#### 3.1.5 Geometrical Interpretation of FEM

The manner in which the FEM was presented in this work makes the geometrical
interpretation of the method a natural consequence. In this section it will be shown that this
concept is intimately connected to the error of the linear space discretization of the FEM.
In order to proceed smoothly through the upcoming discussion, consider that
the linear space is endowed with inner products for real functions defined
by

| (3.19) |

Furthermore, consider the norm induced by the inner product (3.19) as

| (3.20) |

Consequently, the variational form (3.5) and the discrete variational form (3.10) can be
rewritten as

| (3.21) |

and

| (3.22) |

The discrete problem solution () is computed by FEM as seen in the previous sections,
but the exact solution of the BVP problem is given by solving the continuous variational
form. Therefore, it is natural to ask how distant is from . The answer of this question
leads to a geometrical view of the FEM.

Consider . The function also belongs to since . Therefore, for every
(3.22) can be subtracted from (3.21), given by

| (3.23) |

As an inner product, (3.19) must hold the orthogonality property ().
This means that the discretization error () is orthogonal to the space . As a
consequence, the solution is the orthogonal projection of the exact solution in .
Fig. 3.6 illustrates this principle.

As a result of the orthogonal projection, the element is the closest function to in
comparison to all elements of . Hence, the error of discretization is bounded according to
[56]

| (3.24) |

The relation (3.24) will not be proved here, but it is intuitively clear when orthogonal
projections are kept in mind. In conclusion, FEM provides the best approximation
of the exact solution in the discretized space , when the norm (3.20) is
considered.

At this point, the advantage of the original Galerkin’s imposition for the solution and
the test function becomes evident – and , respectively, for the discrete
formulation – to reside in the same linear space. The formulation gains some formal support,
especially regarding the discretization error, since it is guaranteed that the FEM solution is
the best choice in a particular space.