2.3 Galerkin Discretization

In order to solve the Poisson problem numerically we have to discretize the weak formulation of the Poisson equation (Eq. (2.8)) and restrict the solution space of the numerical solution to a finite dimensional subspace of . Accordingly
approximates on . The discretized problem can then be written as: Find such that

If we assume that
is a basis of the
-dimensional space and
a -dimensional subspace
then we can rewrite Eq. (2.9)

(2.11) |

(2.12) |

(2.13) |

where the ``stiffness matrix'' is given by

and the ``right hand side'' by

The stiffness matrix is sparse, symmetric, and positive definite. Thus, Eq. (2.14) has exactly one solution , which gives the Galerkin solution

(2.17) |