The mathematical framework proposed in this paper is the result of several years of research devoted to conceive a Finite Element (FE) programme that would be able to address coupled problems efficiently. Experience in this domain shows that it is worthwhile to avoid notions that are specific to a certain branch of Physics for the benefits of more abstract mathematical notions, such as the ones presented in this paper. Indeed, these unified notions are liable to fit a whole class of particular notions arising from different branches of Physics. They are also more closely related with the mathematical structure of the governing differential equations. Finally, they are richer in well defined mathematical properties, making it possible to implement them as richly endowed objects within an Object-Oriented way of programming. Even if theoretical structures similar to this one are already known [1], it is kept in mind throughout this approach that the main objective is to build a FE programme. The proposed framework is thus also of a practical engineering interest as it is ready at once for implementation, the interpolation and discretisation issues are considered. On the way to acheiving this unifying purpose, one has been led to borrow a set of notions from different mathematical disciplines. It is shown that any Conservation laws can be written in an intrinsic and metric-free manner thanks to notions borrowed from Differential Geometry (vector, covector, tensor, differential forms, exterior calculus and Lie derivative). Particularly, this point is not obvious for equilibrium equations in Elasticity or in Fluid Dynamics and requires to abandon the classical stress tensor for the benefit of a different one with more specific tensorial properties. A set of notions coming from Convex Analysis (Legendre transform, Duality) is then presented as the perfect complement of Differential Geometry to take Constitutive laws into consideration. Particularly, Convex Analysis allows to consider discontinuous and injective constitutive laws. Finally, Thermodynamics is also invoked as it remains the fundamental basis of any coupled problem analysis. The familiarity between thermodynamic potentials and FE functionals is reminded and extended to the case of coupled problems with dissipative effects.

[1] E. Tonti. 'On the mathematical structure of a large class of physical theories'. Rend. Acc. Lincei, 52:48-56, 1972.