Variational approach for the finite element method

Since the advent of computers over half a century ago, and particularly in view of their increasing computing power, digital simulation has replaced direct experimentation, which was too costly and time-consuming to implement. Today, the latter is no more than a means of verifying the calculations carried out on the machine. From a mathematical point of view, digital simulation essentially involves the numerical resolution of partial differential equations, leading to approximate solutions. There are many different approximation methods, all of which have their advantages and disadvantages. Examples include the finite-difference method, the finite-volume method and spectral methods.

In the three articles that make up this set, we focus on the finite element method, which is widely used in industry, particularly in aeronautics, the automotive industry, meteorology and so on. This method is interesting for its flexibility of use, in particular with regard to the approximation of the various operators modeling phenomena in physics-mathematics, and also for taking into account boundary conditions relating to the gradients of the function to be calculated. This flexibility can also be seen in the fact that the domains in which the partial differential equations are defined can be approximated as closely as possible and, in particular, the curved nature of the boundaries of these domains can be taken into account; moreover, the nodes of the discretization, i.e. the points at which the functions to be calculated are approximated, can be distributed in an arbitrary way, allowing a tight mesh in areas where the solution varies greatly and a relatively coarse mesh in regions where this solution varies little; similarly, it is not necessary to use uniform meshes with a constant pitch, as variable-dimension elements can be defined without difficulty; this is particularly useful when studying phenomena defined in heterogeneous media. Finally, from a computing point of view, the finite element method leads to the writing of the most general calculation code possible, which is certainly an advantage but also a disadvantage, given the practical difficulty of programming this algorithm. It should be noted, however, that the code's schematic diagram is relatively simple, the complexity deriving from the innumerable possibilities offered by the method. Moreover, the development of such a code requires many months of programming.

Another difficulty in understanding the finite element method lies in the mathematical formalism that precedes and underlies its algorithmic implementation. Indeed, given the growing complexity of mathematical models for understanding phenomena that are increasingly complicated to explain, it has been necessary to draw on elaborate functional analysis results...