Discretised Euler-Lagrange equations

Let $\mathcal{T}=(T_1,\ldots , T_M)$ be a triangulation of the computational domain $\Omega_h$ . We first want to solve equation for $w$ and $\phi$ and then the equation for $u$ In the case of Navier boundary conditions one can consider piecewise affine elements for $w$ , $\phi$ and $u$ . Let $V_1, \ldots , V_K$ be the nodes of the triangulation with $V_1, \ldots, V_N$ being the nodes in the interior of $\Omega_h$ and $V_{N+1}, \ldots , V_K$ being the nodes on the boundary. Let
$W_h := \left\lbrace u_h \in H^1 (\Omega_h ) \, : \, \left. u_h \right|_T \in \mathcal{P}^1(T) \,\,\, \forall T\in \mathcal{T} \right\rbrace$
be the discretized function space. This is the space corresponding to the the standard piecewise linear finite element basis functions. Let also, for $g\in C(\partial \Omega )$ ,
$W_h^g := \left\lbrace u_h \in W_h \, : \, u_h (V_{N+i}) = g(V_{N+1}) \, \,\, \forall i=1,\ldots K-N \right\rbrace .$
We note that $W_h^0 \subset H^1_0 (\Omega_h )$

Mathematical analysis of the discretisation

We write the discretizations of the problems for $w, \phi$ and for $u$ respectively:

Find $(w_h ,\phi_h ) \in W_h^f \times W_h^p$ such that $\label{wdiscrete}A((w_h , \phi_h ), (v_h,\psi_h )) =0 \hspace{1cm} \forall (v_h,\psi_h ) \in (W_h^0)^2.$
Find $u_h \in W_h^g$ such that $\label{udiscrete}\int_{\Omega_h} \nabla u_h \cdot \nabla v_h \, dx = \int_{\Omega_h } w_h \, v_h \,dx \hspace{1cm} \forall v_h \in W_h^0.$

The existence and uniqueness of solutions $(w_h,\phi_h ) \in W_h^f (\Omega )\times W_h^p$ , $u\in W_h^g$ to the above problems follows by standard argument (note that $W_h^0 (\Omega ) \subset H^1_0 (\Omega )$ )

Convergence rate of the approximation

One can prove the following theorem (see Section 5.2.3 of the RSG report)

Theorem: Suppose that $c^2 < 2 \min \{ \kappa b , \sigma a \}$ , $\Omega = \Omega_h$ , each of the functions $f, g, p: \partial \Omega \to \mathbb{R}$ is constant on each component of $\partial \Omega$ and that the $u, w, \phi \in H^2 (\Omega_h )$ . Then if the functions $u$ , $w$ , $\phi$ solve the system of equations, then the following error bounds hold:
$|| u - u_h ||_{H^1 } & \leq & (1+C) \overline{C} \, h \left( | u |_{H^2(\Omega_h ) } + \frac{C_0}{\epsilon } \sqrt{|w|_{H^2(\Omega_h )}^2 +|\phi |_{H^2(\Omega_h )}^2} \right),$
$\sqrt{ || w-w_h ||^2_{H^1 (\Omega_h ) } + || \phi - \phi_h ||_{H^1 (\Omega_h )}^2 } &\leq & \frac{C_0\, \overline{C}}{\epsilon } h \sqrt{ |w|_{H^2(\Omega_h )}^2 +|\phi |_{H^2(\Omega_h )}^2 }$
for all $\epsilon >0$ such that $\epsilon \leq \frac{1}{2} \min \{ (\kappa +b - \sqrt{(\kappa -b )^2 + 2c^2} ),(\sigma +a - \sqrt{(\sigma -a )^2 + 2c^2} ) \}$ .

This theorem states the $O(h)$ convergence rate of the approximation $u_h, \phi_h$ to the exact solution $u, \phi$ in the $H^1$ norm. We note that the constants depend on $\epsilon >0$ , which measures how much smaller is $c$ from $2 \min \{ \kappa b , \sigma a \}$ . We note, that the constant is fixed for fixed coefficients $\kappa, \sigma , a , b , c$ .