We multiply the equation
with a smooth function
,
integrate over the domain
and apply the proposition (
Green formula
). We
arrive to the following weak formulation (see the section
(
Elliptic PDE section
) for review of
general theory):
Problem
(Poisson equation weak
formulation 1) Find the function
such
that
We introduce the notation
for some basis of
for every
.
Problem
(Galerkin approximation 1) We define the
finite dimensional approximation to the solution of the problem
(
Poisson equation weak
formulation 1
) as the solution
of the
problem
Definition
(Elliptic Ritz projection) We define the
projection
according to the
rule
Proposition
Ritz projection has the following
property:
