Discretised Euler-Lagrange equations
Let be a triangulation of the computational domain
. We first want to solve equation for
and
and then the equation for
In the case of Navier boundary conditions one can consider piecewise affine elements for
,
and
. Let
be the nodes of the triangulation with
being the nodes in the interior of
and
being the nodes on the boundary. Let
be the discretized function space. This is the space corresponding to the the standard piecewise linear finite element basis functions. Let also, for ,
We note that
Mathematical analysis of the discretisation
We write the discretizations of the problems for and for
respectively:
- Find
such that
- Find
such that
The existence and uniqueness of solutions ,
to the above problems follows by standard argument (note that
)
Convergence rate of the approximation
One can prove the following theorem (see Section 5.2.3 of the RSG report)
Theorem: Suppose that ,
, each of the functions
is constant on each component of
and that the
. Then if the functions
,
,
solve the system of equations, then the following error bounds hold:
for all such that
.
This theorem states the convergence rate of the approximation
to the exact solution
in the
norm. We note that the constants depend on
, which measures how much smaller is
from
. We note, that the constant is fixed for fixed coefficients
.