Finite Element Analysis. Barna Szabó

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:

(1.48)

      Proof: Let and let v be an arbitrary function in . Then

      The first term on the right is , the second term is zero on account of Theorem 1.3, the third term is positive for any in . Therefore is minimum.

      Theorem 1.4 states that the error depends on the exact solution of the problem and the definition of the trial space .

      The finite element method is a flexible and powerful method for constructing trial spaces. The basic algorithmic structure of the finite element method is outlined in the following sections.

1.3.1 The standard polynomial space

      The standard polynomial space of degree p, denoted by , is spanned by the monomials defined on the standard element

      (1.49)

      The choice of basis functions is guided by considerations