In Chapter 7 the variational formulation has been introduced to prove the existence of a (weak) solution. Now it will turn out that the variational formulation is extremely important for numerical purposes. It establishes a new, very flexible discretisation method. After historical remarks in Section 8.1 we introduce the Ritz–Galerkin method in Section 8.2. The basic principle is the replacement of the function space V in the variational formulation by an N-dimensional space. This leads to a system of N linear equations (§8.2.1). As described in §8.2.2, the theory from Chapter 7 can be applied. In §8.2.3 two criteria, the inf-sup condition and V-ellipticity are described which are sufficient for solvability. §8.2.4 contains numerical examples. Error estimates are discussed in Section 8.3. The quasioptimality of the Ritz–Galerkin method proved in §8.3.1 shifts the discussion to the approximation properties of the subspace (§8.3.2). The finite elements introduced in Section 8.4 form a special finite-dimensional subspace offering many practical advantages. The corresponding error estimates are given in Section 8.5. Generalisations to differential equations of higher order and to non-polygonal domains are investigated in Section 8.6. An important practical subject are a-posteriori error estimates discussed in Section 8.7. When solving the arising system of linear equations, the properties of the system matrix is of interest which are investigated in Section 8.8. Several other topics are sketched in the final Section 8.9.