably simplifies the problem and makes it possible, in certain cases, to introduce source terms naturally as well as global boundary conditions. It should be noted that the corollary of the use of potentials is the need to impose a gauge condition to have the uniqueness of the solution. The various formulations in potentials being defined, the construction of function spaces is then covered. These spaces will accommodate the scalar and vector fields which constitute the solutions of the problem. This then leads to Tonti diagrams which are an illustration of the series of function spaces. In the case of electrokinetic formulations, the weak formulations, in scalar and vector potential, are developed using the Green formulae.

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After having presented the various formulations in the continuous domain, we focus on the discretization of function spaces and...