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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 forw, \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.