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2

The Finite Element Method

for the Poisson Equation

The finite element method, frequently abbreviated by FEM, was devel-
oped in the fifties in the aircraft industry, after the concept had been
independently outlined by mathematicians at an earlier time. Even today
the notions used reflect that one origin of the development lies structural
mechanics. Shortly after this beginning, the finite element method was ap-
plied to problems of heat conduction and fluid mechanics, which form the
application background of this book.
An intensive mathematical analysis and further development was started
in the later sixties. The basics of this mathematical description and analy-
sis are to be developed in this and the following chapter. The homogeneous
Dirichlet boundary value problem for the Poisson equation forms the
paradigm of this chapter, but more generally valid considerations will be
emphasized. In this way the abstract foundation for the treatment of more
general problems in Chapter 3 is provided. In spite of the importance of the
finite element method for structural mechanics, the treatment of the linear
elasticity equations will be omitted. But we note that only a small expense
is necessary for the application of the considerations to these equations.
We refer to [11], where this is realized with a very similar notation.

2.1 Variational Formulation for the Model Problem

We will develop a new solution concept for the boundary value problem
(1.1), (1.2) as a theoretical foundation for the finite element method. For