# 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 .