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4.1 The Galerkin Method - Optional     continued...

It is understood that this equation must hold for all test functions, , which must vanish at the boundaries where . The Galerkin approximation to the weak form solution in (17) can be expressed as:

The trial functions form a basis for an N+1 dimensional space . We define the Galerkin approximation to be that element which satisfies:

Since our differential operator A is positive definite and self adjoint (i.e., for some non-zero positive constant and , respectively), then we can define a space E with an inner product defined as and norm (the so-called energy norm) equal to: