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4.1 The Galerkin Method - Optional     continued...

The solution of (15) satisfies:

and the approximate Galerkin solution obtained by solving (19) satisfies:

Subtracting (21) from (22) yields:

The difference denotes the error between the solution in the infinite dimensional space V and the N+1 dimensional space . Equation (23) states that the error is orthogonal to all basis functions spanning the space of possible Galerkin solutions. Consequently, the error is orthogonal to all elements in and must therefore be the minimum error. Thus the Galerkin approximation is an orthogonal projection of the true solution onto the given finite dimensional space of possible approximate solutions. Therefore, the Galerkin approximation is the best approximation in the energy space E.