From formal solutions to computational methods avoiding passages to the limit

Before the advent of digital computers, the so-called formal solutions were the only available solutions to differential equations. Formal solutions can be closed solutions, or solutions involving infinite algorithms. The latter involve an infinite number of algebraic operations. Truncation becomes thus necessary, and the concepts of truncation error and convergence become vital. Once digital computers became available, other kinds of computational methods could be used and it became convenient to distinguish between computational methods like finite difference and finite element methods, in which numerical analysis starts before integration, and those like classical integral methods and boundary element methods, in which numerical analysis starts after integration. The classical finite difference method, in which a mesh is required, is a particular case of the generalised difference methods, characterised by a local interpolation around each node together with the collocation technique. The generalised difference method may be regarded as a modality of the meshless techniques. The finite element method differs of the finite difference method in that the approximate solution is generated respectively by variational and by collocation techniques. Hybrid and block elements are dual generalisations of the finite element method in which compatibility and equilibrium are respectively allowed within each element. Also in what concerns the methods in which numerical analysis starts after integration, bold steps have been given toward their generalisation, like those avoiding passages to the limit.