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: