3.2 Weak Solutions

In many cases, the problem of integrating partial differential equations may be replaced by an equivalent variational problem, where one seeks a minimum value to some integral. Historically, the variational problem of primary importance was cast in the form of the Dirichlet principle ([Courant and Hilbert, 1962], [Mikhlin, 1964]). According to this principle, of all functions assuming specified values on the boundary of a two-dimensional domain ?, the function that minimizes the Dirichlet integral

is harmonic in ?. In other words, the minimizing function satisfies the Laplace equation in ?.

In the late 1800's, Weierstrass showed that the Dirichlet principle does not hold in general; a minimum value of the functional J[ u] need not occur. The Weierstrass counter-example, reproduced in [Courant and Hilbert, 1962], demonstrates that even in a one-dimensional case the function that gives a minimum value to the integral

and satisfies the boundary conditions

need not exist. Thus, a more rigorous approach to the problems of existence and uniqueness of solutions to elliptic partial differential equations is required.

A variational formulation corresponds to every elliptic boundary value problem. The existence of weak solutions to the problem is thus guaranteed under some general conditions expressed by the Lax-Milgram theorem for functions in an appropriate Sobolev space. These results provide a mathematical basis of the finite element method covered in this and later chapters. A complete discussion of the mathematical theory of the finite element method may be found in...