Fundamentals of Finite Element Method

Abstract

The main advantage of variational formulations of boundary value problems is that they make it possible to search for approximate solutions of problems in a unified, rational way. In the approximate methods, the solution space of an infinite order is replaced by a finite order one. The fundamental difficulty in application of such methods to problems of practical importance is the choice of function classes that approximate the solution in the considered domain, i.e. the global shape functions \(\varPhi _\alpha \). The functions must enable adequate representation of the solution by taking into account the specific geometry of the domain, inhomogeneous material properties, as well as all other characteristic features of the problem (like distribution of mass forces or heat sources, functions imposed on the boundary, etc.). Efficient generation of approximating functions defined in the entire domain (global shape functions) is therefore impossible in general.