alt="upper S subset-of upper X"/>, and seek the function that satisfies for all . Let us return to the introductory example described in Section 1.1 and define
where ϕi () are basis functions. Using the definitions of and given in eq. (1.12), we write the bilinear form as
where ri is defined in eq. (1.12). Therefore we can write in the following form:
(1.45)
Since this must hold for any choice of , it follows that
(1.46)
which is the same system of linear equations we needed to solve when minimizing the integral , see eq. (1.14). Of course, this is not a coincidence. The solution of the generalized problem: “Find such that for all ”, minimizes the error in the energy norm. See Theorem 1.4.
Theorem 1.3 The error e defined by satisfies for all . This result follows directly from
Subtracting the second equation from the first we have,
(1.47)
This equation is known as the Galerkin11 orthogonality condition.
Theorem 1.4 If satisfies for all then un minimizes the error in energy norm where is the exact solution:
alt="upper S subset-of upper X"/>, 